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I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:

Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R > 0$, then $\overline{B(x,R)} = B[x,R]$. Then the space have the Heine-Borel property, that is, every closed and bounded subset of $X$ is compact.

I'm aware of a proof for this result by changing "closure of ball = closed ball" to "$X$ is an length space", which can be found in Burago's, Yuri's and Ivanov's book. However, that proof relies on the $\varepsilon$-midpoint property of length spaces, which seems stronger than what "closure of ball = closed ball" provides. I was unable to adapt that proof for this case, and I'm not even convinced that this should be true anymore.

I was hoping if this result is written or know somewhere.

Thanks in advance.

EDIT: The paper in question is the following:

[1] Sosa, Gerardo. "The Isometry Group of an RCD∗ space is Lie." Potential Analysis 49.2 (2018): 267-286.

The result is stated in a footnote of proposition 3.6.

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    $\begingroup$ Yuri is also Burago:) $\endgroup$
    – Fedor Petrov
    Commented Jan 9, 2022 at 21:12
  • $\begingroup$ I know, but "burago, burago" is kinda confusing to say. $\endgroup$
    – Kaitei
    Commented Jan 9, 2022 at 21:16
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    $\begingroup$ @YuvalPeres I added the paper in the edit. $\endgroup$
    – Kaitei
    Commented Jan 9, 2022 at 21:45
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    $\begingroup$ What about $\mathbb R$ with the metric $d(x,y)=\frac{|x-y|}{|x-y|+1}$? $\endgroup$
    – user473423
    Commented Jan 10, 2022 at 8:01
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    $\begingroup$ @Echo Isn't your comment worth an answer? $\endgroup$
    – Jochen Wengenroth
    Commented Jan 10, 2022 at 12:31

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