Definition:Basis (Topology)/Analytic Basis/Definition 1
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Definition
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 $\BB$.
Also see
- Results about analytic bases can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.2$: Bases: Definition $3.2.1$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction