Definition:Basis (Topology)

Definition
Otherwise known as a base.

Analytic Basis
Let $T = \left({A, \vartheta}\right)$ be a topological space.

Let $\mathcal B \subseteq \vartheta$ such that for all $U \in \vartheta$, $U$ is a union of sets from $\mathcal B$.

Then $\mathcal B$ is an (analytic) basis for $\vartheta$.

Synthetic Basis
Let $A$ be a set.

Let $\mathcal B \subseteq \mathcal P \left({A}\right)$, where $\mathcal P \left({A}\right)$ is the power set of $A$, such that:


 * B1: $A$ is a union of sets from $\mathcal B$;
 * B2: If $B_1, B_2 \in \mathcal B$, then $B_1 \cap B_2$ is a union of sets from $\mathcal B$.

Then $\mathcal B$ is a (synthetic) basis for $A$.

Countable Basis
A countable basis for a topology $\vartheta$ is a basis for $\vartheta$ which consists of a countable number of sets.

Linguistic Note
The pronunciation of bases in this context is bay-seez, not bay-siz.

Also see

 * Sub-basis


 * Filter Basis