It is very much not compact. In fact, It is not Lindelöf, a weaker form of compactness: https://en.wikipedia.org/wiki/Lindel%C3%B6f_space https://en.wikipedia.

It is very much not compact. In fact, It is not Lindelöf, a weaker form of compactness: https://en.wikipedia.org/wiki/Lindel%C3%B6f_space https://en.wikipedia.org/wiki/Lindel%C3%B6f_space It is also not separable, i.e. no countable everywhere dense subset. I've added this to the Wikipedia page.