Let

and

.

Do the spaces X and Y have the same homology groups? Are they homotopy equivalent? Are they Homeomorphic? Do they have homeomorphic one-point compactifications?

First, I was wondering if the space Y is homeomorphic to a cylinder? To me it looks like it's a Torus where one circle

has been cut off. Or maybe, it is a Torus, where only point is missing?

I know that

if

and

otherwise. (Where

is the i-dimensional reduced homology of the space

)

Also I know as an application of the Mayer-Vietoris sequence, that If

is a subspace of

homeomorphic to

for

, then

if

and

otherwise.

Also, I know that

) for

(Where

is the i-dimensional homology group of

) and that

But does any of this help answer the questions? Or is it totally irrelevant in this case? If this doesn't help, how do you go about answering the questions?