In this post, I show that the infinite product of non-discrete space is not metrizable.
There are quite a few questions on math SE about the metrizability of etc. equipped with the box topology (see here or here for example). For me, those were too specific to see what is the key property of a space used to prove non-metrizability.
Recall that for a topological space the box topology on the product is generated by the basic sets of the form
Proof. We consider two cases.
Case 1:
is Hausdorff. We will show that
is not first countable, hence not metrizable. Let
be a non-isolated point in
(which exists by the assumption that
is not discrete). For the sake of contradiction assume
that
is coutable at
where
for each
. For any
we find a basic open
which contains
, namely
where each
is an open
of
. Since
is infinite we take an infinite countable subset
. For convinience relabel elements of
so that
.
For any let be a point distinct from . Such a point exists because we assumed that is not isolated. Since is Hausdorff we can separate from with an open of , so that . Let Set is a basic open of . It is enough to show that for any . Assume that for some natural number we have . For let and define . Clearly . On the other hand by construction of . Hence . We arrive at a contradiction with the inclusion .
Case 2. is not Hausdorff. It is enough to show that with the box topology is not Hausdorff. Let be the points such that for any open of (respectively) we have . Then points where and cannot be separated by basic open sets in the box topology. ◻
Theorem 1.3 in the Handbook of set-theoretic topology by Kunen and Vaughan is similar. But there it is assumed that is infinite, non-discrete, Hausdorff, and completely regular... On top of that proof seemed complicated to me — after skimming it I decided, following advice from Thurston’s On proof and progress in mathematics, that it would be easier to come up with my own proof. It probably wasn’t easier, but for sure it was more fun! If you see any mistakes, please point them out.