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

### Synthetic Basis

Let $S$ be a set.

#### Definition 1

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

 $(B1)$ $:$ $\mathcal B$ is a cover for $S$ $(B2)$ $:$ $\displaystyle \forall U, V \in \mathcal B:$ $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$

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

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Topological Bases"

The following 26 pages are in this category, out of 26 total.