TITLE: Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
QUESTION [5 upvotes]: Can anyone give me a proof of the statement that if $X$ and its closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected?
Thank you.

REPLY [6 votes]: In case you are using the "you can't write it as a union of disjoint open sets" definition, like I grew up with:
Let $O_1$ and $O_2$ be open sets of the universal set whose union contains $Y$ and such that $(O_1\cap Y)\cap(O_2\cap Y)=\emptyset$. 
Since their union also contains $X$, which is connected, one of them, say $O_1$ is such that $X\cap O_1=\emptyset$ (and $X\subseteq O_2$). 
But then by definition of the closure, $\overline{X}\cap O_1=\emptyset$, and hence $Y\cap O_1=\emptyset$. So, $Y\subset O_2$.