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Connected Space

Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. It is usually easy to think about what is not connected. A simple example would be a space consisting of two rectangles, each of which is a space and not adjoined to the other. The space is not connected since two rectangles are disjoint. Another good example is a space with an annulus removed. The space is not connected since you cannot connect two points, one inside the annulus and the other outside; hence the term "connect". Also, in a sense, a connected space is a generalization of an interval on the real number line, just as a topological space is, so to speak, an attempt to generalize an interval.

Formal definition

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors specifically exclude the empty set with its unique topology as a connected space, but this encyclopedia does not follow that practice. For a topological space X the following conditions are equivalent: #X is connected. #X cannot be divided into two disjoint nonempty closed sets (This follows since the complement of an open set is closed). #The only sets which are both open and closed (clopen sets) are X and the empty set. #The only sets with empty boundary are X and the empty set. #X cannot be written as the union of two nonempty separated sets. The maximal nonempty connected subsets of any topological space are called the connected components of the space. The components form a partition of the space (that is, they are disjoint and their union is the whole space). Every component is a closed subset of the original space. The components in general need not be open: the components of the rational numbers, for instance, are the one-point sets. A space in which all components are one-point sets is called totally disconnected. A space X is totally disconnected iff, for any two elements x and y of X, there exist disjoint open neighborhoods U of x and V of y such that X is the union of U and V.

Examples

- The space of real numbers with the usual topology is connected.
- Every discrete topological space is totally disconnected.
- The Cantor set is totally disconnected.

Path connectedness

Cantor set The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. (This function is called a path from x to y.) Every path-connected space is connected. Example of connected spaces that are not path-connected include the extended long line L
- and the topologist's sine curve. However, subsets of the real line R are connected if and only if they are path-connected; these subsets are the intervals of R. Also, open subsets of Rⁿ or Cⁿ are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces. A space X is said to be arc-connected if any two distinct points can be joined by an arc, that is a path f which is a homeomorphism between the unit interval [0,1] and its image f([0,1]). It can be shown any Hausdorff space which is path-connected is also arc-connected. An example of a space which is path-connected but not arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0,∞). One endows this set with a partial order by specifying that 0'<a for any positive number a, but leaving 0 and 0' incomparable. One then endows this set with the order topology, that is one takes the open intervals (a,b)= and the half-open intervals [0,a)=, [0',a)= as a base for the topology. The resulting space is a T₁ space but not a Hausdorff space. Clearly 0 and 0' can be connected by a path but not by an arc in this space.

Local connectedness

A topological space is said to be locally connected if it has a base of connected sets. It can be shown that a space X is locally connected if and only if every component of every open set of X is open. The topologist's sine curve shown above is an example of a connected space that is not locally connected. Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about Rⁿ and Cⁿ, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.

Theorems

- Main theorem: Let X and Y be topological spaces and let f : X → Y be a continuous function. If X is connected (resp. path-connected) then the image f(X) is connected (resp. path-connected). The intermediate value theorem can be considered as a special case of this result.
- If

\

is a family of connected subsets of a topological space X such that

A_i \cap A_

is nonempty for all i, then

\cup A_i

is also connected.
- If

\

is a nonempty family of connected subsets of a topological space X such that

\cap A_\alpha

is nonempty, then

\cup A_\alpha

is also connected.
- Every path-connected space is connected.
- Every locally path-connected space is locally connected.
- A locally path-connected space is path-connected iff it is connected.
- The connected components of a space are disjoint unions of the path-connected components.
- The components of a locally connected space are open (and closed).
- The closure of a connected subset is connected.
- Every quotient of a connected (resp. path-connected) space is connected (resp. path-connected).
- Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
- Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
- Every manifold is locally path-connected.

References

Category:General topology

Topological space

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

Disjoint union (topology)

In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.

Definition

Let be a family of topological spaces indexed by I. Let :

X = \coprod_i X_i

be the disjoint union of the underlying sets. For each i in I, let :

\phi_i : X_i \to X\,

be the canonical injection. The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e. the final topology for the family of functions ). Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage

\phi_i^(U)

is open in X_i for each i ∈ I.

