Homology groups, homotopy equivalence and one point compactification

Let $\displaystyle X= \textbf{R}^2 - \{ a,b \} $ and $\displaystyle Y = S^1 \times S^1 - \{c \} $.

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 $\displaystyle S^1$ has been cut off. Or maybe, it is a Torus, where only point is missing?

I know that $\displaystyle H'_i (S^n) = \textbf{Z} $ if $\displaystyle i=n $ and $\displaystyle 0 $ otherwise. (Where $\displaystyle H'_i (X)$ is the i-dimensional reduced homology of the space $\displaystyle X$)

Also I know as an application of the Mayer-Vietoris sequence, that If $\displaystyle S$ is a subspace of $\displaystyle S^n$ homeomorphic to $\displaystyle S^k$ for $\displaystyle 0 \leq k < n $, then $\displaystyle H'_i ( S^n -S) = \textbf{Z} $ if $\displaystyle i=n-k-1$ and $\displaystyle 0$ otherwise.

Also, I know that$\displaystyle H'_i (X) = H_i (X$) for $\displaystyle i>0$ (Where $\displaystyle H_i(X)$ is the i-dimensional homology group of $\displaystyle X$) and that $\displaystyle H_0(X) = H'_0(X) \oplus \textbf{Z}$

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?