What do you think?

cProve that Y is Hausdorﬀ if and only if X is Hausdorﬀ and A is a closed subset of X .

The key point to notice is that the restriction X-A)\times\{0,1\}\to Y" alt="p^{*}X-A)\times\{0,1\}\to Y" /> is an injection, and thus if is open then so is since

Uhh... is compact and evidently is compact and which is continuous?(2) Consider the above construction for X = [0, 1] and A an arbitrary subset of [0, 1].

Prove that Y is compact.

is compact and both and are closed subspaces and so automatically compact. And since are the continuous images of compact spaces they are trivially compact.Prove that K = p(X × {0}) and L = p(X × {1}) are compact, and that K ∩ L is homeomorphic to A .

Think about the other part. If then and so you can think of as being but treating and as the same point. Doesn't this sound eerily familiar to which is naturally homeomorphic to ?