Category:Topological Bases

This category contains results about Topological Bases in the context of Topology.
Definitions specific to this category can be found in Definitions/Topological Bases.

Analytic Basis

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 $\BB$.

Synthetic Basis

Let $S$ be a set.

Definition 1

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 $S$
\((\text B 2)\)	$:$		\(\displaystyle \forall U, V \in \BB:\)				$\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$

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

Definition 2

A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:

$\mathcal B$ is a cover for $S$

$\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$