Confusion about a basis for a topology

Question

Here is Munkres' definition of a basis for a topology:

I wonder if it is assumed that when one talks about a basis for a topology, then the topology being considered is the topology generated by this basis? The reason I'm confused are these two theorems:

In the highlighted part, he uses that $\tau$ is the topology generated by the basis $\mathcal B$, but he doesn't mention this in hypothesis. So is it implicitly assumed that $\tau$ is the topology generated by $\mathcal B$?

Similarly,

The second parahraph of the prove says that we must prove that $\tau'=\tau$, but this wasn't claimed in the statement of the theorem.

Answer 1

I'll give you that this is confusing here.

You should read the definition at the top as "$\mathcal B$ is a basis for some topology on $X$ iff (1) and (2) are satisfied." The topology that it is a basis for is the one generated by $\mathcal B$ according to the next definition. (And I presume that they show this is in fact a topology shortly after.)

Then Lemma 13.1 shows that there is an equivalent definition for the topology generated by $\mathcal B$: the collection of arbitrary unions of sets in $\mathcal B.$ There is a third common equivalent definition: the topology generated by $\mathcal B$ is the intersection of all topologies $\mathcal T$ with $\mathcal B\subseteq \mathcal T.$

Here is a summary that may be less confusing:

Let $X$ be a set and $\mathcal B\subseteq \mathcal P(X).$ Then the following are equivalent:

1. Munkres' (1) and (2) hold.
2. The collection of all arbitrary unions of sets in $\mathcal B$ is a topology on $X$.

When these hold, the topology $\mathcal T$ in point 2 is called the topology generated by $\mathcal B$ and $\mathcal B$ is called a basis for the topology $\mathcal T.$ The topology $\mathcal T$ has the following two alternative characterizations:

1. $U\in \mathcal T$ iff for all $x\in U$ there is a $B\in \mathcal B$ such that $x\in B\subseteq U$
2. $\mathcal T$ is the smallest topology such that $\mathcal B\subseteq\mathcal T,$ i.e. the intersection of all such topologies.