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Surface

Surface

:For other senses of this word, see surface (disambiguation). surface (disambiguation) In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy, roughness.

Definition

In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E² (Euclidean 2-space) or an open subset of the closed half of E². The set of points which have an open neighbourhood homeomorphic to Eⁿ is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (1)-manifold, or union of closed curves. A surface with empty boundary is said to be closed if it is compact, and open if it is not compact.

Classification of closed surfaces

There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of two infinite collections:
- Spheres with g handles attached (called g-fold tori). These are orientable surfaces with Euler characteristic 2-2g, also called surfaces of genus g.
- Spheres with k projective planes attached. These are non-orientable surfaces with Euler characteristic 2-k. Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism).

Compact surfaces

Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint.

Embeddings in R³

A compact surface can be embedded in R³ if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R⁴.

Differential geometry

A simple review of the embedding of a surface in n dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the article Poincaré metric.

Some models

To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows match: Image:SphereAsSquare.png|sphere Image:ProjectivePlaneAsSquare.png|real projective plane Image:KleinBottleAsSquare.png|Klein bottle Image:TorusAsSquare.png|torus

Fundamental polygon

Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. The above models can be described as follows:
- sphere:

A A^

- projective plane:

A A

- Klein bottle:

A B A^ B

- torus:

A B A^ B^

(See the main article fundamental polygon for details.)

Connected sum of surfaces

Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components. We use the following notation.
- sphere: S
- torus: T
- Klein bottle: K
- Projective plane: P Facts:
- S # S = S
- S # M = M
- P # P = K
- P # K = P # T We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S. Closed surfaces are classified as follows:
- gT (g-fold torus): orientable surface of genus g, for

g \ge 0

.
- gP (g-fold projective plane): non-orientable surface of genus g, for

g \ge 1

Algebraic surface

This notion of a surface is distinct from the notion of an algebraic surface. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a real manifold.

External links

- [http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing] Category:Surfaces Category:Geometric topology ja:表面

Surface (disambiguation)

Surface may mean:
- surface, a two-dimensional manifold in mathematics
- computer representation of surfaces covers the Computer Aided Design/Computer Aided Manufacturing/Computer Aided Engineering definitions of surfaces, including Finite Element Analysis
- In music, surface is the character of the salient immediate detail. It may be thought of as content plus the more general definition of texture, and is often opposed to form
- In painting, surface is the support upon which paint is applied and the character and texture of the visible paint
- Surface (magazine), an American design magazine
- Surface (TV series), an NBC television show

Fluid

A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. Fluids share the properties of not resisting deformation and the ability to flow (also described as their ability to take on the shape of their containers). These properties are typically a function of their inability to support a shear stress in static equilibrium. While in a solid, stress is a function of strain, in a fluid stress is a function of rate of strain. A consequence of this behaviour is Pascal's law which entails the important role of pressure in characterising a fluid's state. Fluids can be characterised as:
- Newtonian fluids; or
- Non-Newtonian fluids, - depending on the way stress depends on strain and its derivatives. The behaviour of fluids is described by a set of partial differential equations, including the Navier-Stokes equations. Fluids are also divided into liquids and gases. Liquids form a free surface (that is, a surface not created by their container) while gases do not. The distinction between solids and fluids is not so obvious. The distinction is made by evaluating the viscosity of the matter: for example Silly Putty can be considered either a solid or a fluid, depending on the time period over which it is observed. The study of fluids is fluid mechanics which is then subdivided into fluid dynamics and fluid statics depending on whether the fluid is in motion or not.

Soap bubble

A soap bubble is a very thin film of soap water that forms a hollow shape with an iridescent surface. Soap bubbles usually last for only a few moments and burst either on their own or on contact with another object. Due to their fragile nature they have also become a metaphor for something that is attractive, yet insubstantial. They are often used as a children's plaything, but their usage in artistic performances shows that they can be fascinating for adults too. Soap bubbles can help to solve complex mathematical problems of space, as they will always find the smallest surface area between points or edges.

Physics

Surface tension and shape

edges.]] Soap bubbles can exist because the surface layer of a liquid (usually water) has a certain surface tension, which causes the layer to behave as an elastic sheet. A common misconception is that soap increases the water's surface tension. Actually soap does the exact opposite, decreasing it to approximately one third the surface tension of pure water. Soap does not strengthen bubbles, it stabilizes them, via an action known as the Marangoni effect. As the soap film stretches, the concentration of soap decreases, which causes the surface tension to increase. Thus, soap selectively strengthens the weakest parts of the bubble and tends to prevent them from stretching further. In addition, the soap reduces evaporation so the bubbles last longer. Their spherical shape is also caused by surface tension. The tension causes the bubble to form a sphere, as a sphere has the smallest possible surface area for a given volume. This shape can be visibly distorted by air currents, and hence by blowing. If a bubble is left to sink in still air, however, it remains very nearly spherical, more so for example than the typical cartoon depiction of a raindrop. When a sinking body has reached its terminal velocity, the drag force acting on it is equal to its weight, and since a bubble's weight is much smaller in relation to its size than a raindrop's, its shape is distorted much less. (The surface tension opposing the distortion is similar in the two cases: The soap reduces the water's surface tension to approximately one third, but it is effectively doubled since the film has an inner and an outer surface.)

Freezing

Soap bubbles blown into air that is below a temperature of minus 15 °C (5 °F) will freeze when they touch a surface. The air inside will gradually diffuse out, causing the bubble to crumple under its own weight. At temperatures below, say, minus 25 °C (minus 13 °F), bubbles will freeze in the air and may shatter when hitting the ground. When, at this low temperature, a bubble is blown with warm breath, the bubble will freeze to an almost perfect sphere at first, but when the warm air cools and thus is reduced in volume there will be a partial collapse of the bubble. A bubble, blown successfully at this low temperature, will always be rather small in size: it will freeze quickly and continuing to blow will shatter the bubble. Soap bubbles will freeze more slowly as the temperature increases to 0°C. Blowing frozen bubbles is a popular seasonal activity in Chagrin Falls, OH.

