Definition:Basis (Topology)

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:


 * 1) $$A$$ is a union of sets from $$\mathcal{B}$$;
 * 2) 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 $$A$$.

Module
Let $$\left({G, +_G: \circ}\right)_R$$ be a unitary $R$-module.

A basis of $$G$$ (plural: "bases") is a linearly independent subset of $$G$$ which is a generator for $$G$$.

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