one point compactification, homeomorphism

[Joolz]

Hi all,

I was wondering how to prove the following:

If X and Y are locally compact hausdorff spaces then (X x Y)* is homeomorphic to a quotient of X* x Y*.

I did show that X x Y is also a locally compact hausdorff space. Can I use this?

Kind regards,

 

[SlipEternal]

Hi all,

I was wondering how to prove the following:

If X and Y are locally compact hausdorff spaces then (X x Y)* is homeomorphic to a quotient of X* x Y*.

I did show that X x Y is also a locally compact hausdorff space. Can I use this?

Kind regards,

Consider what open sets look like in each of \(\displaystyle (X \times Y)^*\text{ and }X^* \times Y^*\). How are they similar? How are they different? Find where they differ, and try to find an equivalence relation on \(\displaystyle X^* \times Y^*\) such that the quotient space has the desired structure. For instance, in \(\displaystyle (X\times Y)^*\), an open set is an open set in \(\displaystyle (X\times Y)\) union with \(\displaystyle \{\infty\}\). An open set in \(\displaystyle X^* \times Y^*\) is an open set in \(\displaystyle X^*\) cross an open set in \(\displaystyle Y^*\) (in the product topology). So, you want a quotient map from \(\displaystyle X^* \times Y^*\) to \(\displaystyle (X\times Y)^*\) that will take \(\displaystyle (\infty,y^*) \mapsto \infty\) and \(\displaystyle (x^*,\infty) \mapsto \infty\). The map should be the identity everywhere else.

(Note: I have never done this type of topology, and this method of solution is my best guess given my understanding after googling Alexandroff one-point compactification for five minutes. For my notation, by \(\displaystyle x^*\), I mean \(\displaystyle x^* \in X^*\) and similarly with \(\displaystyle y^* \in Y^*\).

 

[Joolz]

Consider what open sets look like in each of \(\displaystyle (X \times Y)^*\text{ and }X^* \times Y^*\). How are they similar? How are they different? Find where they differ, and try to find an equivalence relation on \(\displaystyle X^* \times Y^*\) such that the quotient space has the desired structure. For instance, in \(\displaystyle (X\times Y)^*\), an open set is an open set in \(\displaystyle (X\times Y)\) union with \(\displaystyle \{\infty\}\).

Oh, I thought that an open set in \(\displaystyle (X\times Y)^*\) is an open set in \(\displaystyle (X\times Y)\) or a \(\displaystyle V\cup\{\infty\}\) in \(\displaystyle (X\times Y)^*\) such that \(\displaystyle (X\times Y)\) - V is compact.

So, you want a quotient map from \(\displaystyle X^* \times Y^*\) to \(\displaystyle (X\times Y)^*\) that will take \(\displaystyle (\infty,y^*) \mapsto \infty\) and \(\displaystyle (x^*,\infty) \mapsto \infty\). The map should be the identity everywhere else.

I defined such a map q. Then, if A is an open set in \(\displaystyle (X\times Y)^*\) Lets say, A is open in \(\displaystyle (X\times Y)\) then by the definition of q, q^-1(A) is open in \(\displaystyle X^* \times Y^*\). But if A = \(\displaystyle V\cup\{\infty\}\) then q^-1(A) = \(\displaystyle V\cup(X^*\times{\infty_Y})\cup({\infty_X}\times Y^*)\) And V is open in \(\displaystyle X^* \times Y^*\), but what about \(\displaystyle (X^*\times{\infty_Y})\)? I know that \(\displaystyle X^*\) is open in \(\displaystyle X^*\) but is \(\displaystyle {\infty_Y}\) open in \(\displaystyle Y^*\)?

 

[SlipEternal]

Oh, I thought that an open set in \(\displaystyle (X\times Y)^*\) is an open set in \(\displaystyle (X\times Y)\) or a \(\displaystyle V\cup\{\infty\}\) in \(\displaystyle (X\times Y)^*\) such that \(\displaystyle (X\times Y)\) - V is compact.



I defined such a map q. Then, if A is an open set in \(\displaystyle (X\times Y)^*\) Lets say, A is open in \(\displaystyle (X\times Y)\) then by the definition of q, q^-1(A) is open in \(\displaystyle X^* \times Y^*\). But if A = \(\displaystyle V\cup\{\infty\}\) then q^-1(A) = \(\displaystyle V\cup(X^*\times{\infty_Y})\cup({\infty_X}\times Y^*)\) And V is open in \(\displaystyle X^* \times Y^*\), but what about \(\displaystyle (X^*\times{\infty_Y})\)? I know that \(\displaystyle X^*\) is open in \(\displaystyle X^*\) but is \(\displaystyle {\infty_Y}\) open in \(\displaystyle Y^*\)?

What do you know about the empty set? It is both open and closed. So, the empty set is an open set in \(\displaystyle Y\). Thus, in \(\displaystyle Y^*\), \(\displaystyle \emptyset \cup \{\infty\}\) is open.