Category:Definitions/Sub-Bases

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This category contains definitions related to Sub-Bases in the context of Topology.
Related results can be found in Category:Sub-Bases.


Analytic Sub-Basis

Let $\left({S, \tau}\right)$ be a topological space.

Let $\mathcal S \subseteq \tau$.


Define:

$\displaystyle \mathcal B = \left\{{\bigcap \mathcal F: \mathcal F \subseteq \mathcal S, \, \mathcal F \text{ is finite}}\right\}$

That is, $\mathcal B$ is the set of all finite intersections of sets in $\mathcal S$.

Note that $\mathcal F$ is allowed to be empty in the above definition.


Define:

$\displaystyle \tau' = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$

Suppose that $\tau \subseteq \tau'$.


That is, suppose that every $U \in \tau$ is a union of finite intersections of sets in $\mathcal S$, together with $\varnothing$ and $S$ itself.


Then $\mathcal S$ is called an analytic sub-basis for $\tau$.


Synthetic Sub-Basis

Let $X$ be a set.


A synthetic sub-basis on $X$ is any subset $\mathcal S \subseteq \mathcal P \left({X}\right)$ of the power set of $X$.

Subcategories

This category has only the following subcategory.