"Answers for functionally countable compacta"Nyikos, PeterA functionally countable space is one on which every continuous real-valued function has countable range. In this year's STDC, and in a recent paper, Vladimir Tkachuk posed 14 questions having to do with spaces $X$ such that $X^2 \setminus \Delta_X$ is functionally countable. This talk gives a simple example of a space that answers the following question: Question 4.10. Is there a ZFC example of a non-metrizable compact space $X$ such that $(X\times X) \setminus \Delta_X$ is functionally countable? The following example is chosen for easy visualizability. $X^2$ has the open unit square as the underlying set, while $(X\cup\{\infty\})^2$ can be thought of as $(0, 1]^2$ with a very different topology. Example. Let $X$ be the open unit interval $(0, 1)$, with the following topology. Let $Q = \mathbb Q \cap (0,1)$ be a dense set of isolated points, and let each $p \in X - Q$ have a base of neighborhoods consisting of $p$ together with the tails of an ascending sequence of points $q_n$ of $Q$. Theorem. Let $X+1$ denote the one-point compactification of $X$. Then $(X+1)^2 \setminus \Delta_{X+1}$ is functionally countable. There is also a ZFC example that solves Tkachuk's Questions 4.1 through 4.8 that will be described. |
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