It is known in $\mathbb R^n$ we have compact $\iff$ closed and bounded. But someone once told me that this is not true in general. Now I am wondering what is needed for this to be true?
First, for "compact", "closed", "bounded" to be meaningful, the space must have a metric. It follows from this that one direction, namely "compact$\implies$bounded and closed" (prop 1.4.2 in Conway's A Course in Point Set Topology), holds.
Questions:
(1) What property of the metric space is needed for the converse to be true? (I mean what is needed for every closed and bounded subset to be compact?)
(2) Is there an example such that closed and bounded $\nRightarrow$ compact?
Best Answer
First let me note you don't need a metric to talk about a subset of a topological space being closed or compact.
The Heine-Borel Theorem states that a subset of an euclidean space $\mathbb{R}^n$ is compact if and only if it is closed and bounded.
Meanwhile a metric space is compact if and only if it is complete and totally bounded.
As an example imagine an infinite discrete space. It is trivially closed and also bounded, but not compact. It is also not complete and not totally bounded.
It turns out a subset of a euclidean space is bounded if and only if it is totally bounded, and complete if and only if closed.