Category:Definitions/Sub-Bases
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:
- $\ds \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$
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:
- $\ds \tau' = \set {\bigcup \AA: \AA \subseteq \BB}$
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 $S$ be a set.
A synthetic sub-basis on $S$ is any subset $\SS \subseteq \powerset S$ of the power set of $S$.
Subcategories
This category has only the following subcategory.
Pages in category "Definitions/Sub-Bases"
The following 7 pages are in this category, out of 7 total.