Definition:Basis (Topology)

Definition

Otherwise known as a base.

Analytic Basis

Let $T = \left({A, \vartheta}\right)$ be a topological space.

Let $\mathcal B \subseteq \vartheta$ such that for all $U \in \vartheta$, $U$ is a union of sets from $\mathcal B$.

Then $\mathcal B$ is an (analytic) basis for $\vartheta$.

Synthetic Basis

Let $A$ be a set.

Let $\mathcal B \subseteq \mathcal P \left({A}\right)$, where $\mathcal P \left({A}\right)$ is the power set of $A$, such that:

B1: $A$ is a union of sets from $\mathcal B$;

B2: If $B_1, B_2 \in \mathcal B$, then $B_1 \cap B_2$ is a union of sets from $\mathcal B$.

Then $\mathcal B$ is a (synthetic) basis for $A$.

See also

• Synthetic Basis and Analytic Basis are Compatible

• Union from Synthetic Basis is a Topology

It follows from this result that a topology can be specified using a statement of the form:

Let $\vartheta$ be the topology with the sets $B_1, B_2, \ldots$ as basis.

Of course, $B_1, B_2, \ldots$ all need to satisfy B1 and B2.

Countable Basis

A countable basis for a topology $\vartheta$ is a basis for $\vartheta$ which consists of a countable number of sets.

Linguistic Note

The pronunciation of bases in this context is bay-seez, not bay-siz.

Also see

• Sub-basis

• Filter Basis

• Results about bases can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$