Topological properties that the real line does not have The following question is kind of strange, but I would like to know what topological properties $\mathbb{R}$ (with the standard metric topology) does not posses? I know this question sounds a bit broad. But I'm only looking for a partial list of topological properties that come to mind, which do not apply to the real line (standard metric topology ?). As a starter, clearly, $\mathbb{R}$ is not compact. Any ideas of other common (or even more exotic) properties? Thanks.

Here are a few, all easy to verify from their definitions. $\Bbb R$ is not:

* compact
* pseudocompact
* totally disconnected
* zero-dimensional
* extremally disconnected
* non-Archimedeanly metrizable (= ultrametrizable)
* scattered
* irreducible (= hyper-connected)



(If I think of more later, I’ll add them.)