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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
, a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is usually defined by declaring its
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Biography and studies Kazimierz Kuratowski was born in Warsaw, (t ...
's well-known textbook on
point-set topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom. Likewise, the neighborhood-based axioms (in the context of
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s) can be retraced to
Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and f ...
's original definition of a topological space in
Grundzüge der Mengenlehre ''Grundzüge der Mengenlehre'' (German for "Basics of Set Theory") is a book on set theory written by Felix Hausdorff. First published in April 1914, ''Grundzüge der Mengenlehre'' was the first comprehensive introduction to set theory. Besides t ...
. Many different textbooks use many different inter-dependences of concepts to develop point-set topology. The result is always the same collection of objects: open sets, closed sets, and so on. For many practical purposes, the question of which foundation is chosen is irrelevant, as long as the meaning and interrelation between objects (many of which are given in this article), which are the same regardless of choice of development, are understood. However, there are cases where it can be useful to have flexibility. For instance, there are various natural notions of
convergence of measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharin ...
, and it is not immediately clear whether they arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence.


Standard definitions via open sets

A topological space is a set X together with a collection S of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of X satisfying: * The
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
and X are in S. * The
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
of any collection of sets in S is also in S. * The
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of any pair of sets in S is also in S. Equivalently, the intersection of any finite collection of sets in S is also in S. Given a topological space (X, S), one refers to the elements of S as the open sets of X, and it is common to only refer to S in this way, or by the label topology. Then one makes the following secondary definitions: * Given a second topological space Y, a function f : X \to Y is said to be continuous if and only if for every open subset U of Y, one has that f^(U) is an open subset of X. * A subset C of X is closed if and only if its complement X \setminus C is open. * Given a subset A of X, the closure is the set of all points such that any open set containing such a point must intersect A. * Given a subset A of X, the interior is the union of all open sets contained in A. * Given an element x of X, one says that a subset A is a neighborhood of x if and only if x is contained in an open subset of X which is also a subset of A. Some textbooks use "neighborhood of x" to instead refer to an open set containing x. * One says that a net converges to a point x of X if for any open set U containing x, the net is eventually contained in U. * Given a set X, a
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
is a collection of nonempty subsets of X that is closed under finite intersection and under supersets. Some textbooks allow a filter to contain the empty set, and reserve the name "proper filter" for the case in which it is excluded. A topology on X defines a notion of a filter converging to a point x of X, by requiring that any open set U containing x is an element of the filter. * Given a set X, a filterbase is a collection of nonempty subsets such that every two subsets intersect nontrivially and contain a third subset in the intersection. Given a topology on X, one says that a filterbase converges to a point x if every neighborhood of x contains some element of the filterbase.


Definition via closed sets

Let X be a topological space. According to
De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
, the collection T of closed sets satisfies the following properties: * The
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
and X are elements of T * The
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of any collection of sets in T is also in T. * The
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
of any pair of sets in T is also in T. Now suppose that X is only a set. Given any collection T of subsets of X which satisfy the above axioms, the corresponding set \ is a topology on X, and it is the only topology on X for which T is the corresponding collection of closed sets. This is to say that a topology can be defined by declaring the closed sets. As such, one can rephrase all definitions to be in terms of closed sets: * Given a second topological space Y, a function f : X \to Y is continuous if and only if for every closed subset U of Y, the set f^(U) is closed as a subset of X. * a subset C of X is open if and only if its complement X \setminus C is closed. * given a subset A of X, the closure is the intersection of all closed sets containing A. * given a subset A of X, the interior is the complement of the intersection of all closed sets containing X \setminus A.


Definition via closure operators

Given a topological space X, the closure can be considered as a map \wp(X) \to \wp(X), where \wp(X) denotes the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is p ...
of X. One has the following
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first form ...
: * A \subseteq \operatorname(A) * \operatorname(\operatorname(A)) = \operatorname(A) * \operatorname(A \cup B) = \operatorname(A) \cup \operatorname(B) * \operatorname(\varnothing) = \varnothing If X is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of cl satisfies the previous axioms for closed sets, and hence defines a topology; it is the unique topology whose associated closure operator coincides with the given cl. As before, it follows that on a topological space X, all definitions can be phrased in terms of the closure operator: *Given a second topological space Y, a function f : X \to Y is continuous if and only if for every subset A of X, one has that the set f(\operatorname(A)) is a subset of \operatorname(f(A)). * A subset A of X is open if and only if \operatorname(X \setminus A) = X \setminus A. * A subset C of X is closed if and only if \operatorname(C) = C. * Given a subset A of X, the interior is the complement of \operatorname(X \setminus A).


