topology generated by basis

Overview

Given a topological basis, we can “generate” a topology for that basis.

Definition

For a set X and a basis $\mathcal{B}$, a set $U\subset X$ is an open set in the topology generated by the basis $\mathcal{B}$ (call it $\tau$) if $\forall x\in U,\exists B\in \mathcal{B}x\in B\subset U$

Note that the important part is that $B\subset U$. This is the condition that is ‘hard’, or not generally automatic.

Proof that $\tau$ is a topology

1. $\varnothing$ and X come automatically.
2. Arbitrary union: let $U={\bigcup }_{\alpha \in I}{U}_{\alpha }$ be an arbitrary union of open sets (i.e. $U_{\alpha }$ each follow above conditions) then $\forall x\in U$ $x\in B\subset U_{\alpha }\subset U$ so $U\in \tau$
3. Finite intersection: For $x\in U_1\cap U_2$ open sets, so thus $x\in B_1\subset U_1$ and $x\in B_2\subset U_2$. Then by the 2nd condition of the definition of being a basis, $x\in B_3\subset B_1\cap B_2\subset U_1\cap U_2$