Let X be an uncountable set, let τf be the finite complement topology on X, and...

Let X be an uncountable set, let τf be the finite complement topology on X, and let τ_c be the countable complement topology; namely, we have
τ_f ={U⊂X : X\U is finite}∪{∅},

τc={U⊂X : X\U is countable}∪{∅},

where “countable” means that the set is either finite or countably infinite (in bijection with the natural numbers).

(a) What are the compact subspaces of (X, τ_f )? Are all compact subspaces closed in (X, τ_f )?

(b) What are the compact subspaces of (X,τ_c)? Are all compact subspaces closed in (X,τ_c)?

(c) What have we learned about the nature of compact subspaces when (X,τ) is not Hausdorff ?

Expert Solution

Colby Messinger answered 7 months ago

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