Definition:Basis (Topology)/Analytic Basis

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Definition 1

Let $\left({S, \tau}\right)$ 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 $\mathcal B$.

Definition 2

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

Let $\mathcal B \subseteq \tau$.

Then $\mathcal B$ is an analytic basis for $\tau$ if and only if:

$\forall U \in \tau: \forall x \in U: \exists V \in \mathcal B: x \in V \subseteq U$

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

  • Results about bases can be found here.

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.