Let $X$ be a topological space. Is the following claim true: If $X$ is uncountable, Hausdorff, and every countable subset of $X$ is closed, then $X$ is discrete.
This claim is false without the Hausdorff assumption (If every countable subset of a space is closed, is the topology discrete?), but the example given in the link does not work since it is not Hausdorff. I think my claim is not true (but not sure) and I cannot found a counterexample. Any hints?
Best Answer
Let $X=\Bbb R$; every point of $\Bbb R\setminus\{0\}$ is isolated, and open nbhds of $0$ are co-countable. Thus, the topology on $X$ is
$$\wp(\Bbb R\setminus\{0\})\cup\{\Bbb R\setminus C:C\subseteq\Bbb R\setminus\{0\}\text{ is countable}\}\,.$$
Every countable set is closed, and $X$ is Hausdorff, but $0$ is a limit point of $X\setminus\{0\}$.