# 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:

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

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 $\mathcal B \subseteq \mathcal P \left({S}\right)$ 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 30 pages are in this category, out of 30 total.

### B

### E

- Equivalence of Definitions of Analytic Basis
- Equivalence of Definitions of Synthetic Basis
- Equivalence of Definitions of Topology Generated by Synthetic Basis
- Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2
- Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3
- Equivalent Definitions of Synthetic Basis