Let $X$ be a topological space and $A \subseteq X$ be a subset. A limit point of $A$ is a point $x \in X$ such that any open neighbourhood $U$ of $x$ intersects $A$. Show that $A$ is closed if and only if it contains all its limit points. Explain what is meant by the interior Int $(A)$ and the closure $\bar{A}$ of $A$. Show that if $A$ is connected, then $\bar{A}$ is connected.