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Continuous Function (topology)

Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. Intuitively, a function is continuous if it maps nearby points to nearby points. For metric spaces, nearness is measured in terms of distance, leading to the ε-δ definition used in real analysis. For more general topological spaces, nearness is measured less directly in terms of open sets, leading to the definition below. If a topological space has the metric topology, the two definitions coincide. Given a set X a partial ordering can be defined on the possible topologies on X. A continuous functions between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space. Thus we can consider the continuity of a given function a topological property, depending only on the topologies of its domain and codomain spaces. A continuous function can be visualized as weakening the topology of the domain space. In real analysis continuity of functions is commonly defined using the ε-δ definition which builds on the property of the real line being a metric space. As topological spaces generally do not have this property a more general definition is needed which reduces to the ε-δ definition in case of the real line.

Definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Open and closed set definition

The most common one defines continuous functions as those functions where the preimages of open sets are open. Similar to the open set formulation is the closed set formulation, which says that preimages of closed sets are closed.

Neighborhood definition

Definition based on preimages are often difficult to use directly. Instead, suppose we have a function f from X to Y, where X,Y are topological spaces. We say f is continuous at x for some

x \in X

if for any neighborhood V of f(x), there is a neighborhood U of x such that

f(U) \subseteq V

. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can find a small U containing x that will map inside it. If f is continuous at every

x \in X

, then we simply say f is continuous. Continuity of a function at a point In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. This leads to the standard ε-δ definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance.

Closure and interior operator definition

Given two topological spaces (X,cl) and (X ' ,cl ') where cl and cl ' are two closure operators then a function :

f:(X,\mathrm) \to (X' ,\mathrm')

is continuous if for all subsets A of X :

f(\mathrm(A)) \subseteq \mathrm'(f(A)).

Similarly given two topological spaces (X,int) and (X ' ,int ') where int and int ' are two interior operators then a function :

f:(X,\mathrm) \to (X' ,\mathrm')

is continuous if for all subsets A of X :

f(\mathrm(A)) \subseteq \mathrm'(f(A)).

Closeness relation definition

Given two topological spaces (X,δ) and (X ' ,δ ') where δ and δ ' are two closeness relations then a function :

f:(X,\delta) \to (X' ,\delta')

is continuous if for all points x and y of X :

x \delta y \Leftrightarrow f(x)\delta'f(y).

Useful properties of continuous maps

Some facts about continuous maps between topological spaces:
- If f : X → Y and g : Y → Z are continuous, then so is the composition g o f : X → Z.
- If f : X → Y is continuous and
- X is compact, then f(X) is compact.
- X is connected, then f(X) is connected.
- X is path-connected, then f(X) is path-connected.
- If f : X → Y is continuous and a sequence (x_n) in X converges to a limit x, then the sequence (f(x_n)) obtained by applying f to each element converges to f(x). We say continuous functions take limits to limits. This also holds if sequences are replaced by general nets.
- If X is a first-countable space, then the converse also holds: any function taking limits of sequences to limits of sequences is continuous. In particular, the converse holds if X is a metric space. When using nets instead of sequences, this converse holds for a general topological space X.

Other notes

If a set is given the discrete topology, all functions with that space as a domain are continuous. If the domain set is given the indiscrete topology and the range set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous. Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection need not be continuous, but if it is, this special function is called a homeomorphism. A continuous bijection is a homeomorphism if its domain is compact and its codomain is Hausdorff. Category:General topology

Topological structure

Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. This article is technical. For a general overview of the subject, see the article on topology.

Definition

A topological space is a set X together with a collection T of subsets of X satisfying the following axioms: # The empty set and X are in T. # The union of any collection of sets in T is also in T. # The intersection of any pair of sets in T is also in T. The set T is a topology on X. The sets in T are the open sets, and their complements in X are the closed sets. The elements of X are called points. The requirement that the union of any collection of open sets be open is more stringent than simply requiring that all pairwise unions be open, as the former includes unions of infinite collections of sets. By induction, the intersection of any finite collection of open sets is open. Thus, the third axiom can be also formulated as: The intersection of any finite collection of sets in T is also in T. An alternate axiom equivalent to axiom 3 is that the topology be closed under all finite intersections instead of just pairwise intersections. This has the benefit that we need not explicitly require that X be in T, since the empty intersection is (by convention) X. Similarly, we can conclude that the empty set is in T by using axiom 2 and taking a union over the empty collection. Nevertheless, it is conventional to include axiom 1 even when it is redundant.

Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology T₁ is also in a topology T₂, we say that T₂ is finer than T₁, and T₁ is coarser than T₂. A proof which relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading. The collection of all topologies on a given fixed set X forms a complete lattice: if F = is a collection of topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of all topologies on X which contain every member of F.

Continuous functions

A function between topological spaces is said to be continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "breaks" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are said to be homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. The category of topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and K-theory, to name just a few.

Alternative definitions

There are many other equivalent ways to define a topological space. (In other words, each of the following defines a category equivalent to the category of topological spaces above.)
- Using de Morgan's laws, the axioms defining open sets become axioms defining closed sets: # The empty set and X are closed. # The intersection of any collection of closed sets is also closed. # The union of any pair of closed sets is also closed.
- The Kuratowski closure axioms determine the closed sets as the fixed points of an operator on the power set of X.
- A neighbourhood of a point x is any set that contains an open set containing x. The neighbourhood system at x consists of all neighbourhoods of x. A topology can be determined by a set of axioms concerning all neighbourhood systems. Equivalently, a topology can be determined by a nearness relation between sets and points.
- A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X the set of its accumulation points is specified.

Examples of topological spaces

- Any set can be given the discrete topology in which every set is open. The only convergent sequences or nets in this topology are those that are eventually constant.
- Any set can be given the trivial topology in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique.
- The set of real numbers R is a topological space: the open sets are generated by the base of open intervals. This means a set is open if it is the union of (possibly infinitely many) open intervals. This is in many ways the most basic topological space and the one that guides most of our human intuition. However, relying on the real line as an intuitive guide for the general concept of topological space can often be dangerous.
- More generally, the Euclidean spaces Rⁿ are topological spaces, and the open sets are generated by open balls.
- Every metric space is a topological space if one defines the open sets to be generated by the set of all open balls. In particular, every normed vector space is a topological space.
- Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
- Any local field has a topology native to it, and this can be extended to vector spaces over that field.
- Every manifold is a topological space.
- Every simplex is a topological space. Simplexes are convex objects that are very useful in computational geometry. In Euclidean space of dimensions 0, 1, 2, and 3, the simplexes are the point, line segment, triangle and tetrahedron, respectively.
- Every simplicial complex is a topological space. A simplicial complex is made up of many simplices. Many geometric objects can be modeled by simplicial complexes — see also Polytope.
- The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety. On Rⁿ or Cⁿ, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.
- A linear graph is a topological space that generalises many of the geometric aspects of graphs with vertices and edges.
- Sierpinski space is the simplest non-trivial, non-discrete topology. It has important relations to the theory of computation and semantics.
- Any infinite set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T₁ topology on the set.
- The real line can also be given the lower limit topology. Here, the open sets consist of the empty set, the whole real line, and all sets generated by half-open intervals of the form [a, b). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.
- If Γ is an ordinal number, then the set [0, Γ] is a topological space, generated by the intervals (a, b], where a and b are elements of Γ.

Topological constructions

- Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset.
- For any nonempty collection of topological spaces, the product can be given the product topology. For finite products, the open sets are generated by the products of open sets.
- A quotient space is defined as follows. If X is a topological space and Y is a set, and if f : X → Y is a surjective function, then Y gets a topology where a set is open if and only if its inverse image is open. A common example comes when an equivalence relation is defined on the topological space X: the map f is then the natural projection onto the set of equivalence classes.
- The Vietoris topology on the set of all non-empty subsets of a topological space X is generated by the following basis: for every n-tuple U₁, ..., U_n of open sets in X, we construct a basis set consisting of all subsets of the union of the U_i which have non-empty intersection with each U_i.

Classification of topological spaces

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property which is not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms. See the article on topological properties for more details and examples.

