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 $\struct {S, \tau}$ be a topological space.

Let $\SS \subseteq \tau$.


Define:

$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$

That is, $\BB$ is the set of all finite intersections of sets in $\SS$.

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


Define:

$\displaystyle \tau' = \set {\bigcup \AA: \AA \subseteq \BB}$

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


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


Then $\SS$ 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 $X$.