Strong Colorings Yield Κ-bounded Spaces with Discretely Untouchable Points

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 for ω-bounded regular spaces. In fact, there are κ-bounded counterexamples for every infinite cardinal κ. The proof makes essential use of the so-called strong colorings that were invented by the second author.