Continuous Deformation Of Punctured Torus

To give you an alternative way to see this, you certainly know how to obtain a torus by taking a square of paper and gluing the edges together in couples. Now the punctured torus can be obtained the same way by making a hole in the square of paper. If you deform the hole enough and leave only a small strip around the edges, when you glue them together you'll get you figure.

The following intermediate picture, which is taken from a step in a video I found on YouTube uploaded by user esterdalvit, was sufficient to help me see that there is a continuous deformation between spaces X and Y:

This is probably clear, given your answer, but just in case a verbal description is helpful for you or others:

I like to think of this the following way: put your hands in the puncture, one on either side, and begin to stretch the puncture around the torus; once you do this, you can imagine that the torus is mostly puncture, with just two small "ribs" left, as in the picture in your answer.