Merging

diffuse diffuse diffuse diffuse diffuse diffuse diffuse When two bubbles merge, the same physical principles apply, and the bubbles will adopt the shape with the smallest possible surface area. Their common wall will bulge into the larger bubble, as smaller bubbles have a higher internal pressure. If the bubbles are of equal size, the wall will be flat. At a point where two or more bubbles meet, they sort themselves out so that only three bubble walls meet along a line, separated by angles of 120°. This is the most efficient choice, again, which is also the reason why the cells of a beehive use the same 120° angle, thus forming hexagons. Only four bubble walls can meet at a point, with the lines where triplets of bubble walls meet separated by 109.47°.

Interference and reflection

The iridescent colours of soap bubbles are caused by interfering light waves. As light impinges on the film, some of it is reflected off the outer surface while some of it enters the film and reemerges after being reflected back and forth between the two surfaces. The total reflection observed is determined by the interference of all these reflections. Since each traversal of the film incurs a phase shift proportional to the thickness of the film and inversely proportional to the wavelength, the result of the interference depends on these two quantities. Thus, at a given thickness, interference is constructive for some wavelengths and destructive for others, so that white light impinging on the film is reflected with a hue that changes with thickness. A change in colour can be observed while the bubble is thinning due to evaporation. Thicker walls cancel out red (longer) wavelengths, thus causing a blue-green reflection. Later, thinner walls will cancel out yellow (leaving blue light), then green (leaving magenta), then blue (leaving yellow). Finally, when the bubble's wall becomes much thinner than the wavelength of visible light, all the waves in the visible region cancel each other out and no reflection is visible at all. When this state is observed, the wall is thinner than about 25 nanometres, and is probably about to pop. Interference effects also depend upon the angle at which the light strikes the film, an effect called iridescence. So, even if the wall of the bubble were of uniform thickness, one would still see variations of color due to curvature and/or movement. However, the thickness of the wall is continuously changing as gravity pulls the liquid downwards, so bands of colours that move downwards can usually also be observed.

Mathematical properties

Soap bubbles are also physical illustrations of the problem of minimal surfaces, an area of intense mathematical and scientific study over the past 15 years. For example, while it has been known since 1884 that a spherical soap bubble is the least-area way of enclosing a given volume of air (a theorem of H. A. Schwarz), it was only recently proved in the year 2000 that two merged soap bubbles provide the optimum way of enclosing two given volumes of air with the least surface area. This has been termed the double bubble theorem. Soap films seek to minimise their surface area, that is, to minimise their surface energy. The optimum shape for an isolated bubble is thus a sphere. Many bubbles packed together in a foam have much more complicated shapes. See Weaire-Phelan structure for a discussion of this (called the Kelvin problem), and Plateau's Rules for a discussion of the structure of the films.

How to make soap bubbles

The easiest ways are to use commercially produced soap bubble fluid (marketed as a toy) or to simply put some dish washing soap in water. However, this latter might not work as well as expected, and there are several tricks to improve the soap sud formula:

Additives

- Something to reduce the water's surface tension, such as liquid soap or baby shampoo. These may work better the more pure (devoid of perfume or other additives) the soap is, or perhaps with more expensive soaps.
- Something to thicken the water. Most commonly used is glycerin (available at the pharmacy), which makes the bubbles more colourful, too. Sugar, icing sugar or corn syrup have similar effects. It may be advantageous to dissolve the sugar in hot water. However, the soap sud can also be too thick and heavy, so it is important not to add too much of these thickening substances.
- Distilled water. As tap water contains calcium ions, and these bind the soap, distilled water works better.

Procedure

- Leaving the soap sud in an open container overnight makes it thicker, too. But again, if the solution becomes too heavy it will be harder to make soap bubbles.
- Bubbles or foam on the surface of the soap sud should be avoided by stirring gently, skimming them away or simply waiting until they are gone.
- How easy it is to make soap bubbles depends on a vast number of factors. Every soap is different, and environmental conditions influence performance, too. For example, dusty air is unfavourable, and so is wind. Also, the more humid the air is, the better, which means making soap bubbles is easier on rainy days. Altogether, the best procedure for finding the perfect solution is the trial and error method.

Bubble blowers

The easiest way is to use one of the plastic blowers that are sold with most commercial soap bubble solutions. However, as the blower's diameter determines the size of the soap bubble, it might be necessary to build a blower. Most closed-ring structures will work. A blower can be made by bending a wire a into loop with a handle, where the wire should be thick enough so the ring remains stiff. It can be improved by wrapping a thread or bandages around the wire so the soap water can stick better to the ring. Klutz Press popularized a "giant bubble" blower, which used a cloth loop attached to a plastic wand, with a slide permitting the loop to be gently opened or closed. Klutz sells bubble books which offer howtwos and fun ideas, usually with a ready-to-use bubble loop. Bubbles can be blown by using a bubble pipe, which is made of plastic and usually takes the shape of a smoking pipe, sometimes containing multiple bowls. The bubble solution is poured into the bowl of the pipe; when someone blows into the mouthpiece, bubbles rise from the bowl.

