Closure

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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

  1. Munkres, James R. Topology, 2015.