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If not, what is a counterexample?

I'm trying to prove Theorem 19.2 from Munkres's Topology.

Suppose the topology on each $X_a$ is given by a basis $B_a$. The collection $B$ of all sets of the form $\prod b_a$, where $b_a \in B_a$ for all $a$, will serve as a basis for the box topology on $\prod X_a$.

At first, I thought it was enough just to show that $B$ satisfies the basis definition. Do I also need to show that the topology generated by $B$ equals the box topology?

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You will need to show that for every element $x$ in the product and every box topology neighborhood $U$ of $x$, there is an element of $B$ such that $x\in B\subseteq U$.

Edit: Once you've done that, you need only confirm that every element of $B$ is open in the box topology on $X$--which should be a fairly simple task, as elements of a basis $B_a$ of $X_a$ are necessarily open in $X_a$. Then by Lemma 13.2 of Munkres, $B$ will be a basis for the box topology on $X$, as desired.

Note that this last detail is actually essential to include! (I neglected to mention it because it should be clear that the elements of $B$ are open in the box topology on $X$, but in hindsight, I shouldn't have omitted that detail. Apologies for any confusion!) For example, if we take $B'$ to be the set of all singletons $\{x\}$ (for $x\in X$), the set $B'$ rather trivially satisfies the condition I mentioned in the first place, but while it is a basis, it is a basis for the discrete topology--strictly a finer topology (in general) than the box topology.

All the condition I mentioned at first really shows is that if $B$ is a basis for some topology, then it generates a topology finer than the box topology on $X$ (possibly strictly finer). Knowing that the elements of $B$ are open in the box topology allows us to conclude that (1) $B$ is a basis for a topology on $X$, and that (2) the generated topology is coarser than (so equivalent to, since also finer than) the box topology on $X$.

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    @sli: I hear you on that. It's fairly standard practice, and it's something you'll get used to. Often, the text'll make brief mention in passing ("hereinafter referred to as 'blah'"), but sometimes there's just a shortening with no warning.2012-05-22