Sample formulae

#General purpose formula: #
- ²/₃ cup dishwashing detergent #
- 1 gallon water #
- 2 to 3 tablespoons of glycerin #Another general purpose formula: #
- 100 g sugar #
- 2 to 3 tablespoons salt #
- 1.4 L water (distilled water is better) #
- 150 ml dish washing detergent #
- 12 ml glycerin #Yet another general purpose formula: #
- 1 part of washing-up detergent #
- 2 parts of glycerin #
- 3 parts of water #For long living bubbles: #
- ¹/₃ cup commercial bubble solution #
- ¹/₃ cup water #
- ¹/₃ cup glycerin #For no-tears soap bubbles: #
- 60 ml baby shampoo #
- 200 ml water #
- 3 tablespoons corn syrup

Performance art

Soap bubble performances combine entertainment with artistic achievement. They require a high degree of skill as well as perfect bubble suds. Some artists create giant bubbles or tubes, often enveloping objects or even humans. Others manage to create bubbles forming cubes, tetrahedra and other shapes or sculptures. Bubbles are often handled with bare hands. To add to the visual experience, they are sometimes filled with smoke or helium and combined with laser lights or fire. Soap bubbles can be filled with a flammable gas such as natural gas and then ignited. Of course, this destroys the bubble.

References

- [http://www.exploratorium.edu/ronh/bubbles/bubbles.html A more detailed scientific explanation]
- [http://www.ugr.es/~ritore/bubble/bubble.htm The proof paper on the Double Bubble Theorem]
- A book about soap bubbles and mathematics: Oprea, John (2000). The Mathematics of Soap Films – Explorations with Maple. American Mathematical Society (1st ed.). ISBN 0-82-182118-0
- Boys, C. V. (1890) Soap-Bubbles and the Forces that Mould Them; (Dover reprint) ISBN 0-48-620542-8. Classic Victorian exposition, based on a series of lectures originally delivered "before a juvenile audience".
- Isenberg, Cyril (1992) The Science of Soap Films and Soap Bubbles ; (Dover) ISBN 0486269604. Category:Fluid dynamics Category:Minimal surfaces ja:シャボン玉 simple:Soap bubble

Surface tension

In physics, surface tension is an effect within the surface layer of a liquid that causes the layer to behave as an elastic sheet. It is the effect that allows insects (such as the water strider) to walk on water, and causes capillary action, for example. capillary action capillary action Surface tension is caused by the attraction between the molecules of the liquid, due to various intermolecular forces. In the bulk of the liquid each molecule is pulled equally in all directions by neighbouring liquid molecules, resulting in a net force of zero. At the surface of the liquid, the molecules are pulled inwards by other molecules deeper inside the liquid, but there are no liquid molecules on the outside to balance these forces, so the surface molecules are subject to an inward force of molecular attraction which is balanced by the resistance of the liquid to compression. There may also be a small outward attraction caused by air molecules, but as air is much less dense than the liquid, this force is negligible. Surface tension is measured in newtons per meter (N·m^-1), is represented by the symbol σ or γ or T and is defined as the force along a line of unit length perpendicular to the surface, or work done per unit area. Dimensional analysis shows that the units of surface tension (N·m^-1) are equivalent to joules per square metre (J·m^-2). This means that surface tension can also be considered as surface energy. If a surface with surface tension σ is expanded by a unit area, then the increase in the surface's stored energy is also equal to σ. A related quantity is the energy of cohesion, which is the energy released when two bodies of the same liquid become joined by a boundary of unit area. Since this process involves the removal of a unit area of surface from each of the two bodies of liquid, the energy of cohesion is equal to twice the surface energy. A similar concept, the energy of adhesion, applies to two bodies of different liquids. See also Cassie's law.

Measuring methods

- Du Noüy Ring method: The traditional method used to measure surface or interfacial tension. Wetting properties of the surface or interface have little influence on this measuring technique. Maximum pull exerted on the ring by the surface is measured.
- Wilhelmy Plate method: A universal method especially suited to check surface tension over long time intervals. A vertical plate of known perimeter is attached to a balance, and the force due to wetting is measured.
- Spinning Drop method: This technique is ideal for measuring low interfacial tensions. The diameter of a drop within a heavy phase is measured while both are rotated.
- Pendant Drop method: Surface and interfacial tension can be measured by this technique, even at elevated temperatures and pressures. Geometry of a drop is analyzed optically.
- Bubble Pressure method: A measurement technique for determining surface tension at short surface ages. Maximum pressure of each bubble is measured.
- Drop Volume method: A method for determining interfacial tension as a function of interface age. Liquid of one density is pumped into a second liquid of a different density and time between drops produced is measured.

External links

- [http://www.kruss.info/techniques/surface_tension_e.html Theory of surface tension measurements] Category:Fluid mechanics ja:表面張力

Surface chemistry

Surface chemistry is the study of chemical phenomena that occur at the interface of two phases, usually between a gas and a solid or between a liquid and a solid. One important field of surface chemical research is related to surface functionalization, which aims at modifying the chemical composition of a surface by incorporation of selected chemical elements or functional groups. Another important aspect of surface chemistry studies is to determine whether a molecule attaches itself to a surface by chemisorption or by physisorption. Surface chemistry is of particular importance to the field of heterogeneous catalysis. The advent of scanning probe microscopies like atomic force microscopy (AFM) and scanning-tunneling microscopy (STM) has stimulated a considerable increase in research activity in surface chemistry. This increase is part of a more general interest in nanotechnology. Behaviour in solution surface chemistry and colloid chemistry is dependent on the surface charge and the potential distribution in the surrounding electrical double layer. Irving Langmuir was one of the founders of this field, and a scientific journal on surface chemistry bears his name. The Langmuir adsorption equation is used to model monolayer adsorption where all surface adsorption sites have the same affinity for the adsorbing species.

