Definition:Basis (Topology)/Synthetic Basis/Definition 1
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Definition
Let $S$ be a set.
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 $\BB$.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.2$: Bases: Definition $3.2.2$