TITLE: Is real line disconnected in discrete topology?
QUESTION [6 upvotes]: My professor told me that the real line is connected in any topology. But i am thinking that if for example we consider the discrete topology in $\mathbf R$ i.e every subset is open then for any $x$ in $\mathbf R$ $(-\infty,x) \cup [x,\infty)$ would be a partition of open sets such that their union is $\mathbf R$ and thus $\mathbf R$ is disconnected in discrete topology. Is what i am saying wrong? why?

REPLY [4 votes]: You are correct. The only discrete spaces that are connected are those with at most one point, since otherwise, they have a non-empty proper subset that is both open and closed.