Neighborhood spaces

D C Kent,Won Keun Min

doi:10.1155/s0161171202202203

Abstract

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more general p‐stacks.

Highlights

In his original definition of topology, Hausdorff [7] assigned to each point x in a set X a system of neighborhoods subject to certain axioms
Of particular significance is the fact that pretopologies, supratopologies, and neighborhood structures on X are all uniquely determined by their associated interior operators, which, in each case, can be axiomatically described by generalizing in different ways the Kuratowski interior operator that characterizes a topology
To better understand the preceding statements, we recall that a topological space (X, τ) has an interior operator I which satisfies the following Kuratowski interior axioms: (i) I(X) = X; (ii) I(A) ⊆ A, for all A ⊆ X; (iii) I(A ∩ B) = I(A) ∩ I(B), for all A, B ∈ 2X ; (iv) I(I(A)) = I(A), for all A ⊆ X