Axiomatic foundations of topological spaces

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.

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.

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:

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.

Definition via open sets

Objects: all topological spaces, i.e., all pairs (X,T) of set X together with a collection T of subsets of X satisfying:

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.

Comments: This is the ordinary category of topological spaces.

Definition via closed sets

Objects: all pairs (X,T) of set X together with a collection T of subsets of X satisfying:

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.

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.

Definition via closure operators

Objects: all pairs (X,cl) of set X together with a closure operator cl : P(X) → P(X) satisfying the Kuratowski closure axioms:

1. $A\subseteq \operatorname {cl} (A)\!$ (Extensivity)
2. $\operatorname {cl} (\operatorname {cl} (A))=\operatorname {cl} (A)\!$ (Idempotence)
3. $\operatorname {cl} (A\cup B)=\operatorname {cl} (A)\cup \operatorname {cl} (B)\!$ (Preservation of binary unions)
4. $\operatorname {cl} (\varnothing )=\varnothing \!$ (Preservation of nullary unions)

Morphisms: all closure-preserving functions, i.e., all functions f between two closure spaces

$f:(X,\operatorname {cl} )\to (X',\operatorname {cl} ')$

such that for all subsets $A$ of $X$

$f(\operatorname {cl} (A))\subset \operatorname {cl} '(f(A))$

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.

Definition via interior operators

Objects: all pairs (X,int) of set X together with an interior operator int : P(X) → P(X) satisfying the following dualisation of the Kuratowski closure axioms:

1. $A\supseteq \operatorname {int} (A)\!$
2. $\operatorname {int} (\operatorname {int} (A))=\operatorname {int} (A)\!$ (Idempotence)
3. $\operatorname {int} (A\cap B)=\operatorname {int} (A)\cap \operatorname {int} (B)\!$ (Preservation of binary intersections)
4. $\operatorname {int} (X)=X\!$ (Preservation of nullary intersections)

Morphisms: all interior-preserving functions, i.e., all functions f between two interior spaces

$f:(X,\operatorname {int} )\to (X',\operatorname {int} ')$

such that for all subsets $A$ of $X'$

$f^{-1}(\operatorname {int} '(A))\subset \operatorname {int} (f^{-1}(A))$

Comments: The interior operator assigns to each subset its topological interior, in the same way the closure operator assigns to each subset its topological closure.

Definition via neighbourhoods

Objects: all pairs (X,N) of set X together with a neighbourhood function N : X → F(X), where F(X) denotes the set of all filters on X, satisfying for every x in X:

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 U 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 if V is in N(f(x)), then there exists U in N(x) such that f(U) is contained in V. This is equivalent to asking that whenever V is in N(f(x)), then f⁻¹(V) 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. These axioms (coupled with the Hausdorff condition) can be retraced to Felix Hausdorff's original definition of a topological space in Grundzüge der Mengenlehre.

Definition via convergence

The category of topological spaces can also be defined via a convergence relation between filters on X and points of x. This definition demonstrates 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.

Similarly, Top can be described via net convergence. As for filters, 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.

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., Introduction 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
• Wyler, Oswald (1996). Convergence axioms for topology. Ann. N. Y. Acad. Sci. 806, 465-475