NOETHERIAN BASES IN SPACES WITH AN ORTHOBASE

Abstract

A collection C of subsets of a set X is said to be Noetherian if C does not contain a strictly increasing infinite chain. A space X is Noetherianly refinable, or more briefly, N - refinable, if each open covering of X has a Noetherian open refinement which covers X. A base B for a topological space X is an ortho - base if for each B' subset-or-equal-to B, either intersectB' is an open set of X or intersectB' = {p} and B' is a local base at p. We show that every T1 - space with an ortho-base has a Noetherian base if and only if each of its open subspaces is N - refinable.