Spaces with a s-wHCP base.
by
Dennis Burke
Miami University
Coauthors: Sheldon Davis

A collection W of subsets of a space X is said to be weakly hereditarily closure-preserving (wHCP) if every choice x(W) ∈ W, W ∈ W yields a closed discrete set {x(W):W ∈ W}. Spaces with a s-wHCP base were introduced by Burke-Engelking-Lutzer in the last century and were more recently studied by C. Lui and L. Ludwig.

Proposition. If c(x, X) denotes the character of x in X and Y(x, X) denotes the pseudo-character then c(x, X) has countable cofinality for all non-isolated x and Y(x, X) < c(x, X) whenever c(x, X) is uncountable.

This helps explain the nature of non-metrizable examples. Previous known examples of spaces with a s-wHCP base were hereditarily paracompact. Answering a question by Lui-Ludwig we have the following.

Example. There is an example of a space X with a s-wHCP base which is not meta-Lindelöf. In fact, for any cardinal k there is such a space where every "canonical" open cover has point-order ≥ k.

Date received: February 24, 2008