what is the exact meaning of compact hausdorff space ? what are example of compact hausdorff ?
My thinking : compact = closed + bounded = finite subcover
and hausdorff = disconnected
Im confused why they (munkre) combine compact hausdorff?
Any hints /solution
No, not "closed and bounded" - that's not a definition of compactness, that's a theorem characterizing compact subsets of $\mathbb{R}^n$. And not "disconnected", either; the Hausdorff separation property has nothing to do with connectedness.
The standard topological definitions:
A set $K$ in a topological space $S$ is compact when, for any collection of open sets that cover $K$ (their union includes $K$), there is some finite subset of them that covers $K$.
A topological space $S$ is Hausdorff when any two points $x$ and $y$ can be separated by neighborhoods - that is, there are open sets $G_x$ and $G_y$ containing $x$ and $y$ respectively such that $G_x\cap G_y$ is empty.
What does this get us? Well, here are two standard results:
In a Hausdorff space, single-point sets are closed.
Every compact subset of a Hausdorff space is closed.
Talking about compact sets/spaces without that Hausdorff property gets weird. For example, consider the cofinite topology on some infinite set $S$; a set is open iff it's empty or its complement is finite. In that topology, every subset of $S$ is compact.