Proving Homeomorphism between Topological Spaces (X,T) and (delta,U_delta)

In summary, the conversation discusses proving that two topological spaces, (X,T) and (delta,U_delta), are homeomorphic. This involves showing that the projections of these spaces onto each other are continuous and invertible. The conversation also mentions constructing open sets and finding the inverse of pi_1.
  • #1
tylerc1991
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0

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!
 
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  • #2
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?
 

What is a homeomorphism?

A homeomorphism is a type of mapping between two topological spaces that preserves the topological properties of the spaces. Essentially, it is a continuous, bijective function where both the function and its inverse are continuous.

How is a homeomorphism different from an isomorphism?

An isomorphism is a mapping between two mathematical structures that preserves their algebraic or geometric structure. While a homeomorphism preserves the topological structure of a space, an isomorphism preserves a different type of structure, such as algebraic or geometric properties.

What are some examples of homeomorphisms?

Some examples of homeomorphisms include stretching, bending, and twisting a space without tearing or gluing any parts. For instance, a circle and a square are homeomorphic because you can continuously deform a square into a circle without tearing or gluing any edges.

How is a homeomorphism useful in topology?

Homeomorphisms are useful in topology because they allow us to define and classify topological spaces based on their properties. By comparing two spaces and determining if they are homeomorphic, we can gain a better understanding of their topological properties and relationships.

Can two spaces be homeomorphic but have different dimensions?

Yes, two spaces can be homeomorphic but have different dimensions. For example, a line segment and a circle are homeomorphic, but one is one-dimensional while the other is two-dimensional. This is because homeomorphisms only preserve topological properties, not geometric properties like dimension.

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