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.
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:
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:
"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:
"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:
"Morphisms": all "closure-preserving functions", i.e., all functions "f" between two closure spaces
such that for all subsets formula_6 of formula_7
"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 a binary relation between points and subsets.
Similar to the Kuratowski closure axioms approach, one can also define a topological space as a set formula_7 together with a relation formula_10 between points and subsets (formula_11 intuitively expresses that using the elements of formula_6 one can get arbitrarily close to formula_13) satisfying
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:
"Morphisms": all "interior-preserving functions", i.e., all functions "f" between two interior spaces
such that for all subsets formula_6 of formula_31
"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":
"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, the category of topological spaces can also 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.