External links

- The American Chemical Society journal, [http://pubs.acs.org/journals/langd5/ Langmuir]
- The Institute for Surface Chemistry [http://www.yki.se]

Surface energy

Surface energy quantifies the disruption of chemical bonds that occurs when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favourable than the bulk of a material; otherwise there would be a driving force for surfaces to be created, and surface is all there would be (see sublimation (physics)). Cutting a solid body into pieces disrupts its bonds, and therefore consumes energy. If the cutting is done reversibly (see reversible), then conservation of energy means that the energy consumed by the cutting process will be equal to the energy inherent in the two new surfaces created. The unit surface energy of a material would therefore be half of its energy of cohesion, all other things being equal; in practice, this is true only for a surface freshly prepared in vacuum. Surfaces often change their form away from the simple "cleaved bond" model just implied above. They are found to be highly dynamic regions, which readily rearrange or react, so that energy is often reduced by such processes as passivation or adsorption. As first described by Thomas Young in 1805 in the Philosophical Transactions of the Royal Society of London, it is the interaction between the forces of cohesion and the forces of adhesion which determines whether or not wetting, the spreading of a liquid over a surface, occurs. If complete wetting does not occur, then a bead of liquid will form, with a contact angle which is a function of the surface energies of the system. Category:Energy Category:Condensed matter physics Category:Surface chemistry

Roughness

Roughness or rugosity is a measurement (see surface metrology) of the small-scale variations in the height of a physical surface. This is in contrast to large-scale variations, which may be either part of the geometry of the surface or unwanted 'waviness'. Roughness is sometimes an undesirable property, as it causes friction, wear, drag and fatigue, but it is sometimes beneficial, as it allows surfaces to trap lubricants and prevents them from welding together. It is measured in different ways for different purposes. Here are some examples.

Examples

- International Roughness Index (IRI) - a dimensionless quantity used for measuring road roughness and proposed as a world standard by the World Bank.
- Average roughness (R_a). The average height of the bumps on a surface, measured in micrometres or microinches.
- Root mean square (RMS) roughness. Less common than average roughness. Measured in the same units.
- Roughness numbers, as defined by ISO 1302.
- Manning's n-value - used by geologists to characterise river channels.

Theory

The mathematician Benoit Mandelbrot has pointed out the connection between surface roughness and fractal dimension.

References

- [http://www.predev.com/smg/parameters.htm "Surface Profile Parameters"] at Surface Metrology Guide
- [http://www.predev.com/smg/intro.htm "Surfaces and Profiles"] (ibid.)
- [http://www.umtri.umich.edu/erd/roughness/iri.html "International Roughness Index"] at The University of Michigan Transportation Research Institute (UMTRI)
- [http://www.phoenixmarine.ca/roughness.htm "Propeller Roughness Definitions"] at Phoenix Marine Services
- [http://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/ "Verified Roughness Characteristics of Natural Channels"] at USGS.
- [http://www.edge.org/3rd_culture/mandelbrot04/mandelbrot04_index.html "A Theory of Roughness"] - interview with Mandelbrot at [http://www.edge.org/ edge.org] Category:Metrology

Second countable

In topology, a second-countable space is a topological space satisfying the "second axiom of countability". Specifically, a space is said to be second-countable if its topology has a countable base. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. In general, the finer the topology, the less likely it is to be second-countable. Most "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rⁿ) with its usual topology is second-countable. Although the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a base.

Properties

Second-countability is a stronger notion than first-countability. Recall that a space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x. Thus, if one has a countable base for a topology then one clearly has a countable local base at every point. Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implications do not hold. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent. In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties. Urysohn's metrization theorem states that every second-countable, regular Hausdorff space is metrizable. It follows that every such space is completely normal as well as paracompact. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability. Other properties:
- A continuous, open image of a second-countable space is second-countable.
- Every subspace of a second-countable space is second-countable.
- Quotients of second-countable spaces need not be second countable; however, open quotients always are.
- Any countable product of a second-countable space is second-countable, although uncountable products need not be.
- The topology of a second-countable space has cardinality less than or equal to c (the cardinality of the continuum).
- Any base for a second-countable space has a countable subfamily which is still a base.
- Every collection of disjoint open sets in a second-countable space is countable. Category:General topology

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T₂ space is a topological space in which points can be separated by neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the Hausdorff condition is the most frequently used and discussed . Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology. In fact, Hausdorff's original definition of a topological space included the Hausdorff condition as an axiom.

Definitions

Suppose that X is a topological space. Let x and y be points in X. We say that x and y can be separated by neighbourhoods if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U ∩ V = ).

disjoint

X is a Hausdorff space if any two distinct points of X can be separated by neighborhoods. This is why Hausdorff spaces are also called T₂ spaces or separated spaces. X is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular spaces are also called R₁ spaces. The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

Examples and counterexamples

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. A simple example of a topology that is T₁ but is not Hausdorff is the cofinite topology. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff.

Properties

One of the nicest properties of Hausdorff spaces is that limits of sequences, nets, and filters are unique whenever they exist. In fact, a topological space is Hausdorff if and only if every net (or filter) has at most one limit. Similarly, a space is preregular if all of the limits of a given net (or filter) are topologically indistinguishable. A useful alternative characterization of Hausdorff spaces is the following. A topological space X is Hausdorff iff the diagonal Δ = is closed as a subset of the product space X × X. Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space. Hausdorff spaces are T₁, meaning that all singletons are closed. Similarly, preregular spaces are R₀. Another nice property of Hausdorff spaces is that compact sets are always closed. This may fail for spaces which are non-Hausdorff (there are examples of T₁ spaces where it fails). The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can be separated by neighborhoods. This is an example of the general rule that compact sets often behave like points. Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locally compact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff. The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces. Let f : X → Y be a function and let

\mbox(f) = \

be its kernel regarded as a subspace of X × X.
- If f is continuous and Y is Hausdorff then ker(f) is closed.
- If f is an open surjection and ker(f) is closed then Y is Hausdorff.
- If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff iff ker(f) is closed. If f,g : X → Y are continuous maps and Y is Hausdorff then the equalizer

\mbox(f,g) = \

is closed in X. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets. Let f : X → Y be a closed surjection such that f⁻¹(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y. Let f : X → Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent
- Y is Hausdorff
- f is a closed map
- ker(f) is closed

Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that hold for both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces. There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually still phrased in terms of regularity, since this condition is more well known than preregularity. See History of the separation axioms for more on this issue.

