A set is open if and only if every point in the set has an open neighbourhood contained within the set

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Flesh out and clean up, then demote to grade B
See page 25 in Lee's top manifolds, there's a bunch of equivalent conditions. However I do think there's a more general neighbourhood one, if I could just be bothered to prove, "a set is open if and only if it is neighbourhood to all of its points" or something

Statement

Let $(X,\mathcal{ J })$ be a topological space, then:

• A set, $A\in\mathcal{P}(X)$ is open if and only if every point of $A$ has an open neighbourhood contained in $A$

Proof

Grade: B

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