Topological spaces with algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure which is not finite, we often have a natural topology which is compatible with the algebraic operations in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups, topological vector spaces, topological rings and local fields.

Topological spaces with order structure

- Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).
- Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if and only if c() ⊆ c(). Category:Topology ko:위상공간 (수학) ja:位相空間

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. The Euclidean metric of this space defines the distance between two points as the length of the straight line connecting them. The geometry of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity. A metric space induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22(1906) 1-74.

Definition

A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function :d : X × X → R such that # d(x, y) ≥ 0 (non-negativity) # d(x, y) = 0 if and only if x = y (identity of indiscernibles) # d(x, y) = d(y, x) (symmetry) # d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality). The function d is also called distance function or simply distance. We often omit d and just write X for a metric space if it is clear from the context what metric we are using.

Examples

- The real numbers with the distance function d(x, y) = |y − x| given by the absolute value, and more generally Euclidean n-space with the Euclidean distance, are complete metric spaces.
- Hyperbolic plane.
- Any normed vector space is a metric space by defining d(x, y) = ||y − x||, see also distances based on norms. (If such a space is complete, we call it a Banach space). Example:
- the Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the distances between corresponding coordinates.
- The discrete metric, where d(x,y)=1 for all x not equal to y and d(x,y)=0 otherwise, is a simple but important example, and can be applied to all sets.
- The British Rail metric (also called the Post Office metric) on a normed vector space, given by d(x, y)=||x|| + ||y|| for distinct points x and y, and d(x, x) = 0. The name alludes to the tendency of railway journeys (or letters) to proceed via London, which is identified with the origin.
- The Chessboard distance, the number of moves a chess king would take to travel from x to y.
- If X is some set and M is a metric space, then the set of all bounded functions f : X → M (i.e. those functions whose image is a bounded subset of M) can be turned into a metric space by defining d(f, g) = sup_{x in X} d(f(x), g(x)) for any bounded functions f and g. If M is complete, then this space is complete as well.
- The Levenshtein distance, also called character edit distance, is a measure of the similarity between two strings u and v. The distance is the minimal number of deletions, insertions, or substitutions required to transform u into v.
- If X is a topological (or metric) space and M is a metric space, then the set of all bounded continuous functions from X to M forms a metric space if we define the metric as above: d(f, g) = sup_{x in X} d(f(x), g(x)) for any bounded continuous functions f and g. If M is complete, then this space is complete as well.
- If M is a connected Riemannian manifold, then we can turn M into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
- If G is an undirected connected graph, then the set V of vertices of G can be turned into a metric space by defining d(x, y) to be the length of the shortest path connecting the vertices x and y.
- Similarly (apart from mathematical details):
- For any system of roads and terrains the distance between two locations can be defined as the length of the shortest route. To be a metric there should not be one-way roads. Examples include some mentioned above: the Manhattan norm, the British Rail metric, and the Chessboard distance.
- More generally, for any system of roads and terrains, with given maximum possible speed at any location, the "distance" between two locations can be defined as the time the fastest route takes. To be a metric there should not be one-way roads, and the maximum speed should not depend on direction.
- Similarly, in 3D, the metrics on the surface of a polyhedron include the ordinary metric, and the distance over the surface; a third metric on the edges of a polyhedron is one where the "paths" are the edges. For example, the distance between opposite vertices of a unit cube is √3, √5, and 3, respectively.
- If M is a metric space, we can turn the set K(M) of all compact subsets of M into a metric space by defining the Hausdorff distance d(X, Y) = inf. In this metric, two elements are close to each other if every element of one set is close to some element of the other set. One can show that K(M) is complete if M is complete.
- The set of all (isometry classes of) compact metric spaces form a metric space with respect to Gromov-Hausdorff distance.

Metric spaces as topological spaces

In any metric space M we can define the open balls as the sets of the form ::B(x; r) = , where x is in M and r is a positive real number, called the radius of the ball. A subset of M which is a union of (finitely or infinitely many) open balls is called an open set. The complement of an open set is called closed. Every metric space is automatically a topological space, the topology being the set of all open sets. A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details. Since metric spaces are topological spaces, one has a notion of continuous function between metric spaces. Without referring to the topology, this notion can also be directly defined using limits of sequences; this is explained in the article on continuous functions.

