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.