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Topology: Homeomorphism

  • Thread starter tylerc1991
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Homework Statement



Let (X,T) be a topological space and let U denote the product topology on X x X. Let delta = {(x,y) in X x X : x = y} and let U_delta be the subspace topology on delta determined by U. Prove that (X,T) is homeomorphic to (delta,U_delta)

The Attempt at a Solution



Since a homeomorphism is a continuous function between two topological spaces, I am assuming that I have to prove that the projections (call them pi_1 and pi_2), as well as their inverses are continuous? Well if I show that pi_1 is continuous then I have shown that pi_2 is also continuous (since they project to the same topological space).

So take an open set in (X,T), call it V. Since (delta,U_delta) was created from the product of (X,T) with itself, I can take some other open set in (delta,U_delta), call it V_delta such that pi_1(V_delta) maps inside of V. Hence pi_1 is continuous and hence pi_2 is continuous as well. I can do something similar for the inverses of the projection functions. Is this anywhere close to being correct/on the right track? Thank you for your help!
 

Answers and Replies

  • #2
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Yes, this is correct, but I think you'd have to give some more details. Such as:
- How did you construct V_delta. You said that you can find it, but how do you do so?
- Why is pi_1 invertible? What is it's inverse?
- Is the continuity of the inverse of pi_1 really analogous to the continuity of pi_1?
 

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