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Nik Weaver has the following exercise in Chapter 1 of his Measure Theory and Functional Analysis:

1.21. Find an example of a closed equivalence relation on a separable metric space such that the quotient space is not metrizable.

By a closed equivalence relation $R$ on a space $X$, he means that $R$ is closed in $X\times X$.

The standard counterexamples in these matters like the line with two origins, $\mathbb Q$'s action on $\mathbb R$, or $\mathbb Z$'s action on $\mathbb R^2\setminus\{0\}$ (as mentioned here) don't work since the relations are all non-closed.

Question: Can someone help me coming up with an exmaple?

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1 Answers1

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Note that identifying the points of $\mathbb Z\subseteq \mathbb R$ to a single point, the space $\mathbb R/\mathbb Z$, is obtained by quotienting via the closed relation $\mathbb Z^2\cup\Delta\subseteq\mathbb R^2$.

This space is not metrizable, because $[0]$ is not first-countable, as noted by Brian Scott here - essentially, one may use a diagonalization argument to find an open neighborhood of $[0]$ not generated by any countable collection of open neighborhoods of $[0]$.

See also https://topology.pi-base.org/spaces/S000139/properties/P000187 for a proof that this space is not a W-space, a property held by all first-countable (and thus all metrizable) spaces.

Steven Clontz
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