Definition:Basis (Topology)/Analytic Basis

Definition

Definition 1

Let $\struct {S, \tau}$ be a topological space.

An analytic basis for $\tau$ is a subset $\BB \subseteq \tau$ such that:

$\ds \forall U \in \tau: \exists \AA \subseteq \BB: U = \bigcup \AA$

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

Definition 2

Let $\struct {S, \tau}$ be a topological space.

Let $\BB \subseteq \tau$.

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

$\forall U \in \tau: \forall x \in U: \exists V \in \BB: x \in V \subseteq U$

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

• Equivalence of Definitions of Analytic Basis
• Definition:Synthetic Basis

• Synthetic Basis and Analytic Basis are Compatible

• Results about bases can be found here.

Linguistic Note

The plural of basis is bases.

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