# Closure

A subset $A$ of a topological space $X$ is said to be closed if the set $X - A$ is open.

## Proof

**THEOREM:** A subset of a topological space is closed if and only if it contains all its limit points.

*Proof:*

The set $A$ is closed if and only if $A = \bar{A}$, and the latter holds if and only if $A' \subset A$. █

## Implications

The concept of closure is significant in Topology because topological spaces in which one-point sets are not closed impose limits which make them less interesting than those in which one-point sets are closed. The distinction between these two categories of topological spaces is known as the Hausdorff condition.^{[1]}

## References

- ↑ Munkres, James R. Topology, 2015.