Properties

The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and f_i : X_i → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute: commute This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff f_i = f o φ_i is continuous for all i in I. In addition to being continuous, the canonical injections φ_i : X_i → X are open and closed maps. It follows that the injections are topological embeddings so that each X_i may be canonically thought of as a subspace of X.

Examples

If each X_i is homeomorphic to a fixed space A, then the disjoint union X will be homeomorphic to A × I where I is given the discrete topology.

Preservation of topological properties

- every disjoint union of discrete spaces is discrete
- Separation
- every disjoint union of T₀ spaces is T₀
- every disjoint union of T₁ spaces is T₁
- every disjoint union of Hausdorff spaces is Hausdorff
- Connectedness
- the disjoint union of two or more topological spaces is disconnected

Topological properties

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

Topological space

Definition

Comparison of topologies

Continuous functions

Alternative definitions

Examples of topological spaces

Topological constructions

Classification of topological spaces

Topological spaces with algebraic structure

Topological spaces with order structure

Disjoint

In mathematics, two sets are said to be disjoint if they have no element in common. For example, and are disjoint sets.

Explanation

Formally, two sets A and B are disjoint if their intersection is the empty set, i.e. if :

A\cap B = \varnothing.\,

This definition extends to any collection of sets. A collection of sets is pairwise disjoint or mutually disjoint if any two distinct sets in the collection are disjoint. Formally, let I be an index set, and for each i in I, let A_i be a set. Then the family of sets is pairwise disjoint if for any i and j in I with i ≠ j, :

A_i \cap A_j = \varnothing.\,

For example, the collection of sets is pairwise disjoint. If is a pairwise disjoint collection, then clearly its intersection is empty: :

\bigcap_ A_i = \varnothing.\,

However, the converse is not true: the intersection of the collection is empty, but the collection is not pairwise disjoint - in fact, there are no two disjoint sets on the collection. A partition of a set X is any collection of non-empty subsets of X such that are pairwise disjoint and :

\bigcup_ A_i = X.\,

NonEmpty

In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. The requirement that a set be non-empty is frequently found in mathematical hypotheses. One reason is that we can easily make logical errors when we make hypotheses about the empty set, which is often nonintuitive and can be tricky to reason about correctly (see Empty set for further discussion of this). Thus, this hypothesis of nonemptiness can often be removed under a more careful treatment. On the other hand, there certainly are times when the empty set is a special case and really does need to be excluded from a hypothesis. A common example where both of these situations arise is the axiom of choice. Although this axiom can be stated in several ways, for each standard way of stating it, there are two places where the term "nonempty" could be used. Often you will find the term used in both places, whereas in fact it is needed in only one. (See Axiom of choice for further discussion of this example). Category:Set theory

Empty set

In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. In axiomatic set theory it is postulated to exist by the axiom of empty set and all finite sets are constructed from it. The empty set is also sometimes called the null set, but because null set means something else in measure theory, that term is generally avoided in current work. Various possible properties of sets are trivially true for the empty set.

Notation

The standard notation for denoting the empty set is the symbol

\varnothing

, ø, or Ø, introduced by the Bourbaki group (specifically André Weil) in 1939[http://members.aol.com/jeff570/set.html]. This should not be confused with the Greek letter Φ. Another common notation for the empty set is .

Properties

(Here we use mathematical symbols.)
- The empty set is NOT 0.
- :

\varnothing

≠ 0
- For any set A, the empty set is a subset of A:
- : ∀A:

\varnothing

⊆ A
- For any set A, the union of A with the empty set is A:
- : ∀A: A ∪

\varnothing

= A
- For any set A, the intersection of A with the empty set is the empty set:
- : ∀A: A ∩

\varnothing

\varnothing

- For any set A, the Cartesian product of A and the empty set is empty:
- : ∀A: A ×

\varnothing

\varnothing

- The only subset of the empty set is the empty set itself:
- : ∀A: A ⊆

\varnothing

⇒ A =

\varnothing

- The number of elements of the empty set (that is its cardinality) is zero; in particular, the empty set is finite:
- : |

\varnothing

| = 0
- For any property:
- for every element of

\varnothing

the property holds (vacuous truth)
- there is no element of

\varnothing

for which the property holds
- Conversely: if, for some property, the following two statements hold:
- for every element of V the property holds
- there is no element of V for which the property holds :then V =

\varnothing

Mathematicians speak of "the empty set" rather than "an empty set". In set theory, two sets are equal if they have the same elements; therefore there can be only one set with no elements. Considered as a subset of the real number line (or more generally any topological space), the empty set is both closed and open. All its boundary points (of which there are none) are in the empty set, and the set is therefore closed; while all its interior points (of which there are again none) are in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the fact that every finite set is compact. The closure of the empty set is empty. This is known as "preservation of nullary unions."

