Sets where Heine-Borel theorem works 
Possible Duplicate:
Are there more general spaces than Euclidean spaces to have the Heine–Borel property? 

By Heine-Borel theorem, a closed and bounded subset of the Euclidean space is compact. If we analyze the proof, the only characteristic of Euclidean space that we need is: every bounded subset is contained in a compact subset. Is there a special name this kind of sets?
 A: This proves that HB is equivalent to the fact that every bounded subset is a subset of a compact set. Clearly, I am doing it on a metric space, only then this makes sense. As pointed out OP seems to understand this equivalence and hence is irrelevant.

Heine-Borel Property:
A metric space $(X, d)$ is said to be Heine-Borel if any closed and 
bounded subset of it is compact.

You'd like to know the name of those spaces in which a bounded set is contained in a compact set (BIC). 
We claim that this property is equivalent to the Heine-Borel (HB) Property. 
  
  
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*$(BIC\implies HB)$  Suppose a space $X$ is $BIC$. Let $B$ bounded subset of $X$. Because $X$ is $BIC$, $B \subseteq K$ where $K$ is compact. If $B$ is also closed, you have closed subset of a compact set which is compact. Hence the space is $HB$.
  
*$(HB \implies BIC)$ Let $B$ bounded subset of $X$ that has $HB$. We need to find a compact subset $K$ such that $B \subseteq K$. Then the closure, $\bar B$ is closed and bounded, hence compact, because of $HB$ property and note that $B \subseteq \bar B$. Hence, $X$ is $BIC$.
  

So, we can as well call these Heine-Borel spaces.
In fact, another characterisation of Heine Borel spaces is that, bounded sets are also totally bounded. 
