Definition:Basis (Topology)/Synthetic Basis

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

• Equivalence of Definitions of Synthetic Basis
• Definition:Topology Generated by Synthetic Basis

• Definition:Analytic Basis
• Synthetic Basis and Analytic Basis are Compatible

• 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.