Definition 10

The closure of a subset $A$ of a topological space $X$ is the set of all points $x \in X$ such that every neighbourhood of $x$ meets $A$, and is denoted by $\overline{A}$.

This definition can be reformulated by saying that a point $x$ lies in the closure of a set $A$ if there are points of $A$ as near $x$ as we please to.

Every point which is not in the closure of $A$ is exterior to $A$, and conversely; thus we have the formulae (which are duals of each other)

Hence, to any proposition on interiors of sets, there corresponds by duality a proposition on closures, and vice versa. In particular, the closure of a set $A$ is the smallest closed set which contains $A$; in other words, if $B$ is a closed set such that $A \subset B$, then $\overline{A} \subset B$. If $A$ and $B$ are two subsets of $X$ such that $A \subset B$, then $\overline{A} \subset \overline{B}$.

A set is closed if and only if it coincides with its closure.

The dual of formula (1) is