Strong colorings yield 𝜅-bounded spaces with discretely untouchable points

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. Szentmiklóssy and the first author, we show that this statement fails for countably compact regular spaces, and even forω\omega-bounded regular spaces. In fact, there areκ\kappa-bounded counterexamples for every infinite cardinalκ\kappa. The proof makes essential use of the so-calledstrong coloringsthat were invented by the second author.