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

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

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

Also see

 * Sub-basis


 * Filter Basis