Characterizations of the category 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.

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:

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

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.

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

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