Proposition: A Necessary Condition of a Neighborhood to be Open

If a set $B$ is a neighborhood of all of its points, then it is open. Conversely, if $B$ is closed, then there is a point $A\in B$ such that $B$ is not a neighborhood of it.

Proofs: 1

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  3. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition