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 $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:
\((B1)\) | $:$ | $\mathcal B$ is a cover for $S$ | ||||||
\((B2)\) | $:$ | \(\displaystyle \forall U, V \in \mathcal B:\) | $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$ |
That is, the intersection of any pair of elements of $\mathcal B$ is a union of sets of $\mathcal B$.
Definition 2
A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:
- $\mathcal B$ is a cover for $S$
- $\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$
Also see
- Equivalent Definitions of Synthetic Basis
- Results about bases can be found here.
Linguistic Note
The plural of basis is bases.
This is properly pronounced bay-seez, not bay-siz.