Boundedness and compactness

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. The smallest possible such r is called the diameter of M. The space M is called precompact or totally bounded if for every r > 0 there exist finitely many open balls of radius r whose union equals M. Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded. The converse does not hold, since any infinite set can be given the discrete metric (the first example above) under which it is bounded and yet not totally bounded. A useful characterisation of compactness for metric spaces is that a metric space is compact if and only if it is complete and totally bounded. Note that compactness depends only on the topology, while boundedness depends on the metric. Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space Rⁿ a bounded set is referred to as "a finite interval" or "finite region". However boundedness should not in general be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely. By restricting the metric, any subset of a metric space is a metric space itself (a subspace). We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.

Separation properties and extension of continuous functions

Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

Distance between points and sets

A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the distance between the point and the set. If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as :d(x,S) = inf Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality: :d(x,S) ≤ d(x,y) + d(y,S) which in particular shows that the map x |-> d(x,S) is continuous.

Equivalence of metric spaces

Comparing two metric spaces one can distinguish various degrees of equivalence. To preserve at least the topological structure induced by the metric, these require at least the existence of a continuous function between them (morphism preserving the topology of the metric spaces). Given two metric spaces (M₁, d₁) and (M₂, d₂):
- They are called topologically isomorphic (or homeomorphic) if there exists a homeomorphism between them.
- They are called uniformly isomorphic if there exists a uniform isomorphism between them.
- They are called isometrically isomorphic if there exists a bijective isometry between them. In this case, the two spaces are essentially identical. An isometry is a function f : M₁ → M₂ which preserves distances: d₂(f(x), f(y)) = d₁(x, y) for all x, y in M₁. Isometries are necessarily injective.
- They are called similar if there exists a positive constant k > 0 and a bijective function f, called similarity such that f : M₁ → M₂ and d₂(f(x), f(y)) = k d₁(x, y) for all x, y in M₁.
- They are called similar (of the second type) if there exists a bijective function f, called similarity such that f : M₁ → M₂ and d₂(f(x), f(y)) = d₂(f(u), f(v)) if and only if d₁(x, y) = d₁(u, v) for all x, y,u, v in M₁. In case of Euclidean space with usual metric the two notions of similarity are equivalent.

Quotient metric space

If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~ with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define :

d'([x],[y]) = \inf\

where the infimum is taken over all finite sequences

(p_1, p_2, \dots, p_n)

and

(q_1, q_2, \dots, q_n)

with

[p_1]=[x]

[q_n]=[y]

[q_i]=[p_], i=1,2,\dots n-1

. In general this will only define a pseudometric, i.e.

d'([x],[y])=0

does not necessarily imply that [x]=[y]. However for nice equivalence relations (e.g., those given by gluing together polyhedra along faces), it is a metric. Moreover if M is a compact space, then the induced topology on M/~ is the quotient topology. The quotient metric d is characterized by the following universal property. If $f:(M,d)\longrightarrow(X,\delta)$ is a short map between metric spaces (that is, $\delta(f(x),f(y))\le d(x,y)$ for all x, y) satisfying f(x)=f(y) whenever $x\sim y,$ then the induced function $\overline:M/\sim\longrightarrow X$ , given by $\overline([x])=f(x)$ , is a short map $\overline:(M/\sim,d')\longrightarrow (X,\delta).$
See also

- Glossary of Riemannian and metric geometry
- topology
- triangle inequality
- Lipschitz continuity
- isometry, contraction mapping and short map
- Category of metric spaces
- Norm (mathematics)
References

- Dmitri Burago, Iu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0821821296.
External link

- [http://www.cut-the-knot.org/do_you_know/far_near.shtml Far and near — several examples of distance functions] at cut-the-knot
- [http://mathworld.wolfram.com/MetricSpace.html Metric Space] — Metric Spaces on Wolfram's MathWorld Category:Metric geometry Category:Topology ko:거리공간 ja:距離空間

