Can you give me an example of a non compact topological space with compact dense subset?
I know a Hausdorff topological space $(X, \tau)$ with compact dense subset must be compact.
Hence, my intuition suggests that there may be a non compact topological space with compact dense subset.
And such space will not be Hausdorff space anymore.
But it is difficult for me to cite this example.
Please explain it in details. Thanks.
Best Answer
Take the included point topology on e..g. $X=\Bbb R$: all open subsets are $\emptyset$ and all $A \subseteq \Bbb R$ with $0 \in A$.
Then $\{0\}$ is compact and dense while $X$ is not compact: consider the open cover $\{\{0,x\}, x \in X\}$.