TITLE: Normal, Non-Metrizable Spaces
QUESTION [11 upvotes]: We know that every metric space is normal. We know also that a normal, second countable space is metrizable.
What is an example of a normal space that is not metrizable?
Thanks for your help.

REPLY [10 votes]: Rather than list specific spaces, I thought that I’d mention a few classes of spaces.
A compact metric space has cardinality at most $2^\omega$, so every compact Hausdorff space of cardinality greater than $2^\omega$ is normal and non-metrizable. In particular, this includes every product of compact Hausdorff spaces with at least $2^\omega$ non-trivial factors.
Let $I$ be an index set, for each $i\in I$ let $X_i$ be a space with at least two points, and for each $i\in I$ let $p_i\in X_i$. Let $$X=\left\{x\in\prod_{i\in I}X_i:|\{i\in I:x_i\ne p_i\}|\le\omega\right\}$$ as a subspace of the Tikhonov product of the $X_i$; such spaces are called $\Sigma$-products. If $I$ is countable, the $\Sigma$-product is just the ordinary Tikhonov product, but if $I$ is uncountable it’s something new.

Proposition: If $I$ is uncountable and each $X_i$ is $T_1$, $X$ is not paracompact (and therefore not metrizable).
Proof: Let $I_0=\{i_\xi:\xi<\omega_1\}$ be a subset of $I$ of cardinality $\omega_1$, and for each $\xi<\omega_1$ fix $q_{i_\xi}\in X_{i_\xi}\setminus\{p_{i_\xi}\}$. For $\eta<\omega_1$ define $x^\eta\in X$ by $$x^\eta_i=\begin{cases}q_{i_\xi},&\text{if }i=i_\xi\text{ and }\xi<\eta\\p_i,&\text{otherwise}\;.\end{cases}$$ It’s not hard to check that $\{x^\eta:\eta<\omega_1\}$ is a closed subspace of $X$ homeomorphic to $\omega_1$ (with the order topology), which is not paracompact. $\dashv$

However, it’s a theorem of Mary Ellen Rudin and, independently, S.P. Gul’ko that $\Sigma$-products of metric spaces are always normal. (The proof is highly non-trivial and can be found in Teodor C. Przymusiński, Products of Normal Spaces, in the Handbook of Set-Theoretic Topology, K. Kunen & J.E. Vaughan, eds., where the result is Theorem 7.4.) Thus, every uncountable $\Sigma$-product of non-trivial metric spaces is an example of a normal, non-metrizable space.
It’s well known that every linearly ordered space [LOTS] is hereditarily normal. This means that every generalized ordered [GO] space is hereditarily normal, since the GO-spaces are precisely the subspaces of linearly ordered spaces. (The actual definition of a GO-space is that it’s a space $X$ equipped with a linear order $\le$ whose topology has a base consisting of $\le$-intervals, not necessarily open, but this is equivalent to being a subspace of a LOTS. An example is the Sorgenfrey line.)
Thus, any non-metrizable GO-space is an example. The most straightforward way for a GO-space to fail to be metrizable is to have a point with uncountable character, i.e., a point that has no countable local base; this automatically includes all ordinal spaces $\alpha$ for $\alpha>\omega_1$ and many of their subspaces. This is far from the only way, of course. The Sorgenfrey line, for example, is first countable but fails to be metrizable for a variety of reasons: it’s separable and Lindelöf but not second countable, and its square is neither normal nor Lindelöf. It’s rather easy to come up with all sorts of variations on this theme.