**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.