Definition via interior operators

Given a topological space X, the interior can be considered as a map \wp(X) \to \wp(X), where \wp(X) denotes the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is p ...
of X. It satisfies the following conditions: * \operatorname(A) \subseteq A * \operatorname(\operatorname(A)) = \operatorname(A) * \operatorname(A \cap B) = \operatorname(A) \cap \operatorname(B) * \operatorname(X) = X If X is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of int satisfies the previous axioms for open sets, and hence defines a topology; it is the unique topology whose associated interior operator coincides with the given int. It follows that on a topological space X, all definitions can be phrased in terms of the interior operator, for instance: * Given topological spaces X and Y, a function f : X \to Y is continuous if and only if for every subset B of Y, one has that the set f^(\operatorname(B)) is a subset of \operatorname(f^(B)). * A set is open if and only if it equals its interior. * The closure of a set is the complement of the interior of its complement.


Definition via neighbourhoods

Recall that this article follows the convention that a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
is not necessarily open. In a topological space, one has the following facts: * If U is a neighborhood of x then x is an element of U. * The intersection of two neighborhoods of x is a neighborhood of x. Equivalently, the intersection of finitely many neighborhoods of x is a neighborhood of x. * If V contains a neighborhood of x, then V is a neighborhood of x. * If U is a neighborhood of x, then there exists a neighborhood V of x such that U is a neighborhood of each point of V. If X is a set and one declares a nonempty collection of neighborhoods for every point of X, satisfying the above conditions, then a topology is defined by declaring a set to be open if and only if it is a neighborhood of each of its points; it is the unique topology whose associated system of neighborhoods is as given. It follows that on a topological space X, all definitions can be phrased in terms of neighborhoods: * Given another topological space Y, a map f : X \to Y is continuous if and only for every element x of X and every neighborhood B of f(x), the preimage f^(B) is a neighborhood of x. * A subset of X is open if and only if it is a neighborhood of each of its points. * Given a subset A of X, the interior is the collection of all elements x of X such that A is a neighbourhood of x. * Given a subset A of X, the closure is the collection of all elements x of X such that every neighborhood of x intersects A.


Definition via convergence of nets

Convergence of nets satisfies the following properties: # Every constant net converges to itself. # Every
subnet A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting. Computers that belong to the same subnet are addressed with an identical ...
of a convergent net converges to the same limits. # If a net does not converge to a point x then there is a subnet such that no further subnet converges to x. Equivalently, if x_ is a net such that every one of its subnets has a sub-subnet that converges to a point x, then x_ converges to x. # '/''Convergence of iterated limits''. If \left(x_a\right)_ \to x in X and for every a \in A, \left(x_a^i\right)_ is a net that converges to x_a in X, then there exists a diagonal net that converges to x (a ''diagonal net'' is a subnet of \left(x_a^i\right)_ where the domain of this net is ordered lexicographically first by A and then by I_a; explicitly, given (a, i), \left(a_2, i_2\right) \in \cup_ A \times I_a, declare that (a, i) \leq \left(a_2, i_2\right) holds if and only if both (1) a \leq a_2, and also (2) if a = a_2 then i \leq i_2). If X is a set, then given any collection of nets and points satisfying the above axioms, a closure operator on X is defined by sending any given set A to the set of all limits of all nets in A; the corresponding topology is the unique topology inducing the given convergences of nets to points. Given a subset A \subseteq X of a topological space X: * A is open in X if and only if every net converging to an element of A is eventually contained in A. * the closure of A in X is the set of all limits of all convergent nets in A. * A is closed in X if and only if there does not exist a net in A that converges to an element of the complement X \setminus A. A subset A \subseteq X is closed in X if and only if every limit point of every convergent net in A necessarily belongs to A. A function f : X \to Y between two topological spaces is continuous if and only if for every x \in X and every net x_ in X that converges to x in X, the net f\left(x_\right) converges to f(x) in Y.


Definition via convergence of filters

A topology can also be defined on a set by declaring which filters converge to which points. One has the following characterizations of standard objects in terms of
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component th ...
and
prefilter In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
s (also known as filterbases): * Given a second topological space Y, a function f : X \to Y is continuous if and only if it preserves convergence of prefilters. * A subset A of X is open if and only if every filter converging to an element of A contains A. * A subset A of X is closed if and only if there does not exist a prefilter on A which converges to a point in the complement X \setminus A. * Given a subset A of X, the closure consists of all points x for which there is a prefilter on A converging to x. * A subset A of X is a neighborhood of x if and only if it is an element of every filter converging to x.


See also

* * * * *


References

Notes Textbooks * * * * * General topology Categories in category theory