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Is there a name for a metric space in which any bounded subset is totally bounded, or equivalently, in which any bounded sequence contains a Cauchy subsequence?

I have seen the name Bolzano-Weierstraß property when the subsequence is convergent, which in the case of complete metric spaces is equivalent to the property I am interested in. Does anyone knows other terminologies in the complete case and/or some terminology in the noncomplete case?

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    $\begingroup$ it is equivalent to the completion enjoying Bolzano-Weierstraß property, right? $\endgroup$
    – Fedor Petrov
    Commented Jan 5, 2023 at 15:46

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Proper space is the a complete space such that any bounded subset is totally bounded,

  • or equivalently, in which any bounded sequence contains a converging subsequence,
  • or equivalently, any bounded closed set is compact,
  • or equivalently, the distance function from one (and therefore any) point is proper; the latter means that invese image of any compact set is compact.

For noncomplete space you may say space with proper completion, or you may call it preproper space by analogy with precompact.

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In Functional Analysis such spaces are called Fréchet-Montel spaces because of Montel's theorem that closed and bounded subsets of the space $H(\Omega)$ of holomorphic functions on an open subset of $\mathbb C$ are compact (this is one of the main ingredients in Riemann's mapping theorem).

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  • $\begingroup$ Does it assume completeness? $\endgroup$
    – Anton Petrunin
    Commented Jan 5, 2023 at 21:38
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    $\begingroup$ Horváth in his book Topological Vector Spaces and Distributions calls a locally convex space semi-Montel if every bounded set is relatively compact. This implies that closed bounded sets are complete and in the metrizable case completeness. For a locally convex Montel space one requires additionally barreledness. In the metrizable case this is automatic by Baire's theorem. $\endgroup$
    – Jochen Wengenroth
    Commented Jan 6, 2023 at 8:15

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