Give an example of a separable topological space where uncountable set does not necessarily have limit point.

Question

In metric space there is a well-known result that $X$ is a separable metric space $\implies$Every uncountable set in $X$ has a limit point.My question is does the result hold if $(X,\tau)$ is arbitrary topological space?I think that it may not hold for Non-Hausdorff spaces where we do not have disjoint open sets around two points.

Answer 1

The Niemitzky plane (a.k.a. the Moore plane or the Moore-Niemitzky plane) is Hausdorff (in fact it is a Tychonoff space) which is separable because $\Bbb Q\times \Bbb Q^+$ is dense. And $E=\Bbb R\times \{0\}$ is an uncountable closed discrete subspace so $E$ has no limit point.

Answer 2

See the example in the following link and take the uncountable set to the the real line. This set has no limit point since basic open sets containig a real number have only one point from the line: https://www.mathcounterexamples.net/a-separable-space-that-is-not-second-countable/