Base (topology)

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A basis or base of a topology is a collection $\mathcal{B}$ of sets in a topological space $(X,\tau)$ that classify the set of open sets of the space.

  1. For any basis $\mathcal{B}$, the union of the sets in $\mathcal{B}$ is equal to $X$. Phrased differently, for any element $x \in X$, there exists a basis set $A \in \mathcal{B}$ such that $x \in A$.
  2. For any two sets $A, B \in \mathcal{B}$, given an element $x \in A \cap B$, then there exists another set $C \in \mathcal{B}$ such that $x \in C \subset A \cap B$.

This definition is very useful for comparing different topologies. In particular, we have the following theorem:

Theorem: In a space $X$, given two topologies $\tau$ and $\tau'$, then $\tau \subset \tau'$ iff for any basis element $B \in \mathcal{B}$ and any element $x \in B$, there exists a basis element $B' \in \mathcal{B}'$ such that $x \in B' \subset B$.

Sub-basis

A sub-basis $\mathcal{S}$ of $X$ is a collection of sets whose union is $X$. The collection of intersection of sets in $\mathcal{S}$ forms a basis $\mathcal{B}$ on $X$.

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