Common problems

The empty set is not the same thing as nothing; it is a set with nothing inside it, and a set is something. This often causes difficulty among those who first encounter it. It may be helpful to think of a set as a bag containing its elements; an empty bag may be empty, but the bag itself certainly exists. Some people balk at the first property listed above, that the empty set is a subset of any set A. By the definition of subset, this claim means that for every element x of , x belongs to A. If it is not true that every element of is in A, there must be at least one element of that is not present in A. Since there are no elements of at all, there is no element of that is not in A, leading us to conclude that every element of is in A and that is a subset of A. Any statement that begins "for every element of " is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."

Axiomatic set theory

In the axiomatization of set theory known as Zermelo-Fraenkel set theory, the existence of the empty set is assured by the axiom of empty set. The uniqueness of the empty set follows from the axiom of extensionality. Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = , which can be defined to be the empty set.

Does it exist or is it necessary?

While the empty set is a standard and universally accepted concept in mathematics, there are those who still entertain doubts. Jonathan Lowe has argued that while the idea "was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object." It is not clear that such an idea makes sense. "All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a set which has no members. We cannot conjure such an entity into existence by mere stipulation." In "To be is to be the value of a variable…", Journal of Philosophy, 1984 (reprinted in his book Logic, Logic and Logic), the late George Boolos has argued that we can go a long way just by quantifying plurally over individuals, without reifying sets as singular entities having other entities as members. In a recent book [http://philosophy.syr.edu Tom McKay] has disparaged the "singularist" assumption that natural expressions using plurals can be analysed using plural surrogates, such as signs for sets. He argues for an anti-singularist theory which differs from set theory in that there is no analogue of the empty set, and there is just one relation, among, that is an analogue of both the membership and the subset relation.

Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) can also be confusing. (Such operations are nullary operations.) For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). This may seem odd, since there are no elements of the empty set, so how could it matter whether they are added or multiplied (since “they” do not exist)? Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication.

Bounds

Since the empty set has no members, when it is considered as a subset of any ordered set, then any member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set. When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely "negative infinity", denoted

-\infty\!\,,

which is defined to be less than every other extended real number, and "positive infinity", denoted

+\infty\!\,,

which is defined to be greater than every other extended real number, then: :

\sup\varnothing=\min(\ \cup \mathbb)=-\infty,

and :

\inf\varnothing=\max(\ \cup \mathbb)=+\infty.

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity.

The empty set and zero

It was mentioned earlier that the empty set has zero elements, or that its cardinality is zero. The connection between the two concepts goes further however: in the standard set-theoretic definition of natural numbers, zero is defined as the empty set.

Category theory

If A is a set, then there exists precisely one function f from to A, the empty function. As a result, the empty set is the unique initial object of the category of sets and functions. The empty set can be turned into a topological space in just one way (by defining the empty set to be open); this empty topological space is the unique initial object in the category of topological spaces with continuous maps. Category:Set theory ko:공집합 ja:空集合

Closed set

:For closed manifolds, see closed manifold. For closed orbits, see Closed orbit. In topology and related branches of mathematics, a closed set is a set whose complement is open. This implies that a closed set contains its own boundary. Intuitively, if you are outside the set, and you "wiggle" a little bit, you will still be outside the set. Note that this notion depends on the concept of "outside", the surrounding space with respect to which the complement is taken. For instance, the unit interval [0,1] is closed in the real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the real numbers. Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers. In functional analysis, a point set is closed if it contains all its accumulation points. The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets. One value of this characterisation is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces. Notice that this characterisation also depends on the surrounding space X, because whether or not a sequence or net converges in X depends on what points are present in X. Any intersection of arbitrarily many closed sets is closed, and any union of finitely many closed sets is closed. In particular, the empty set and the whole space are closed. In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A can be constructed as the intersection of all of these closed supersets. Sets that can be constructed as the union of countably many closed sets are denoted F_σ sets. These sets need not be closed. We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in. However, the compact Hausdorff spaces are "absolutely closed" in a certain sense. To be precise, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here. In fact, this property characterizes the compact Hausdorff spaces. Stone-Čech compactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space. Category:General topology ja:閉集合

Complement (set theory)

In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement.

