Axiomatic foundations of topological spaces

A topological space is a set X together with a collection S of subsets of X satisfying: • The empty set and X are in S. • The union of any collection of sets in S is also in S. • The intersection 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 only to 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^{-1}(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 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, the collection T of closed sets satisfies the following properties: • The empty set and X are elements of T • The intersection of any collection of sets in T is also in T. • The union 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 \{U : X \setminus U \in T\} 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^{-1}(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 of X. One has the following Kuratowski closure axioms: • A \subseteq \operatorname{cl}(A) • \operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) • \operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) • \operatorname{cl}(\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{cl}(A)) is a subset of \operatorname{cl}(f(A)). • A subset A of X is open if and only if \operatorname{cl}(X \setminus A) = X \setminus A. • A subset C of X is closed if and only if \operatorname{cl}(C) = C. • Given a subset A of X, the interior is the complement of \operatorname{cl}(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 of X. It satisfies the following conditions: • \operatorname{int}(A) \subseteq A • \operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A) • \operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B) • \operatorname{int}(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^{-1}(\operatorname{int}(B)) is a subset of \operatorname{int}(f^{-1}(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 exterior operators==

Given a topological space X, the exterior can be considered as a map \wp(X) \to \wp(X), where \wp(X) denotes the power set of X. It satisfies the following conditions: • \operatorname{ext}(\varnothing) = X • A \cap \operatorname{ext}(A) = \varnothing • \operatorname{ext}(X \setminus \operatorname{ext}(A)) = \operatorname{ext}(A) • \operatorname{ext}(A \cup B) = \operatorname{ext}(A) \cap \operatorname{ext}(B) If X is a set equipped with a mapping satisfying the above properties, then we can define the interior operator and vice versa. More precisely, if we define \operatorname{int}_\operatorname{ext} : \wp(X) \to \wp(X), \operatorname{int}_\operatorname{ext} = \operatorname{ext}(X \setminus A), \operatorname{int}_\operatorname{ext} satisfies the interior operator axioms, and hence defines a topology. Conversely, if we define \operatorname{ext}_\operatorname{int} : \wp(X) \to \wp(X), \operatorname{ext}_\operatorname{int} = \operatorname{int}(X \setminus A), \operatorname{ext}_\operatorname{int} satisfies the above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space X, all definitions can be phrased in terms of the exterior operator, for instance: • The closure of a set is the complement of its exterior, \operatorname{cl}_\operatorname{ext} : \wp(X) \to \wp(X), \operatorname{cl}_\operatorname{ext}(A) = X \setminus \operatorname{ext}(A). • 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(X \setminus \operatorname{ext}(A)) is a subset of X \setminus \operatorname{ext}(f(A)). Equivalently, f is continuous if and only if for every subset B of Y, one has that the set f^{-1}(\operatorname{ext}(X \setminus B)) is a subset of \operatorname{ext}(X \setminus f^{-1}(B)). • A set is open if and only if it equals the exterior of its complement. • A set is closed if and only if it equals the complement of its exterior. ==Definition via boundary operators==

Given a topological space X, the boundary can be considered as a map \wp(X) \to \wp(X), where \wp(X) denotes the power set of X. It satisfies the following conditions: • \partial A = \partial(X \setminus A) • \partial(\partial(A)) \subseteq \partial(A) • \partial(A \cup B) \subseteq \partial(A) \cup \partial(B) • A \subseteq B \Rightarrow \partial A \subseteq B \cup \partial B • \partial(\varnothing) = \varnothing If X is a set equipped with a mapping satisfying the above properties, then we can define closure operator and vice versa. More precisely, if we define \operatorname{cl}_\partial : \wp(X) \to \wp(X), \operatorname{cl}_\partial(A) = A \cup \partial(A), \operatorname{cl}_\partial satisfies closure axioms, and hence boundary operation defines a topology. Conversely, if we define \partial_\operatorname{cl} : \wp(X) \to \wp(X), \partial_\operatorname{cl} = \operatorname{cl}(A) \cap \operatorname{cl}(X \setminus A), \partial_\operatorname{cl} satisfies above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space X, all definitions can be phrased in terms of the boundary operator, for instance: • \operatorname{int}_\partial : \wp(X) \to \wp(X), \operatorname{int}_\partial(A) = A \setminus \partial(A) • A set is open if and only if \partial(A) \cap A = \varnothing. • A set is closed if and only if \partial(A) \subseteq A. == Definition via derived sets ==

The derived set of a subset S of a topological space X is the set of all points x \in X that are limit points of S, that is, points x such that every neighbourhood of x contains a point of S other than x itself. The derived set of S \subseteq X, denoted S^*, satisfies the following conditions: the derived set uniquely defines a topology. It follows that on a topological space X, all definitions can be phrased in terms of derived sets, for instance: • \operatorname{cl}(A) = A \cup A^*. • Given topological spaces X and 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(A^*) is a subset of {f(A)}^* \cup f(A). It is worth while noting that it is also possible to define a topological space via isolated sets - the set of all isolated points of a set. ==Definition via neighbourhoods==

Recall that this article follows the convention that a neighborhood 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^{-1}(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 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_{\bull} is a net such that every one of its subnets has a sub-subnet that converges to a point x, then x_{\bull} converges to x. • ''/Convergence of iterated limits''. If \left(x_a\right)_{a \in A} \to x in X and for every index a \in A, \left(x_a^i\right)_{i \in I_a} is a net that converges to x_a in X, then there exists a diagonal (sub)net of \left(x_a^i\right)_{a \in A, i \in I_a} that converges to x. ::* A refers to any subnet of \left(x_a^i\right)_{a \in A, i \in I_a}. ::* The : notation \left(x_a^i\right)_{a \in A, i \in I_a} denotes the net defined by (a, i) \mapsto x_a^i whose domain is the set {\textstyle\bigcup\limits_{a \in A}} A \times I_a ordered lexicographically first by A and then by I_a; explicitly, given any two pairs (a_1, i_1), \left(a_2, i_2\right) \in {\textstyle\bigcup\limits_{a \in A}} A \times I_a, declare that (a_1, i_1) \leq \left(a_2, i_2\right) holds if and only if both (1) a_1 \leq a_2, and also (2) if a_1 = a_2 then i_1 \leq i_2. If X is a set, then given a notion of net convergence (telling what nets converge to what points) satisfying the above four axioms, a closure operator on X is defined by sending any given set A to the set of all limits of all nets valued 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 valued 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_{\bull} in X that converges to x in X, the net f\left(x_{\bull}\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 and prefilters (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==