In General Topology Class we proved that if X is a metric space, X is compact iff is sequentially compact.
In particular in the first implication we assumed there is some sequence that doesn't have any convergent subsequence converging to a point in X; let Z be the set of the elements of the sequence then for every point x belonging to X exists an open set containing x that either has no intersection with Z o has only x(if x belongs to Z);
then X\Z is open (so Z is closed) and every point in Z is an isolated point;
X is compact an Z is closed then Z is compact(for some prop.);
Z is a compact discret topology subspace so |Z| is finite and this is impossible because we assumed there is some sequence that doesn't have any convergent subsequence converging to a point in X
My question is:
when do i use the fact that X is a metric space? When i say that there is an open set etc. i can say that for every Hausdorff space but i know there are known counterexample to this
can you help me?
If something is not clear enough please ask me some clarification