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**Let (X, τ) be a Hausdorff space and τ**

Show that:

1) τ

2) τ

3) (X, τ

_{0}= {X\K: so that K is compact in (X, τ)}Show that:

1) τ

_{0}is a topology of X.2) τ

_{0}is rougher than τ (i.e. τ_{0}is a genuine subset of τ).3) (X, τ

_{0}) is compact.This was a question in a recent exam that I took (I failed). I was especially clueless with 3).

Thanks for any help.