# Category:Topological Bases

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This category contains results about **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

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 $\BB$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Topological Bases"

The following 35 pages are in this category, out of 35 total.

### B

### C

### E

### O

### P

### S

- Space is Compact iff exists Basis such that Every Cover has Finite Subcover
- Sub-Basis for Initial Topology in terms of Sub-Bases of Target Spaces
- Sub-Basis for Real Number Line
- Sub-Basis for Topological Subspace
- Synthetic Basis and Analytic Basis are Compatible
- Synthetic Basis formed from Synthetic Sub-Basis
- Synthetic Sub-Basis and Analytic Sub-Basis are Compatible