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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.

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.

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.

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.

It is not exactly the same: even if all bounded subsets are totally bounded, there might be bounded sequence contains a Cauchy subsequence. But if you assume that the space is complete, then it is the same.

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

• or equivalently, in which any bounded sequence contains a Cauchy subsequence,
• or equivalently, any bounded closed set is compace,
• 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.

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.