Definition:Sub-Basis/Analytic Sub-Basis

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Definition

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 $\tau$.


Also known as

Some sources do not distinguish between an analytic sub-basis and a synthetic sub-basis, and instead use this definition and call it a sub-basis.

Some sources do not hyphenate sub-basis but instead render it as subbasis.

Some sources use the term sub-base (or subbase).


Also see

  • Results about Sub-Bases can be found here.


Linguistic Note

The plural of sub-basis is sub-bases.

This is properly pronounced sub-bay-seez, rather than sub-bay-siz, deriving as it does from the Greek plural form of nouns ending -is.


Sources