I need two counter examples, that show the following two theorems don't/B] hold:
Let X be a topological space.
1. If from the closeness of any subset A in X follows compactness of A, then X is compact.
2. If from the compactness of a subset A in X follows closeness of A, then X is housdorff.
I proved the *opposite* theormes which do hold, but I cannot seem to find counter examples.
That means, I need to find a non-compact space, in which every closed subset is compact, and a non-housdorff space, in which every compact subset in closed.
Are there any equations in Topology?
The Attempt at a Solution
I just tried to take for "1" spaces I know that are not compact, but then couldn't find a space in which every closed subset is a compact one...
In "2" I thought of non-hausdorff spaces I know, but couldn't directly see whether indeed every compact set is closed.
I'd really appreciate your help!