Characterizations of The Category of Topological Spaces - Definition Via Neighbourhoods

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.