Product Topology and Compactness

In summary, the proof uses the fact that if X and Y are compact, then their product is compact to create a non-empty set which satisfies the condition.
  • #1
Bleys
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Please if someone could help me understand something I saw in a proof. It's about proving that if X,Y is compact then their product (with product topology) is compact.

Suppose that X and Y are compact. Let F be an open cover for XxY. Then, for y in Y, F is an open cover for Xx{y}, which is compact. Hence F has a finite subcover
Fy = [tex]\{ U^{y}_{1}\times V^{y}_{1},...,U^{y}_{n}\times V^{y}_{n} \} [/itex] where y is in all [tex]V^{y}_{i}[/itex]

This is the step I don't get. Why is y in all those sets? I understand y must be in at least one (to be covered) but why all? The proof uses this fact to construct a non-empty set so it's pretty crucial, but for the life of me I don't understand how to deduce it :(
I was thinking maybe it's because of the axiom of choice, but I don't know much about that to even be sure it involves it.
 
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  • #2
There is a subtle point here. Not every element of a subcover of F needs to contain y, but the point is that we can pick a subcover of F such that the property holds.

Indeed, F is a cover of the compact space [tex]X\times\{y\}[/tex], thus it has a compact subcover [tex]\{U_1\times V_1,...,U_n\times V_n\}[/tex]. This subcover does not need to satisfy our assumptions, indeed it is possible that y is not in Vi. But if this is the case, then [tex](U_i\times V_i)\cap (X\times \{y\})=\empty[/tex]. Thus [tex]U_i\times V_i[/tex] is not really important in our cover, and it can be safely removed.

After removing all such sets, we end up with a finite subcover of F, which does satsify the condition. We call this cover Fy.
 
  • #3
aah, thanks a bunch!
I had thought about "picking" certain elements of the subcover such that y was in [itex]V_{i}[/itex], but then I was wondering if the remaining [itex]U_{i}[/itex] would still cover X, so I was getting confused (forgetting slightly that I was dealing with Cartesian products!).

Thanks for your swift reply
 

1. What is the product topology?

The product topology is a way of defining a topology on the Cartesian product of two or more topological spaces. It is defined as the smallest topology that makes all the projection maps continuous.

2. How is the product topology different from the box topology?

The product topology is generally finer than the box topology, meaning that it has more open sets. This allows for more flexibility in defining continuous maps and making the product space behave more like the individual spaces. However, the product topology may not always be the most convenient for certain types of convergence, in which case the box topology may be preferred.

3. What is the significance of compactness in the context of product topology?

Compactness is an important concept in topology that describes a space where every open cover has a finite subcover. In the context of product topology, compactness allows for certain nice properties, such as the finite product of compact spaces being compact and the Tychonoff theorem, which states that the product of compact spaces is compact in the product topology.

4. Can a product of compact spaces be non-compact?

Yes, it is possible for a product of compact spaces to be non-compact. This can happen if the product space is not Hausdorff, meaning that there exist points that cannot be separated by disjoint open sets. In this case, the product may not satisfy the Tychonoff theorem and therefore not be compact.

5. How is the compactness of a product space related to the compactness of its individual spaces?

The compactness of a product space is closely related to the compactness of its individual spaces. As mentioned before, the Tychonoff theorem states that the product of compact spaces is compact in the product topology. Additionally, if one of the individual spaces is not compact, then the product space cannot be compact. However, the converse is not true, as a product of compact spaces can be non-compact if the product space is not Hausdorff.

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