Definition:Basis (Topology)

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Definition

Analytic Basis

Let $\left({S, \tau}\right)$ be a topological space.


An analytic basis for $\tau$ is a subset $\mathcal B \subseteq \tau$ such that:

$\displaystyle \forall U \in \tau: \exists \mathcal A \subseteq \mathcal B: U = \bigcup \mathcal A$


That is, such that for all $U \in \tau$, $U$ is a union of sets from $\mathcal B$.


Synthetic Basis

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 $S$
$\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$


Countable Basis

A countable (analytic) basis for a topology $\tau$ is an analytic basis for $\tau$ which is a countable set.


Also known as

A basis can also be seen referred to as a base.


Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.


Also see

  • Results about bases can be found here.