Real analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. It is a sophisticated theory of the 'numerical function' idea, and contains modern theories of generalized functions. The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction. Then the real numbers are either introduced axiomatically, or they are constructed from Cauchy sequences or Dedekind cuts of rational numbers. Initial consequences are derived, most importantly the properties of the absolute value such as the triangle inequality and Bernoulli's inequality. The concept of convergence, central to analysis, is introduced via limits of sequences. Several laws governing the limiting process can be derived, and several limits can be computed. Infinite series, which are special sequences, are also studied at this point. Power series serve to cleanly define several central functions, such as the exponential function and the trigonometric functions. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next. The concept of continuity may now be defined via limits. One can show that the sum, product, composition and quotient of continuous functions is continuous, excluding at points where the denominator function has value zero, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem. Then one can do integration (Riemann and Lebesgue) and prove the fundamental theorem of calculus, typically using the mean value theorem. At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated. We can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here.

External links

- [http://www.math.unl.edu/~webnotes/contents/chapters.htm Analysis WebNotes] by John Lindsay Orr
- [http://www.shu.edu/projects/reals/index.html Interactive Real Analysis] by Bert G. Wachsmuth
- [http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html A First Analysis Course] by John O'Connor
- ja:%E5%AE%9F%E8%A7%A3%E6%9E%90

Open set

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U; it can't be on the edge of U. As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1]. Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers. Note also that "open" is not the opposite of "closed". First, there are sets which are both open and closed (called clopen sets); in R and other connected spaces, only the empty set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as (0,1] in R.

Definitions

The concept of open sets can be formalized in various degrees of generality.

Function-analytic

A point set in Rⁿ is called open when every point P of the set is an inner point.

Euclidean space

A subset U of the Euclidean n-space Rⁿ is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rⁿ whose Euclidean distance from x is smaller than ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) Intuitively, ε measures the size of the allowed "wiggles". An example of an open set in R² (on a plane) would be all the points within a circle radius r, which satisfy the equation

r>\sqrt

. Because the distance of any point p in this set from the edge of the set is greater than zero:

r-\sqrt>0

, we can set ε to half of this distance, which means ε is also greater than zero, and all the points that are within a distance of ε to p are also in the set, thus satisfying the conditions for an open set.

Metric spaces

A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological spaces

In topological spaces, the concept of openness is taken to be fundamental. One starts with an arbitrary set X and a family of subsets of X satisfying certain properties that every "reasonable" notion of openness is supposed to have. (Specifically: the union of open sets is open, the finite intersection of open sets is open, and in particular the empty set and X itself are open.) Such a family T of subsets is called a topology on X, and the members of the family are called the open sets of the topological space (X,T). Note that infinite intersections of open sets need not be open. Sets that can be constructed as the intersection of countably many open sets are denoted G_δ sets. The topological definition of open sets generalises the metric space definition: If you start with a metric space and define open sets as before, then the family of all open sets will form a topology on the metric space. Every metric space is hence in a natural way a topological space. (There are however topological spaces which are not metric spaces.)

Uses

Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The map f is called open if the image of every open set in X is open in Y. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

Manifolds

A manifold is called open if it is a manifold without boundary and if it is not compact. This notion differs somewhat from the openness discussed above. Category:general topology ja:開集合

Metric topology

History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22(1906) 1-74.

Definition

Examples

Metric spaces as topological spaces

Boundedness and compactness

Separation properties and extension of continuous functions

Distance between points and sets

Equivalence of metric spaces

Quotient metric space

d'([x],[y]) = \inf\

where the infimum is taken over all finite sequences

(p_1, p_2, \dots, p_n)

and

(q_1, q_2, \dots, q_n)

with

[p_1]=[x]

[q_n]=[y]

[q_i]=[p_], i=1,2,\dots n-1

. In general this will only define a pseudometric, i.e.

d'([x],[y])=0

Stronger topology

In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set. This order relation can be used to compare the different topologies.