Variants

The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces). As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T₀ condition. These are also the spaces in which completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete iff every Cauchy net has at least one limit, while a space is Hausdorff iff every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).

Joke

There is a mathematicians' joke that serves as a reminder of the meaning of this term: In a Hausdorff space, points can be "housed off" from one another. Atiyah used to draw house-shaped sets on the blackboard. (In an old-fashioned British accent, off could be orf, phonetically, which all helps.) Category:Topology Category:Separation axioms ko:하우스도르프 공간 ja:ハウスドルフ空間

Homeomorphism

:This word should not be confused with homomorphism. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. Two spaces with a homeomorphism between them are called homeomorphic. From a topological viewpoint they are the same. Roughly speaking a topological space is a geometric object and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus a square and a circle are homeomorphic. The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. Intuitively a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. Topology is the study of those properties of objects that do not change when homeomorphisms are applied.

Definition

A function f between two topological spaces X and Y is called a homeomorphism if it has the following properties
- f is a bijection,
- f is continuous,
- the inverse function f⁻¹ is continuous. If such a function exists we say X and Y are homeomorphic. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes.

Examples

- The unit 2-disc D² and the unit square in R² are homeomorphic.
- The open interval (-1, 1) is homeomorphic to the real numbers R.
- The product space S¹ × S¹ and the two-dimensional torus are homeomorphic.
- Every uniform isomorphism and isometric isomorphism is a homeomorphism.

Notes

The third requirement, that f⁻¹ be continuous, is essential. Consider for instance the function f : [0, 2π) → S¹ defined by f(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism. Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X forms a group, called the homeomorphism group of X, often denoted Homeo(X). For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one can reduce this group to the mapping class group.

Properties

- two homeomorphic spaces share the same topological properties. For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces which are homeomorphic even though one of them is complete and the other is not.
- a homeomorphism is an open mapping and a closed mapping, that is it maps open sets to open sets and closed sets to closed sets.
- Every self-homeomorphism in

S^1

can be extended to a self-homeomorphism of the whole disk

D^2

(Alexander's Trick).

Informal discussion

The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly — it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts. This characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y — one just follows them as X deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence. There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between the identity map on X and the homeomorphism from X to Y.

Euclidean space

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclid's concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the "standard" example of a finite-dimensional, real, inner product space. A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness. An inner product space is a generalization of a Euclidean space. Both inner product spaces and metric spaces are explored within functional analysis. Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and non-Euclidean geometry. One mathematical motivation for defining a distance function is the ability to define an open ball around points in the space. This fundamental concept justifies a differential calculus between a Euclidean space and other manifolds. Differential geometry brings such a differential calculus into play, together with a technique of launching a mobile, local Euclidean space, to explore the properties of non-Euclidean manifolds.

Real coordinate space

Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R sometimes called real coordinate space and denoted Rⁿ. An element of Rⁿ is written x = (x₁, x₂, …, x_n) where each x_i is a real number. The vector space operations on Rⁿ are defined by :

\mathbf + \mathbf = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)

a\,\mathbf = (a x_1, a x_2, \ldots, a x_n)

Real coordinate space Rⁿ comes with a standard basis: :

\mathbf_1 = (1, 0, \ldots, 0)

\mathbf_2 = (0, 1, \ldots, 0)

\vdots

\mathbf_n = (0, 0, \ldots, 1)

An arbitrary vector in Rⁿ can then be written in the form :

\mathbf = \sum_^n x_i \mathbf_i

Real coordinate space is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rⁿ. This isomorphism is not canonical however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rⁿ in V). The reason for working with arbitrary vector spaces instead of Rⁿ is that it is often preferable to work in a coordinate-free manner (i.e. without choosing a preferred basis).

Euclidean structure

Euclidean space is more than just real coordinate space. In order to do Euclidean geometry one needs to be able to talk about the distance between points and the angles between lines or vectors. The natural way in which to do this is to introduce what is called an inner product or dot product on Rⁿ. This product is defined by :

\mathbf\cdot\mathbf = \sum_^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.

The dot product of any two vectors x and y gives a real number. This product allows us to define the "length" of a vector x in the following way :

\|\mathbf\| = \sqrt = \sqrt

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rⁿ. The (interior) angle θ between x and y is then given by :

\theta = \cos^\left(\frac\right)

where cos⁻¹ is the arccosine function. Finally, one can use the norm to define a distance function (or metric) on Rⁿ in the following manner :

d(\mathbf, \mathbf) = \|\mathbf - \mathbf\| = \sqrt.

The form of this distance function is based on the Pythagorean theorem, and is called the Euclidean metric. Real coordinate space together with the above Euclidean structure (dot product and the associated norm and metric) is called Euclidean space often denoted by Eⁿ. (Many authors refer to Rⁿ itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure on Eⁿ gives it the structure of an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.

Euclidean topology

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on Eⁿ is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on Rⁿ considered as a product of n copies of the real line R (with its standard topology). An important result on the topology of Rⁿ, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rⁿ (with its subspace topology) which is homeomorphic to another open subset of Rⁿ is itself open. An immediate consequence of this is that R^m is not homeomorphic to Rⁿ if m ≠ n — an intuitively "obvious" result which is nonetheless difficult to prove. Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Rⁿ is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4-spaces. Euclidean space is also known as linear manifold. An m-dimensional linear submanifold of Rⁿ is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higher-dimensional Euclidean space is a 1-dimensional linear submanifold of that space.

