Definition:Basis (Topology)/Analytic Basis

Definition
Let $\left({X, \tau}\right)$ be a topological space.

An analytic basis for $\tau$ is a subset $\mathcal B \subseteq \tau$ such that:
 * $\displaystyle \forall U \in \tau: \exists \mathcal A \subseteq \mathcal B: U = \bigcup \mathcal A$

That is, such that for all $U \in \tau$, $U$ is a union of sets from $\mathcal B$.

Also known as
Some sources do not distinguish between an analytic basis and a synthetic basis, and instead use this definition and call it a basis.

Also see

 * Synthetic Basis


 * Synthetic Basis and Analytic Basis are Compatible which, among other things, shows that $\tau$ is uniquely determined by $\mathcal B$.
 * Equivalent Definitions of Analytic Basis