Strong colorings yield kappa-bounded spaces with discretely untouchable points

TL;DR

This paper demonstrates that in certain countably compact spaces, non-isolated points can be discretely untouchable, using strong colorings to construct counterexamples for various infinite cardinals.

Contribution

It provides the first counterexamples to a known property in countably compact spaces, utilizing strong colorings to construct kappa-bounded spaces with discretely untouchable points.

Findings

Counterexamples for countably compact spaces

Discretely untouchable points in kappa-bounded spaces

Use of strong colorings in topological constructions

Abstract

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklossy and the first author, we show that this statement fails for countably compact regular spaces, and even for omega-bounded regular spaces. In fact, there are kappa-bounded counterexamples for every infinite cardinal kappa. The proof makes essential use of the so-called 'strong colorings' that were invented by the second author.