Torus 2-manifold

The torus,

Tcan be obtained as an identification space as
follows: Let X = [0,1] x [0,1] denote the square, and define an equivalence relation on X by (0,t) ~(1,t) and (s,0) ~(s,1) for all s,t in [0,1].

Prove carefully that X/~is a closed 2-manifold

Prove that X is homeomorphic to S1 x S1 where S1 is the circle

Any help would be great. Thanks

Quote:

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Originally Posted by ejgmath

The torus,

Tcan be obtained as an identification space as
follows: Let X = [0,1] x [0,1] denote the square, and define an equivalence relation on X by (0,t) ~(1,t) and (s,0) ~(s,1) for all s,t in [0,1].

Prove carefully that X/~is a closed 2-manifold

Prove that X is homeomorphic to S1 x S1 where S1 is the circle

Any help would be great. Thanks

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Let's see some work. How much knowledge do you have of quotient spaces?

Well to be a closed 2-manifold (without boundary) it must be hausdorff and be homeomorphic to an open set in R^n. But I am at a loss as how to show prove this.

I am see how X/~ is hausdorff, but dont know how to prove this, and I am completely stuck with the homeomorphic part.