While working on infinite dimensional Hilbert spaces, I came up with the following question: under what conditions is the existence of an open dense countable subset assured? If we assume that the space is separable, we are certain that a part of this question is answered. But what about an open dense countable subset? For instance (this example is not related to infinite dimensional spaces), $\mathbb{Q}$ is countable and dense in $\mathbb{R}$ (under the usual topology), but it is neither open nor closed. Is there any condition that assures the existence of at least one such open subset?
Infinite dimensional separable Hilbert spaces having an open countable dense subset
functional-analysisgeneral-topology
Best Answer
A non-empty open set in $\mathbb R$ (or any Hilbert space $\neq \{0\}$) is necessarily uncountable. Hence you can never have such a set.