Monotonically D-spaces
by
Strashimir G. Popvassilev
University of Louisiana at Lafayette
Coauthors: John E. (Ted) Porter

An open neighborhood assignment (ONA) for a topological space X is a function U with domain X such that U(x) is a neighborhood of x for every x in X. If for every ONA there is a closed discrete subset of X such that the neighborhoods assigned to points in this closed discrete set cover X then X is called a D-space. We call X monotonically D if there is an operator that assigns a closed discrete set (generating a cover) to each ONA, such that larger closed discrete sets correspond to finer ONA's. This is a rather restrictive property: There is a countable space which is not monotonically D, the Cantor set and uncountable subsets of the Sorgenfrey line are not monotonically D, although the Michael line is. We also defined cofinally monotonically D-spaces (as a less restrictive version, when there is a monotone operator defined for cofinally many ONA's) but these are the same as D-spaces.

Date received: February 18, 2005