Compact space

In mathematics, a subset of Euclidean space Rⁿ is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed). A more modern approach is to call a topological space compact if all its open covers have a finite subcover. The Heine–Borel theorem affirms that this coincides with "closed and bounded" for subsets of Euclidean space. Note: Some authors such as Bourbaki use the term "quasi-compact" instead and reserve the name "compact" for topological spaces that are Hausdorff and compact.

History and motivation

The term compact was introduced by Fréchet in 1906. It has been recognized for a long time that a property like compactness was needed to prove many useful results. At one time, when primarily metric spaces were studied, compact was taken to mean sequentially compact (every sequence has a convergent subsequence). The definition based on open coverings has surpassed it by allowing many useful results that could be proven about metric spaces using the old definition to be proven in general. One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets. In other words, there are many results which are easy to show for finite sets, the proofs of which carry over with minimal change to compact spaces. It is often said that "compactness is the next best thing to finiteness". Here is an example:
- Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. Then we can separate x and A by neighbourhoods: for each a in A, let U(x) and V(a) be disjoint neighbourhoods containing x and a, respectively. Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A. Note that if A is infinite, the proof fails, because the intersection of arbitrarily many neighbourhoods of x might not be a neighbourhood of x. The proof can be "rescued", however, if A is compact: we simply take a finite subcover of the cover of A. In this way, we see that in a Hausdorff space, any point can be separated by neighbourhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighbourhoods -- note that this is precisely what we get if we replace "point" (i.e. singleton set) with "compact set" in the Hausdorff separation axiom. Many of the arguments and results involving compact spaces follow such a pattern.

Definitions

Compactness of subsets of Rⁿ

For any subset of Euclidean space Rⁿ, the following four conditions are equivalent:
- Every open cover has a finite subcover. This is the definition most commonly used.
- Every sequence in the set has a convergent subsequence, the limit point of which belongs to the set.
- Every infinite subset of the set has an accumulation point in the set.
- The set is closed and bounded. This is the condition that is easiest to verify, for example a closed interval or closed n-ball. In other spaces, these conditions may or may not be equivalent, depending on the properties of the space.

Compactness of topological spaces

The "finite subcover" property from the previous paragraph is more abstract than the "closed and bounded" one, but it has the distinct advantage that it can be given using the subspace topology on a subset of Rⁿ, eliminating the need of using a metric or an ambient space. Thus, compactness is a topological property. In a sense, the closed unit interval [0,1] is intrinsically compact, regardless of how it is embedded in R or Rⁿ. The general definition goes as follows. A topological space is called compact iff all its open covers have a finite subcover. Formally, this means that :for every arbitrary collection

\_

of open subsets of

X

such that

\cup_ U_i = X

, there is a finite subset

J\subset I

such that

\cup_ U_i = X

. An often used equivalent definition is given in terms of the finite intersection property: if any collection of closed sets satisfying the finite intersection property has nonempty intersection, then the space is compact. This definition is dual to the usual one stated in terms of open sets. Some authors require that a compact space also be Hausdorff, and the non-Hausdorff version is then called quasicompact.

Examples of compact spaces

- The empty set.
- The closed unit interval [0, 1] is compact. (But not the half-open interval [0, 1)).
- For every natural number n, the n-sphere is compact.
- The Cantor set is compact. Since the p-adic integers are homeomorphic to the Cantor set, they also form a compact set.
- Any finite topological space, including the empty set, is compact. Slightly more generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
- Any space carrying the cofinite topology is compact.
- The spectrum of any continuous linear operator on a Hilbert space is a compact subset of C.
- The spectrum of any commutative ring or Boolean algebra is compact.
- The Hilbert cube is compact.
- The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpinski space is compact.

Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions):
- A continuous image of a compact space is compact.
- A closed subset of a compact space is compact.
- A compact subset of a Hausdorff space is closed.
- A nonempty compact subset of the real numbers has a greatest element and a least element.
- A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine–Borel theorem)
- A metric space (or uniform space) is compact if and only if it is complete and totally bounded.
- The product of any collection of compact spaces is compact. (Tychonoff's theorem -- this is equivalent to the axiom of choice)
- A compact Hausdorff space is normal.
- Every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
- A metric space is compact if and only if every sequence in the space has a convergent subsequence.
- A topological space is compact if and only if every net on the space has a convergent subnet.
- A topological space is compact if and only if every filter on the space has a convergent refinement.
- A topological space is compact if and only if every ultrafilter on the space is convergent.
- A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.
- Every topological space X is a dense subspace of a compact space which has at most one point more than X. (Alexandroff one-point compactification)
- A metric space X is compact if and only if every metric space homeomorphic to X is complete.
- If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (Lebesgue's number lemma)
- If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact. (Alexander's Sub-base Theorem)
- Two compact Hausdorff spaces X₁ and X₂ are homeomorphic if and only if their rings of continuous real-valued functions C(X₁) and C(X₂) are isomorphic.

Other forms of compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.
- Sequentially compact: Every sequence has a convergent subsequence.
- Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
- Pseudocompact: Every real-valued continuous function on the space is bounded.
- Weakly countably compact (or limit point compact): Every infinite subset has an accumulation point. While all these conditions are equivalent for metric spaces, in general we have the following implications:
- Compact spaces are countably compact.
- Sequentially compact spaces are countably compact.
- Countably compact spaces are pseudocompact and weakly countably compact. Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology. Not every compact space is sequentially compact; an example is the infinite product space 2 ^{[0, 1]} with the product topology. A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces. For complete metric spaces this is equivalent to compactness. See relatively compact for the topological version. Another related notion that is usually strictly weaker than compactness is local compactness.

