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Basis

Definition

Given a topological space $(X, \mathcal{T})$, a basis is a set $\mathcal{B} \subset \mathcal{T}$ such that every set in $\mathcal{T}$ can be expressed as a union of sets in $\mathcal{B}$.

Given sets $X$ and $\mathcal{B} \subset \mathcal{P}(X)$, the topology with basis $\mathcal{B}$ is the set of unions of sets in $\mathcal{B}$, provided that this is a valid topology on $X$.

Criterion

(TODO tidy this up - this is the criterion for the "top'gy with basis ~B~ existing) Given sets $X$ and $\mathcal{B} \subset \mathcal{P}(X)$, if $X$, and any intersection $B_1 \cap B_2: B_1, B_2 \in \mathcal{B}$ can be expressed as a union of sets in $\mathcal{B}$, then the set of unions of members of $\mathcal{B}$ is a topology on $X$

all these pages adapted with probably insufficient credit from my university's lecture notes