Question

If Y is a connected subspace of a space X ,then prove that Y closure is connected

Answer 1

Suppose that $\bar Y$ is covered by two disjoint open sets $U, V$ such that $\bar Y \subseteq U\cup V$. As $Y\subseteq \bar Y$, we also have $Y \subseteq U \cup V$. As $Y$ is connected, one of these two open sets does not intersect $Y$, i.e. $U\subseteq X\setminus Y$ or $V\subseteq X\setminus Y$. For convenience, let us suppose that $U\subseteq X\setminus Y$. But as $U$ is open, $U$ is contained in the interior of $X\setminus Y$, which is $X\setminus \bar Y$ and therefore $\bar Y\cap U = \empty$. As it is impossible to cover $\bar Y$ by two disjoint open sets such that their intersection with $\bar Y$ is not empty, we conclude that $\bar Y$ is connected.