Collapsing longitudinal and meridian circles of a torus.

Question

What will be the quotient space obtained from torus by identifying the longitudinal and the meridian circles of it?

The problem is that I can't visualize the space but in Hatcher's book it is claimed that the quotient space is homeomorphic to the $2$-sphere $\Bbb S^2.$ Is there any easier way to visualize the quotient space? Since torus is obtained from the closed square by identifying it's opposite edges it follows that the longitudinal and meridian circles of a torus are precisely the opposite edges of the closed square. So what I think is that the quotient will be homeomorphic to the open square $(0,1) \times (0,1)$ adjoined with a single point (the identification point). The resultant space being the quotient of a compact set is compact. So by the uniqueness of one point compactification it follows that the quotient space is homeomorphic to the one point compactification of the unit square. But the unit square is homeomorphic to $\Bbb R^2$ and by the virtue of stereographic projection it follows that the quotient space is homeomorphic to the one point compactification of $\Bbb R^2$ which is $\Bbb S^2.$

Is the above reasoning correct? Please have a look at it.

Thanks in advance.

Answer 1

From your reasoning I suppose that you mean collapsing meridians into points. Then it is correct and that's the easiest way to tackle the problem, since the fact that $S^2 \simeq D^2/\partial D^2$ is used on every step in topology and is considered kinda obvious.

A straightforward way to picture what happens with the torus is hinted at in #1 here. It is easy to see that if you collapse one meridian you get "a chubby bagel", which is shown to be a sphere with a loop wedged. Collapsing this loop you thus get a sphere.

Now, if you really meant "identifying the longitudinal and the meridian circles" then I don't think that you get a sphere. First of all, you need to specify the homeomorphism whereby you perform the identification. But for instance the way shown on the picture does not give a sphere because the drawn vicinity of the point A on meridians does not look like disk. For a sphere all vicinities are locally disks.