Example of a Baire metric space which is not completely metrizable

Question

I know that some Baire metric spaces are not complete metric spaces but all examples, that I know, are completely metrizable. Help me to find an example of Baire metric space which is not completely metrizable. $[$Please give some short proofs or references$]$

See Completely Metrizable Space and Baire Theorem. – Dave L. Renfro Nov 18 '18 at 15:15 — Dave L. Renfro, Nov 18 '18 at 15:15

score 13 · Accepted Answer · answered Nov 18 '18 at 17:30

13

A classic example is the open upper half plane with the rationals on the $x$-axis:

$X = \{(x,y) \in \mathbb{R}^2: y >0 \text{ or } y=0, x \in \mathbb{Q}\}$ in the Euclidean metric.

This is Baire as it has an open dense Baire subspace $\mathbb{R} \times (0,\infty)$ (which is completely metrisable) and not completely metrisable as it has a closed homeomorphic copy of $\mathbb{Q}$.

answered Nov 18 '18 at 17:30

Henno Brandsma

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If I'm not wrong, $X=\mathbb{Q}\times (0,\infty)$, then considering $U_n=X\setminus {r_n}\times (0, \infty)$ we got $\cap U_n = \phi$ – Offlaw Nov 19 '18 at 13:44
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@Offlaw $\mathbb{Q}\times (0,\infty)$ is irrelevant. – Henno Brandsma Nov 19 '18 at 21:06
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I got it. $X=\mathbb{R} \times (0,\infty) \cup \mathbb{Q} \times {0}$. – Offlaw Nov 20 '18 at 01:29
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@Offlaw indeed. That’s the space – Henno Brandsma Nov 20 '18 at 04:24

Example of a Baire metric space which is not completely metrizable

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