Definition:Basis (Topology)/Synthetic Basis
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Definition
Let $S$ be a set.
Definition 1
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 $\BB$.
Definition 2
A synthetic basis on $S$ is a subset $\BB \subseteq \powerset S$ of the power set of $S$ such that:
- $\BB$ is a cover for $S$
- $\forall U, V \in \BB: \forall x \in U \cap V: \exists W \in \BB: x \in W \subseteq U \cap V$
Also see
Linguistic Note
The plural of basis is bases.
This is properly pronounced bay-seez, not bay-siz.