Relative complement

If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A.

setThe relative complement
of A in B

The relative complement of A in B is usually written B − A (also B \ A). Formally: :

B - A = \.

Examples: :
- − = :
- − = :
- If

\mathbb

is the set of real numbers and

\mathbb

is the set of rational numbers, then

\mathbb-\mathbb

is the set of irrational numbers. The following proposition lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection. PROPOSITION 1: If A, B, and C are sets, then the following identities hold: :
- C − (A ∩B)  =  (C − A) ∪(C − B) :
- C − (A ∪B)  =  (C − A) ∩(C − B) :
- C − (B − A)  =  (A ∩C) ∪(C − B) :
- (B − A) ∩C  =  (B ∩C) − A  =  B ∩(C − A) :
- (B − A) ∪C  =  (B ∪C) − (A − C) :
- A − A  =  Ø :
- Ø − A  =  Ø :
- A − Ø  =  A

Absolute complement

identitiesThe complement of A in U

If a universal set U is defined, then the relative complement of A in U is called the absolute complement (or simply complement) of A, and is denoted by A^C, that is: :A^C  = U − A For example, if the universal set is the set of natural numbers, then the complement of the set of odd numbers is the set of even numbers. The following proposition lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection. PROPOSITION 2: If A and B are subsets of a universal set U, then the following identities hold: :De Morgan's laws: ::
- (A ∪ B)^C = A^C ∩ B^C ::
- (A ∩ B)^C = A^C ∪ B^C :Complement laws: ::
- A ∪ A^C  =  U ::
- A ∩ A^C  =  Ø ::
- Ø^C  =  U ::
- U^C  =  Ø ::
- If A⊆B, then B^C⊆A^C (this follows from the equivalence of a conditional with its contrapositive) :Involution or double complement law: ::
- A^C^C  =  A. :Relationships between relative and absolute complements: ::
- A − B = A ∩ B^C ::
- (A − B)^C = A^C ∪ B The first two complement laws above shows that if A is a non-empty subset of U, then is a partition of U.

Boundary (topology)

:For a different notion of boundary related to manifolds, see that article. In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd(S), fr(S), and

\partial S

. There are several common (and equivalent) definitions to the boundary of S:
- the intersection of the closure of S with the closure of its complement: ::

\partial S = \bar \bigcap \overline.

- the closure of S without the interior of S:

\partial S = \bar\setminus S^o

.
- a point p in X is a boundary point of S if every neighborhood of p contains at least one point of S and at least one point not in S. The boundary of S is the set of all boundary points of S.

Examples

Consider the real line R with the usual topology, that is, the open sets are the open intervals (a, b). One has
-

\partial [0,5) = \

\partial \emptyset = \emptyset

\partial \mathbb = \mathbb

\partial \big(\mathbb\cap\left[0,1\right]\big) = \left[0,1\right]

The last two examples above illustrate the fact that the boundary of a dense set with empty interior is its closure. One should keep in mind the boundary of S is a topological notion, therefore, one changes the topology, the boundary of a set may change. For example, given the usual topology on R² and the closed disk :Ω=, one has that ∂Ω = . If the same disk is viewed as a set in R³ with its own usual topology, :Ω=, then ∂Ω = Ω.

Properties

- The boundary of a set is closed.
- The boundary of a set is the boundary of the complement of the set:

\partial S = \partial\bar

. Hence:
- p is a boundary point of a set iff every neighborhood of p contains at least one point in the set and at least one point not in the set.
- A set is closed iff it contains its boundary, and open iff it is disjoint from its boundary.
- The closure of a set equals the union of the set with its boundary.

