Category:Definitions/Topological Bases
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This category contains definitions related to bases in the context of topology.
Related results can be found in Category: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 $S$ | |||||||
\((\text B 2)\) | $:$ | \(\ds \forall U, V \in \BB:\) | $\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$ |
That is, the intersection of any pair of elements of $\BB$ is a union of sets of $\BB$.
Pages in category "Definitions/Topological Bases"
The following 12 pages are in this category, out of 12 total.
B
- Definition:Basic Open Set
- Definition:Basis (Topology)
- Definition:Basis (Topology)/Analytic Basis
- Definition:Basis (Topology)/Analytic Basis/Definition 1
- Definition:Basis (Topology)/Analytic Basis/Definition 2
- Definition:Basis (Topology)/Specification of Topology
- Definition:Basis (Topology)/Synthetic Basis
- Definition:Basis (Topology)/Synthetic Basis/Definition 1
- Definition:Basis (Topology)/Synthetic Basis/Definition 2