TITLE: countable topological spaces of uncountable weight
QUESTION [5 upvotes]: I read somewhere the following stament:
There are countable normal $T_1$ spaces of uncountable weight.
Can someone give an example or a reference?

REPLY [4 votes]: Another favourite example of mine: the Hewitt-Marczewski-Pondiczery theorem says that the product of continuum or fewer separable spaces is separable. So $\mathbb{R}^\mathbb{R}$ has a countable dense subset $D$ and it is clear any countable regular space is normal (from being Lindelöf), and it's not hard to show that every point of $D$ has a local base of $\mathfrak{c}$ many open sets (and not fewer). So it's not like the other examples mentioned, in that there are no isolated points.