\bar = S \bigcup\partial S

.
- The boundary of a set is empty iff the set is both closed and open (i.e. a clopen set).
- In

\mathbb^n

, every closed set is the boundary of some set.

Boundary of a boundary

For any set S, ∂S⊇∂∂S, with equality holding if the set S is either closed or open (a sufficient, but not necessary condition for equality. A half-open interval, for example, is neither closed nor open in the real line, but still satisfies the equality). One also has the equality ∂∂S=∂∂∂S for any set, so the boundary operator satisfies a weakened kind of idempotence. In particular, the boundary of the boundary of a set will in general be nonempty.

Relationship to boundary of manifolds

As mentioned above, the boundary of a set depends on what topological space the set is a subset of, and what topology that space carries. For example, every subset of any space can be regarded as a topological space in its own right when endowed with the subspace topology, in which its boundary will be empty. The boundary of a manifold is defined to be those points of the manifold which have neighborhoods homeomorphic to half of Rⁿ and no neighborhood homeomorphic to Rⁿ. This boundary need not agree with the boundary of the manifold as a topological space in its underlying topology, which is always empty. If the manifold can be embedded in Rⁿ, however, then the manifold boundary will agree with the topological boundary in Rⁿ. In general, only fairly topologically trivial n-dimensional manifolds can be embedded in Rⁿ, so this does not make a very useful definition. Alternatively, the boundary of the manifold can be described as those points which map to topological boundaries of halves of Rⁿ under the local homeomorphisms. Similar difficulties arise when understanding how the combinatorial boundaries of simplexes and simplicial complexes are related to their topological boundaries, though all simplexes can be (and usually are) defined to be subsets of Rⁿ, where those difficulties are averted. In discussing boundaries of manifolds or complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically on this fact. The apparent incongruity is due to the differences in topology: in these contexts the boundary is endowed with its subspace topology. To illustrate the point, consider the unit disc D in R². Its topological boundary is the unit circle which agrees with its boundary as a manifold. The unit circle is its own topological boundary in R², but as a manifold its boundary is empty. Thus ∂∂D = ∂D, as expected since D is open, and is nonempty. On the other hand, the unit circle, endowed with its subspace topology, is the whole space, which always has empty boundary. On the other hand, a cylinder C of finite length is a 2-dimensional manifold which cannot be embedded in R². It is its own topological boundary in R³, but its boundary as a manifold is the disjoint union of two circles. Category:General topology ko:경계 (위상수학)

Maximal element

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. Formally, given a partially ordered set (P, ≤), an element m of a subset S of P is a maximal element of S if :

m \leq s

for some

s\in S

implies that

m = s.

The definition for minimal elements is obtained by using ≥ instead of ≤. What is important to note about maximal elements is that they are in general not the greatest elements of a subset S, i.e. while they are not smaller than any other element of S, they do not have to be greater than all other elements either. Indeed, consider the set of all subsets of the natural numbers (i.e. the power set) ordered by subset inclusion. The subset S of all one-element sets of natural numbers consists only of maximal elements, but has no greatest element. This example also shows that maximal elements are usually not unique and that it is possible for an element to be both maximal and minimal at the same time. If a subset has a greatest element, then this is the unique maximal element. Conversely, even if a set has only one maximal element, it is not necessarily the greatest one. Take the set of natural numbers in their usual order, which obviously has no maximal elements, and add a single new element a which can only be compared to itself, i.e. it is neither smaller nor greater than any natural number. Then the whole set has a as a single maximal element that is not the greatest element. Yet, in a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation does not only apply to totally ordered subsets of any poset, but also to their order theoretic generalization via directed sets. In a directed set, every pair of elements (especially pairs of incomparable elements) has a common upper bound within the set. It is easy to see that any maximal element of such a subset will be unique (unlike in a poset). Furthermore, this unique maximal element will also be the greatest element. Similar conclusions are true for minimal elements. Further introductory information is found in the article on order theory. Category:Order theory

Disjoint

In mathematics, two sets are said to be disjoint if they have no element in common. For example, and are disjoint sets.

Explanation

Formally, two sets A and B are disjoint if their intersection is the empty set, i.e. if :

A\cap B = \varnothing.\,

A_i \cap A_j = \varnothing.\,

For example, the collection of sets is pairwise disjoint. If is a pairwise disjoint collection, then clearly its intersection is empty: :

\bigcap_ A_i = \varnothing.\,

\bigcup_ A_i = X.\,

Rational number

In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. Each rational number can be written in infinitely many forms, for example

3/6 = 2/4 = 1/2

. The simplest form is when

a

and

b

have no common divisors, and every non-zero rational number has exactly one simplest form of this type with positive denominator. The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above 1. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not rational is called an irrational number. In mathematics, the term "rational something" means that the underlying field considered is the field

\mathbb

of rational numbers. For example, rational polynomials or rational prime ideals. The set of all rational numbers is denoted by Q, or in blackboard bold

\mathbb

. Using the set-builder notation

\mathbb

is defined as such: :

\mathbb = \left\

Arithmetic

\frac + \frac = \frac

\frac \cdot \frac = \frac

Two rational numbers

\frac

and

\frac

are equal if and only if

ad =  bc

Additive and multiplicative inverses exist in the rational numbers. :

- \left( \frac \right) = \frac

\left(\frac\right)^ = \frac \mbox a \neq 0

History

Egyptian fractions

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance,

\frac = \frac + \frac + \frac

For any positive rational number, there are infinitely many different such representations. These representations are called Egyptian fractions, because the ancient Egyptians used them. The Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals.

Formal construction

Mathematically we may define them as an ordered pair of integers

\left(a, b\right)

, with

b

not equal to zero. We can define addition and multiplication of these pairs with the following rules: :

\left(a, b\right) + \left(c, d\right) = \left(ad + bc, bd\right)

\left(a, b\right) \times \left(c, d\right) = \left(ac, bd\right)

To conform to our expectation that

2/4 = 1/2

, we define an equivalence relation

\sim

upon these pairs with the following rule: :

\left(a, b\right) \sim \left(c, d\right) \mbox ad = bc

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. (This construction can be carried out in any integral domain, see quotient field.) We can also define a total order on Q by writing :

\left(a, b\right) \le \left(c, d\right) \mbox (bd>0\mbox ad \le bc)\mbox(bd<0\mbox ad \ge bc)

Properties

The set

\mathbb

, together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers

\mathbb

. The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of

\mathbb

. The algebraic closure of

\mathbb

, i.e. the field of roots of rational polynomials, is the algebraic numbers. The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set. The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. As a totally ordered set, the rationals are uniquely characterized by being countable, dense (in the above sense), and having no least or greatest element.

Real numbers

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expressions of continued fraction. By virtue of their order, the rationals carry an order topology. The rational numbers also carry a subspace topology. The rational numbers form a metric space by using the metric

d\left(x, y\right) = |x - y|

, and this yields a third topology on

\mathbb

. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metric space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of

\mathbb

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn

\mathbb

into a topological field: let

p

be a prime number and for any non-zero integer

a

let

|a|_p = p^

, where

p^n

is the highest power of

p

dividing

a

; in addition write

|0|_p = 0

. For any rational number

\frac

, we set

\left|\frac\right|_p = \frac

. Then

d_p\left(x, y\right) = |x - y|_p

defines a metric on

\mathbb

. The metric space

\left(\mathbb, d_p\right)

is not complete, and its completion is the p-adic number field

\mathbb_p

. Category:Elementary mathematics Category:Field theory Category:Fractions Category:Real numbers Category:Set theory ko:유리수 ja:有理数 simple:Rational number th:จำนวนตรรกยะ

Real number

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end. Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively,

\Bbb

, the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rⁿ refers to an n-dimensional space of real numbers; for example, a value from R³ consists of three real numbers and specifies a location in 3-dimensional space. In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

History

Vulgar fractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 AD, and then possibly reinvented in China shortly after. They were not used in Europe until the 1600s, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.

Definition

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.

Axiomatic approach

Let R denote the set of all real numbers. Then:
- The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
- The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
- if x ≥ y then x + z ≥ y + z;
- if x ≥ 0 and y ≥ 0 then xy ≥ 0.
- The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object.

Properties

Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other. A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x. It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example, the standard series of the exponential function :

\mathrm^x = \sum_^ \frac

converges to a real number because for every x the sums :

\sum_^ \frac

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a chara