Definition

Let τ₁ and τ₂ be two topologies on a set X such that τ₁ is contained in τ₂: :

\tau_1 \subseteq \tau_2

. That is, every set open under τ₁ is also open under τ₂. Then the topology τ₁ is said to be a coarser (weaker or smaller) topology than τ₂, and τ₂ is said to be a finer (stronger or larger) topology than τ₁. If additionally :

\tau_1 \neq \tau_2

we say τ₁ is strictly coarser than τ₂ and τ₂ is strictly finer than τ₁. The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on X. N.B. There are some authors, especially analysts, who use the terms weak and strong with opposite meaning.

Examples

The finest topology on X is the discrete topology. The coarsest topology on X is the trivial topology. In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships. All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.

Properties

Let τ₁ and τ₂ be two topologies on a set X. Then the following statements are equivalent:
- τ₁ ⊆ τ₂
- the identity map id_X : (X, τ₂) → (X, τ₁) is a continuous map.
- the identity map id_X : (X, τ₁) → (X, τ₂) is an open map (or, equivalently, a closed map) Two immediate corollaries of this statement are
- A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
- An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser. One can also compare topologies using neighborhood bases. Let τ₁ and τ₂ be two topologies on a set X and let B_i(x) be a local base for the topology τ_i at x ∈ X for i = 1,2. Then τ₁ ⊆ τ₂ if and only if for all x ∈ X, each open set U₁ in B₁(x) contains some open set U₂ in B₂(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

Lattice of topologies

The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice. That is, any collection of topologies on X have a meet (or infimum) and a join (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union. Every complete lattice is also a bounded lattice, which is to say that is has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.

Weaker topology

Definition

Let τ₁ and τ₂ be two topologies on a set X such that τ₁ is contained in τ₂: :

\tau_1 \subseteq \tau_2

\tau_1 \neq \tau_2

Examples

Properties

Lattice of topologies

Codomain space

In mathematics, the domain of a function is the set of all input values to the function. X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs . Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. Given a function f : A → B, the set A is called the domain, or domain of definition of f. The set of all values in the codomain that f maps to is called the range of f, written f(A). A well-defined function must map every element of the domain to an element of its codomain. For example, the function f defined by : f(x) = 1/x has no value for f(0). Thus, the set R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R\, or the "gap" is plugged by specifically defining f(0). If we extend the definition of f to : f(x) = 1/x, for x ≠ 0 : f(0) = 0, then f is defined for all real numbers and we can choose its domain to be R. Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where S ⊆ A, is written g |_S : S → B. Some well-known domains are as follows (note that each successive domain includes those above it):

Category theory

In category theory, instead of functions, one deals with morphisms, which are simply arrows from one object to another. The domain of any morphism is then simply the object where the arrow starts. In this context, many set theoretic ideas about domains have to be abandoned, or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more.

Complex analysis

In complex analysis, a domain is an open connected subset of the complex numbers.

Topological property

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.

Common topological properties

Separation of points

For a detailed treatment, see separation axiom. Some of these terms are defined differently in older mathematical literature; see history of the separation axioms.
- T₀ or Kolmogorov. A space is T₀ if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x.
- T₁ or Fréchet. A space is T₁ if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T₀; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T₁ if all its singletons are closed. T₁ spaces are always T₀.
- Sober. A space is sober if every irreducible closed set C has a unique generic point p. In other words, if C is not the (possibly nondisjoint) union of two smaller closed subsets, then there is a p such that the closure of equals C, and p is the only point with this property.
- T₂ or Hausdorff. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. Hausdorff spaces are always T₁.
- Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
- T₃ or Regular Hausdorff. A space is regular Hausdorff if it is a regular T₀ space. (A regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.)
- Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and are functionally separated.
- T_3½, Tychonoff, Completely regular Hausdorff or Completely T₃. A Tychonoff space is a completely regular T₀ space. (A completely regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
- Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.
- T₄ or Normal Hausdorff. A normal space is Hausdorff if and only if it is T₁. Normal Hausdorff spaces are always Tychonoff.
- Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.
- T₅ or Completely normal Hausdorff. A completely normal space is Hausdorff if and only if it is T₁. Completely normal Hausdorff spaces are always normal Hausdorff.
- Discrete space. A space is discrete if all of its points are completely isolated, i.e. if any subset is open.

