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A Set is Closed iff it contains all Net Limits

Last updated Nov 1, 2022

# Statement

Suppose $(X, \tau)$ is a Topological Space. Then $K \subset X$ is Closed If and Only If for all Nets ${x_{\alpha}}{\alpha \in A} \subset K$ s.t. $x{\alpha}\to x \in X$, we have that $x \in K$.

# Proof

This follows quickly from A set is Closed iff it contains all its Limit Points and A point is a Limit Point iff it is a Net Limit. $\blacksquare$