Definition:Basis (Topology)

Definition

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 $\BB$.

A synthetic basis on $S$ is a subset $\BB \subseteq \powerset S$ of the power set of $S$ such that:

$(\text B 1)$	$:$							$\BB$ is a cover for $S$
$(\text B 2)$	$:$			$\ds \forall U, V \in \BB:$				$\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$

That is, the intersection of any pair of elements of $\BB$ is a union of sets of $\BB$.

As a way of specifying a particular topology on a set, we can say:

Let $\tau$ be the topology which has the sets ... as a basis.

In such a case, those sets should all satisfy the axioms for a synthetic basis $(\text B 1)$ and $(\text B 2)$.

A basis can also be seen referred to as a base.

The plural of basis is bases.

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