References

- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (1978) Springer-Verlag, New York Category:Topology Category:General topology Category:Mathematical theorems ja:コンパクト (数学)

Sphere

:For other uses, see sphere (disambiguation). sphere (disambiguation) A sphere is a perfectly symmetrical geometrical object. In mathematics, the term refers to the surface or boundary of a ball, but in non-mathematical usage, the term is used to refer either to a three-dimensional ball or to its surface. This article deals with the mathematical concept of sphere.

Geometry

In three-dimensional Euclidean geometry, a sphere is the set of points in R³ which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.

Equations

real number In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the set of all points (x, y, z) such that :

(x - x_0 )^2 + (y - y_0 )^2 + ( z -  z_0 )^2 =  r^2 \,

The points on the sphere with radius r can be parametrized via :

x = x_0 + r \sin \theta \; \cos \phi

y = y_0 + r \sin \theta \; \sin \phi  \qquad (0 \leq \theta \leq \pi \mbox -\pi < \phi \leq \pi) \,

z = z_0 + r \cos \theta  \,

(see also trigonometric functions and spherical coordinates). A sphere of any radius centered at the origin is described by the following differential equation: :

x \, dx + y \, dy + z \, dz = 0.

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other. The surface area of a sphere of radius r is: :

A = 4 \pi r^2 \,

and its enclosed volume is: :

V = \frac

The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area. surface tension for the Gravity Probe B experiment which differs from a perfect sphere by no more than a mere 40 atoms of thickness as it refracts the image of Einstein in the background. It is thought that only neutron stars are smoother.]] The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes. A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.

Terminology

Pairs of points on a sphere that lie on a straight line through its center are called antipodal points. A great circle is a circle on the sphere that has the same center as the sphere. If a particular point on a sphere is designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis. Circles on the sphere that are parallel to the equator are lines of latitude. This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).

Generalization to other dimensions

Spheres can be generalized to other dimensions. For any natural number n, an n-sphere is the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number:
- a 0-sphere is a pair of points

(-r, r)

- a 1-sphere is a circle of radius r
- a 2-sphere is an ordinary sphere
- a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sⁿ and is often referred to as "the" n-sphere.

Generalization to metric spaces

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set :S(x;r) = . If the center is a distinguished point considered as origin of E, e.g. in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken equal to one, i.e. in the case of a unit sphere. In contrast to a ball, a sphere may be empty. For example, in Zⁿ with Euclidean metric, a sphere of radius r is nonempty only if r² can be written as sum of n squares of integers.

Topology

In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere described above under Geometry, but perhaps lacking its metric.
- a 0-sphere is a pair of points with the discrete topology
- a 1-sphere is a circle (up to homeomorphism); thus, for example, (the image of) any knot is a 1-sphere
- a 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid is a 2-sphere The n-sphere is denoted Sⁿ. It is an example of a compact topological manifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere. The Heine-Borel theorem is used in a short proof that a Euclidean n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is closed. Sⁿ is also bounded. Therefore it is compact.

External links

- [http://mathworld.wolfram.com/Hypersphere.html Mathworld website]
- [http://www.mathsisfun.com/geometry/sphere.html More Sphere Images] Math is Fun Category:Surfaces Category:Differential geometry Category:Topology Category:Elementary geometry ja:球 ja:球面 simple:Sphere

Orientable

:This article discusses orientability and orientation on surfaces and manifolds. For orientation of vector spaces see orientation (mathematics). For alternate uses, see orientation. In geometry and topology, a surface in

\mathbb^3

is called non-orientable, if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed. Otherwise the surface is said to be orientable. For an abstract surface (i.e., a two-dimensional manifold), it is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous (intuitively, locally constant) manner. This turns out be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius band. Thus, for surfaces, the Möbius band may be considered the source of all non-orientability. A surface that is embedded in

\mathbb^3

will be orientable in the letter "R" sense if and only if it is orientable as an abstract surface.

Examples in low dimensions

Surfaces we normally encounter in every day life are orientable. For example, sphere, plane, torus. Example of non-orientable surfaces are Möbius strip, real projective plane, Klein bottle. These surfaces as visualized in 3-dimensions all have just one-side. Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a one-sided surface would think there is an "other side". The essence of one-sidedness is that the ant can crawl from one side of the surface to the "other" without going through the surface or flipping over an edge, but simply by crawling far enough. (Caveat: the real projective plane and Klein bottle can't be embedded in

\mathbb R^3

, only immersed with nice intersections.) In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as

\mathbb^3

above) is orientable. For example, a torus embedded in

K^2 \times S^1

can be one-sided, and a Klein bottle in the same space can be two-sided; here

K^2

refers to the Klein bottle.

Orientation by a triangulation

Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that the arrows go from head to tail as we go around the boundary of the triangle. If we can do this so that in addition triangles sharing an edge have arrows on that edge going in opposite directions, then we call what we've done an orientation for the surface. Note that whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise. This rather precise definition is based on intuition gathered from observing the following phenomenon: Imagine a capital "R" written on the surface, that can freely slide along the surface but cannot be lifted off the surface (that letter is chosen because of its asymmetry). If the surface is a Möbius band, and the letter slides all the way around the band and returns to its starting point, then it will look like a mirror-image of an "R" rather than the "R" it looked like originally. If the surface is a sphere, on the other hand, that cannot happen. The relation to the definition above is that sliding the "R" around from triangle to triangle in a triangulation gives an orientation for each triangle; the "R" in a triangle induces an obvious choice of arrow for each edge. The only obstruction to consistently orienting all the triangles is that when the "R" returns to its original starting triangle, it may induce choices of arrows going opposite to the original choice. Clearly, if this never happens, then we want the surface to be orientable, whereas if this does happen, then we want to call the surface non-orientable. The definition above can be generalized to an n-manifold that has a triangulation, but there are problems with that approach: some 4-manifolds do not have a triangulation, and in general for n > 4 some n-manifolds have triangulations that are inequivalent.

