Let $(X,\tau)$ be some topological space. Munkres defines a basis $\mathcal{B}$ of $\tau$ as a collection of subsets of $X$ such that:
$\mathcal{B}$ covers $X$, and given $B_1, B_2 \in \mathcal{B}$, for all $x \in B_1 \cap B_2$, there exists some $B_3 \in \mathcal{B}$ such that $x \in B_3 \subset B_1 \cap B_2$.
Given a collection of subsets $\mathcal{B}$ with the above properties, Munkres defines the topology generated by $\mathcal{B}$ as all sets $U$ such that for all $x \in U$, there exists some $B \in \mathcal{B}$ such that $x \in B \subset U$.
My question is, given some topological space $(X,\tau)$ with basis $\mathcal{B}$, is $\tau$ necessarily the topology generated by $\mathcal{B}$? I thought they were different concepts, but in the proof of certain lemmas, they seem to imply one another.
Thanks!
Best Answer
Quoting from Munkres:
This definition is not the definition of "$\mathcal{B}$ is a basis of the topology $\tau$" (note that no topology $\tau$ appears in the definition!). Instead, it's the definition of "$\mathcal{B}$ is a basis for some topology on $X$". Which topology? The topology generated by $\mathcal{B}$ (which is the next part of the definition in Munkres).
So when you read "$\mathcal{B}$ is a basis for $\tau$", it means that $\mathcal{B}$ is a basis, and $\tau$ is the topology generated by the basis $\mathcal{B}$.