# The basis criterion (topology)

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## Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]\mathcal{B}\in\mathcal{P}(\mathcal{P}(X))[/ilmath] be a topological basis for [ilmath](X,\mathcal{ J })[/ilmath]. Then[1]:

• [ilmath]\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][/ilmath][Note 1]

If a subset [ilmath]U[/ilmath] of [ilmath]X[/ilmath] satisfies[Note 2] [ilmath]\forall p\in U\exists B\in\mathcal{B}[p\in B\subseteq U][/ilmath] we say it satisfies the basis criterion with respect to [ilmath]\mathcal{B} [/ilmath][1]

## Proof

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Can't be bothered to do now, it's a minor restatement of the second property of a basis.

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## Notes

1. Note that when we write [ilmath]p\in B\subseteq U[/ilmath] we actually mean [ilmath]p\in B\wedge B\subseteq U[/ilmath]. This is a very slight abuse of notation and the meaning of what is written should be obvious to all without this note
2. This means "if a [ilmath]U\in\mathcal{P}(X)[/ilmath] satisfies...