Countability conditions

- Separable. A space is separable if it has a countable dense subset.
- Lindelöf. A space is Lindelöf if every open cover has a countable subcover.
- First-countable. A space is first-countable if every point has a countable local base.
- Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

Connectedness

- Connected. A space X is connected if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.
- Locally connected. A space is locally connected if every point has a local base consisting of connected sets.
- Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.
- Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.
- Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
- Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S¹ → X is homotopic to a constant map.
- Locally simply connected. A space X is locally simply connected if every point x in X has a local base of neighborhoods U that is simply connected.
- Semi-locally simply connected. A space X is semi-locally simply connected if every point has a local base of neighborhoods U such that every loop in U is contractible in X. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a universal cover.
- Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.
- Hyper-connected. A space is hyper-connected if no two non-empty open sets are disjoint. Every hyper-connected space is connected.
- Ultra-connected. A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected.
- Indiscrete or Trivial. A space is indiscrete if the only open sets are the whole space and the empty set. Such a space is said to have the trivial topology.

Compactness

- Compact. A space is compact if every open cover has a finite subcover. Some authors call these spaces quasicompact and reserve compact for Hausdorff spaces where ever open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.
- Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.
- Countably compact. A space is countably compact if every countable open cover has a finite subcover.
- Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Locally compact Hausdorff spaces are always Tychonoff.
- Relatively compact. A relatively compact subspace is one whose closure is compact. Every subspace of a compact space (including the compact space itself) is relatively compact. This is, strictly speaking, not a topological invariant of the space, but depends on how it is embedded as a subspace.
- Ultra-connected compact. In an ultra-connected compact space X every open cover must contain X itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a monolith.

Metrizability

- Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.
- Polish. A space is called Polish if it is metrizable with a separable and complete metric.
- Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.

Miscellaneous

- Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.
- Homogeneous. A space X is homogeneous if for every x and y in X there is a homeomorphism f : X -> X such that f(x) = y. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.
- Finitely generated or Alexandrov. A space X is Alexandrov if arbitrary intersections of open sets in X are open. Equivalently if arbitrary unions of closed sets are closed. These are precisely the finitely generated members of the category of topological spaces and continuous maps.
- Zero-dimensional. A space is zero-dimensional if it has a base of clopen sets. These are precisely the spaces with a topological dimension of 0.
- Almost discrete. A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
- Boolean. A space is Boolean if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the Stone spaces of Boolean algebras.
- Torsion (topology) Category:Topology

Real line

In mathematics, the real line is simply the set R of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. The real line has been studied at least since the days of the ancient Greeks, but it was not rigorously defined until 1872. Before and since that date, it has been a prolific example that has played a significant role in many branches of mathematics. The real line carries a standard topology which can be introduced in two different, equivalent ways. First, since the real numbers are totally ordered, they carry an order topology. Second, the real numbers can be turned into a metric space by using the metric given by the absolute value:

d(x,y) := |y - x|

. This metric induces a topology on R equal to the order topology. As a topological space, the real line is a topological manifold of dimension . It is paracompact and second-countable as well as contractible and locally compact. It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.) Indeed, R was historically the first example to be studied of each of these mathematical structures, so that it serves as the inspiration for these branches of modern mathematics. (Indeed, many of the terms above can't even be defined until R is already in place.) As a vector space, the real line is a vector space over the field R of real numbers (that is, over itself) of dimension . It has a standard inner product, making it an Euclidean space. (The inner product is simply ordinary multiplication of real numbers.) As a vector space, it is not very interesting, and thus it was in fact 2-dimensional Euclidean space that was first studied as a vector space. However, we can still say that R inspired the field of linear algebra, since vector spaces were first studied over R. R is also a premier example of a ring, even a field. It is in fact a real complete field, and was the first such field to be studied, so that it inspired that branch of abstract algebra as well. However, in such purely algebraic contexts, R is rarely called a "line". For more information on R in all of its guises, see real number. Category:Real numbers Category:topological spaces

Metric space

History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22(1906) 1-74.