In this paper of M. L. Wage, W. G. Fleissner, and G. M. Reed, the authors claimed that having a zeroset diagonal does not guarantee submetrizable by showing Example 2. However, the example is very obscure.
The construction of Example 2 is done in a similar manner by considering Heath's "V" space defined on a Q-set.
Could somebody help me to explain it?
Added by Arthur Fischer. In
Heath, R. W., Screenability, pointwise paracompactness, and metrization of Moore spaces, Canad. J. Math. 16 (1964), 763–770, MR0166760, link
Heath's "V" space is constructed as follows.
Let $X = \{ \langle x , y \rangle \in \mathbb{R} \times \mathbb{R} : y \geq 0 \}$ be the closed upper half-plane, and give it the topology generated by the following basic open neighbourhoods:
- every $\langle x ,y \rangle \in X$ with $y > 0$ is isolated;
- for $x \in \mathbb{R}$, given $n \geq 1$ the set $V_{x,n} = \{ \langle\, x+y , |y| \,\rangle : \frac{-1}{n} \leq y \leq \frac{1}{n} \}$ is a basic open neighbourhood of $\langle x , 0 \rangle$.
(Note that $V_{x,n}$ is a "$\mathsf{V}$" with vertex $\langle x,0 \rangle$, slopes $\pm 1$ and height $\frac 1n$.)
This is a pointwise paracompact Moore space which is not screenable.
The construction is given in much more detail in G.M. Reed, ‘On normality and countable paracompactness’, Fundamenta Mathematicae, Vol. $110$, Issue $2$, pp. $145-152$, freely available here.