Category:Topological Bases

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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

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 $\BB$.


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