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## Homework Statement

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?

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!

Thanks,

Tomer.

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.

## Homework Equations

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!

Thanks,

Tomer.

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