In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... The category Top has topological spaces as objects and continuous maps as morphisms. ...

Definitions

Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be concretely isomorphic. This means that for every pair of categories defined below, there is an isomorphism of categories, for which corresponding objects have the same underlying set and corresponding morphisms are identical as set functions. In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ... In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i. ... A forgetful functor is a type of functor in mathematics. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces Top. This would involve the following: The category Top has topological spaces as objects and continuous maps as morphisms. ...

1. Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set.
2. Checking that a set function is "continuous" (i.e., a morphism) in the given category if and only if it is continuous (a morphism) in Top.

↔ ⇔ ≡ logical symbols representing iff. ...

Definition via open sets

Top The category Top has topological spaces as objects and continuous maps as morphisms. ...

Objects: all topological spaces, i.e., all pairs (X,T) of set X together with a collection T of subsets of X satisfying: Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

1. The empty set and X are in T.
2. The union of any collection of sets in T is also in T.
3. The intersection of any pair of sets in T is also in T.

The sets in T are the open sets.

Morphisms: all ordinary continuous functions, i.e. all functions such that the inverse image of every open set is open. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

Comments: This is the ordinary category of topological spaces. The category Top has topological spaces as objects and continuous maps as morphisms. ...

Definition via closed sets

Top-Clos

Objects: all pairs (X,T) of set X together with a collection T of subsets of X satisfying: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

1. The empty set and X are in T.
2. The intersection of any collection of sets in T is also in T.
3. The union of any pair of sets in T is also in T.

The sets in T are the closed sets.

Morphisms: all functions such that the inverse image of every closed set is closed. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

Comments: This is the category that results by replacing each lattice of open sets in a topological space by its order-theoretic dual of closed sets, the lattice of complements of open sets. The relation between the two definitions is given by De Morgan's laws. The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ... In logic, De Morgans laws (or De Morgans theorem) are rules in formal logic relating pairs of dual logical operators in a systematic manner expressed in terms of negation. ...

Definition via closure operators

Clos

Objects: all closure spaces, i.e., all pairs (X,cl) of set X together with a function cl : P(X) → P(X) satisfying the Kuratowski closure axioms: In topology and related branches of mathematics, the Kuratowski closure axioms is a set of axioms that allows one to define a topology on a set. ...

1. (Extensivity)
2. (Idempotence)
3. (Preservation of binary unions)
4. (Preservation of nullary unions)

Morphisms: all closure-preserving functions, i.e., all functions f between two closure spaces In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

such that for all subsets A of X

Comments: The Kuratowski closure axioms abstract the properties of the closure operator on a topological space, which assigns to each subset its topological closure. This topological closure operator has been generalized in category theory; see Categorical Closure Operators by G. Castellini in "Categorical Perspectives", referenced below. In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Definition via interior operators

Int

Objects: all interior spaces, i.e., all pairs (X,int) of set X together with a function int : P(X) → P(X) satisfying the following dualisation of the Kuratowski closure axioms: In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ... In topology and related branches of mathematics, the Kuratowski closure axioms is a set of axioms that allows one to define a topology on a set. ...

1. 
2. (Idempotence)
3. (Preservation of binary intersections)
4. (Preservation of nullary intersections)

Morphisms: all interior-preserving functions, i.e., all functions f between two interior spaces In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

such that for all subsets A of X'

Comments: This category is obtained in much the same way that Top-Clos is obtained from Top. The topological interior operator satisfies the Kuratowski closure axioms when complements are taken, the subset relation is changed to superset, and when unions and intersections are interchanged.

Definition via convergent filters

PrTop

Objects: all pretopological spaces, i.e., all pairs (X,conv) of set X together with a subset conv of X × F(X). Here, F(X) denotes the set of all filters on X, and conv tell us which filters converge to which points. The relation must satisfy: In mathematics, a pretopological space (X, cl ) in general topology is a set X with a function cl : P(X) → P(X) , where P(X) denotes the power set of X. This function has to be extensive and finitely additive; that is, it must satisfy the following conditions for all... In mathematics, a filter is a special subset of a partially ordered set. ...

1. For every x in X, the fixed ultrafilter at x converges to x.
2. If a filter converges to x, then so does every finer filter.
3. If each member of a collection of filters converges to x, then so does the intersection of that collection.

(disputed — see talk page) In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...

Morphisms: all convergence-preserving functions, i.e., all functions f : (X, conv) → (Y, conv') such that whenever F converges to x, then the filter generated by f(F) converges to f(x).

Comments: This definition shows that convergence of filters can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever F is a filter on A, then A contains all points to which F converges.

Definition via convergent nets

Net

Objects: all net spaces, i.e., all pairs (X,conv) of set X together with a subset conv of X × N(X). Here, N(X) denotes the set of all nets on X, and conv tell us which nets converge to which points. The relation must satisfy: In mathematics the term net has at least two meanings. ...

1. For every x in X, every constant net at x converges to x.
2. If a net converges to x, then so does every subnet of that net.
3. (3rd condition here?)

Morphisms: all convergence-preserving functions, i.e., all functions f : (X, conv) → (Y, conv') such that whenever (x_α) converges to x, then (f(x_α)) converges to f(x).

Comments: This definition shows that convergence of nets can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever (x_α) is a net on A, then A contains all points to which (x_α) converges.

Definition via neighbourhoods

Neigh

Objects: all neighbourhood spaces, i.e., all pairs (X,N) of set X together with a function N : X → F(X), where F(X) denotes the set of all filters on X, satisfying for every x in X: In mathematics, a filter is a special subset of a partially ordered set. ...

1. If U is in N(x), then x is in U.
2. If U is in N(x), then there exists V in N(x) such that V is contained in U and V is in N(y) for all y in V.

Morphisms: all neighbourhood-preserving functions, i.e., all functions f : (X, N) → (Y, N') such that whenever U is in N'(f(x)), then f⁻¹(U) is in N(x).

Comments: This definition axiomatizes the notion of neighbourhood. We say that U is a neighbourhood of x if U is in N(x). The open sets can be recovered by declaring a set to be open if it is a neighbourhood of each of its points; the final axiom then states that every neighbourhood contains an open set. This is a glossary of some terms used in the branch of mathematics known as topology. ...

Definition via nearness relations

Near

Objects: all nearness structures, i.e., all pairs (X,δ) of set X together with a subset δ of X × P(X) satisfying:

1. (no point is near the empty set)
2. (every set is near each of its elements)
3. (a point is near a union of two sets iff it is near at least one of the sets)
4. (if a point is near a set, and each point of that set is near another set, then the original point is near the second set as well)

Morphisms: all nearness-preserving functions, i.e., all functions f : (X, δ) → (Y, γ) such that

Comments: This definition captures the intuitive idea of a point being "close" to a set. In terms of open sets, a point is near a set if it is in the closure of the set.

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
• Joshi, K. D., Intro to General Topology, New Age International, 1983, ISBN 0-85226-444-5
• Koslowsk and Melton, eds., Categorical Perspectives, Birkhauser, 2001, ISBN 0-8176-4186-6