The basis criterion (topology)/Statement

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Statement

Let $(X,\mathcal{ J })$ be a topological space and let $\mathcal{B}\in\mathcal{P}(\mathcal{P}(X))$ be a topological basis for $(X,\mathcal{ J })$. Then:

• $\forall U\in\mathcal{P}(X)\big[U\in\mathcal{J}\iff\underbrace{\forall p\in U\exists B\in\mathcal{B}[p\in B\subseteq U]}_{\text{basis criterion} }\big]$

If a subset $U$ of $X$ satisfies $\forall p\in U\exists B\in\mathcal{B}[p\in B\subseteq U]$ we say it satisfies the basis criterion with respect to $\mathcal{B}$