Pseudocompact space having countable $\pi$-base but not metrizable Can we find a normal Hausdorff space which is a countably compact locally connected space without isolated points and has a countable $\pi$-base but not metrizable?
A collection $\mathcal{B}$ of nonempty open sets in $X$ is called a $\pi$-base for $X$ provided that every nonempty open subset of $X$ contains some member of $\mathcal{B}$.
 A: A nice example is Helly space $H$. This space is compact Hausdorff (so $T_4$, countably compact/pseudocompact), and not metrisable (e.g. as it has a discrete subspace of size $\mathfrak{c}$, see the linked answer). So it is also not submetrisable (a strictly coarser topology cannot be Hausdorff, by standard arguments). 
It is also a convex subset of a locally convex space so locally (path-)connected and (path-)connected (so no isolated points). This why I use this space, and not (its subspace) double arrow (i.e. $[0,1] \times \{0,1\}$ in the order topology, which is hereditarily normal and not just normal ($H$ is not hereditarily normal, as $H \times H$ is homeomorphic to $H$ (I'm pretty sure) and $H$ contains the Sorgenfrey line as a subspace as well...). 
$H$ is separable (see linked answer) and first countable (so $H$ is also sequentially compact) and thus has a countable $\pi$-base (use all local countable bases at all points of a countable dense subset). So $H$ fulfills all requirements (including not being submetrisable).