Orientation by top-dimensional forms

Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in the manifold. Formally, a

n

-dimensional differentiable manifold is called orientable if it possesses a differential form

\omega

of degree

n

which is nonzero at every point on the manifold. Conversely, given such a form

\omega

, we say that the manifold is oriented by

\omega

. The crucial point to observe here is that such a differential form gives a choice of "right handed" basis at each point. A traveler in an orientable manifold will never change his/her handedness by going on a round trip.

Orientation and vector bundles

A real vector bundle, which a priori has a GL(n) structure group, is called orientable when the structure group may be reduced to

GL^(n)

, the group of matrices with positive determinant. A smooth real manifold is orientable if and only if its tangent bundle is. Category:Differential topology Category:Orientation Category:Surfaces

Euler characteristic

In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. It is usually denoted by

\chi

. The Euler characteristic of a 2-dimensional topological polyhedron can be calculated using the following formula: :

\chi=F-E+V

where F,E and V are the numbers of faces, edges and vertices respectively. In particular, for any polyhedron homeomorphic to a sphere we have :

\chi(S^2)=F-E+V=2.

For instance, for a cube we have 6 − 12 + 8 = 2 and for a tetrahedron we have 4 − 6 + 4 = 2. The last formula is also called Euler's formula.

Definitions and properties

For a finite CW-complex and in particular for a finite simplicial complex, the Euler characteristic can be defined as the alternating sum :

\chi=k_0-k_1+k_2-\cdots,

where

k_i

denotes the number of cells of dimension

i

. Then, one can define the Euler characteristic of a manifold as the Euler characteristic of a simplicial complex homeomorphic to it. For example, the circle and torus have Euler characteristic 0 and solid balls have Euler characteristic 1. The Euler characteristic of closed orientable surfaces can be calculated using their genus g :

\chi = 2 - 2g

. The Euler characteristic of closed non-orientable surfaces can be calculated using their (non-orientable) genus k :

\chi = 2 - k

. The Euler characteristic is independent of the triangulation. The formula can also be used for decompositions into arbitrary polygons. For the disk we have

\chi = 1

, for the plane we have

\chi = 2

, counting the outside as a face. For closed manifolds, the Euler characteristic coincides with the Euler number, i.e., the Euler class of its tangent bundle evaluated on the fundamental class of a manifold. For closed Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature--see the Gauss-Bonnet theorem for two-dimensional case and generalized Gauss-Bonnet theorem for general case. A discrete analog of the Gauss-Bonnet theorem is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry). More generally, for any topological space, we can define the nth Betti number

b_n

as the rank of the n-th homology group. The Euler characteristic can then be defined as the alternating sum :

\chi=b_0 - b_1 + b_2 - b_3 +\, \cdots.

This definition makes sense if the Betti numbers are all finite and zero beyond a certain index

n_0

. Two topological spaces which are homotopy equivalent have isomorphic homology groups and hence the same Euler characteristic. From this definition and Poincaré duality, it follows that Euler characteristic of any closed odd-dimensional manifold is zero. If M and N are topological spaces, then the Euler characteristic of their product space M × N is :

\chi(M \times N) = \chi(M) \cdot \chi(N)

Partially ordered set

The concept of the Euler characteristic of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and largest elements, which let us call 0 and 1. The Euler characteristic of such a poset is μ(0,1), where μ is the Möbius function in that poset's incidence algebra.

Proof

The first rigorous proof of Euler's formula, given by a 20-year-old Cauchy, is as follows: Remove one face of the polyhedron. By pulling the edges of the missing face away from each other, deform all the rest into a planar network of points and curves. With no loss of generality it's possible to assume that the deformed edges remained straight line segments. Regular faces cease to be regular polygons if of course they were regular to start with. However, the number of vertices, edges and faces remained the same as those of the given polyhedron (the removed face corresponds to the exterior of the network.) Apply repeatedly a series of additional transformations that would simplify the network without changing its Euler's number (also Euler's characteristic) F − E + V. 1. If there is a polygonal face with more than three sides, we draw a diagonal. This adds one edge and one face. Continue adding edges until all the faces are triangular.
2. Remove (one at a time) all the triangles with two edges shared by the exterior of the network. This removes a vertex, two edges and one face.
3. Remove a triangle with only one edge adjacent to the exterior. This decreases the number of edges and faces by one each and does not change the number of vertices. Carry out steps 2 and 3 repeatedly one after another until only one triangle is left. For a single triangle F = 2 (counting the exterior), E = 3, V = 3. Therefore F − E + V = 2. This proves the theorem. For additional proofs, see [http://www.ics.uci.edu/~eppstein/junkyard/euler/ Nineteen Proofs of Euler's Formula]. Multiple proofs of this, and their flaws and limitations were used as an example in Proofs and Refutations by Lakatos. Category:Algebraic topology Category:Topological graph theory

Sphere

Geometry

Equations

real number In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the set of all points (x, y, z) such that :

(x - x_0 )^2 + (y - y_0 )^2 + ( z -  z_0 )^2 =  r^2 \,

The points on the sphere with radius r can be parametrized via :

x = x_0 + r \sin \theta \; \cos \phi

y = y_0 + r \sin \theta \; \sin \phi  \qquad (0 \leq \theta \leq \pi \mbox -\pi < \phi \leq \pi) \,

z = z_0 + r \cos \theta  \,

(see also trigonometric functions and spherical coordinates). A sphere of any radius centered at the origin is described by the following differential equation: :

x \, dx + y \, dy + z \, dz = 0.

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each

Surface

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Sphere

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