If $B$ is a basis for a topology $T$, does $B$ necessarily generate $T$? 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! 
 A: There is a difference between being a basis and a basis for a particular topology. A basis is a collection of subsets of a set $T$ satisfying certain properties. It generates a topology as you indicate, and we call it a basis for the topology it generates. It's possible that a proof begins "let $(T,\tau)$ be a topological space and $\mathcal{B}$..." then there is an implied "for the topology $\tau$".
A: Quoting from Munkres:

If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis elements) such that

*

*For each $x\in X$, there is at least one basis element $B$ containing $x$.

*If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3\subset B_1\cap B_2$.


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 $\mathcal{B}$.
