Product of compact sets compact in box topology?

In summary, the Tychonoff theorem states that products of compact sets are compact in the product topology. However, this is not true for the box topology as shown by the counterexample \prod_{n\in \mathbb{N}}{[0,1]} where the set {S_n} is an open cover with no finite subcover. This can be further demonstrated by considering a cover by sets of the form \prod_{n\in \mathbb{N}}{A_i} where Ai=[0,0.6[ or Ai=]0.5,1].
  • #1
spicychicken
3
0
So Tychonoff theorem states products of compact sets are compact in the product topology.

is this true for the box topology? counterexample?
 
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  • #2
A counterexample is [itex]\prod_{n\in \mathbb{N}}{[0,1]}[/itex]. Can you show why?
 
  • #3
if S_n is the set with empty sets in each index except n where for index n you have [0,1], then {S_n} is an open cover with no finite subcover...i think
 
  • #4
spicychicken said:
if S_n is the set with empty sets in each index except n where for index n you have [0,1], then {S_n} is an open cover with no finite subcover...i think

Such a sets will always be empty. Try to consider a cover by all sets of the form

[tex]\prod_{n\in \mathbb{N}}{A_i}[/tex]

Where Ai=[0,0.6[ or Ai=]0.5,1]
 
  • #5


Yes, this statement is also true for the box topology. This can be proved using the Tychonoff theorem, which states that the product of compact spaces is compact in the product topology. Since the box topology is a finer topology than the product topology, any space that is compact in the product topology will also be compact in the box topology. Therefore, the product of compact sets will also be compact in the box topology.

As for a counterexample, it is not possible to find one as the statement is true. However, it is important to note that the box topology is not always the same as the product topology, so it is possible to find examples where a product of compact sets is not compact in the box topology. This is because the box topology takes into account all possible open sets in the product space, while the product topology only considers finite intersections of open sets.
 

1. What is the definition of a product of compact sets in box topology?

The product of compact sets in box topology is the set of all possible combinations of elements from each of the individual compact sets. It is defined as the Cartesian product of the compact sets, where each element in the product is a tuple consisting of one element from each of the individual sets.

2. How is the product of compact sets in box topology different from the product of compact sets in other topologies?

The product of compact sets in box topology is different from other topologies because it uses the box topology, which is defined by taking the product of open sets in each of the individual topologies. This results in a topology that is finer than the product topology and coarser than the Tychonoff topology.

3. What is the significance of compact sets being compact in the box topology?

When compact sets are compact in the box topology, it means that the product space is also compact. This is important because compactness implies that the product space is both closed and bounded, which allows for certain theorems and properties to be applied to the product space.

4. How does the box topology affect the convergence of sequences in the product of compact sets?

The box topology can affect the convergence of sequences in the product of compact sets by making it more difficult for sequences to converge. This is because the box topology is finer than the product topology, meaning that there are more open sets in the box topology, making it easier for sequences to escape from a particular set.

5. Can the product of compact sets in box topology be infinite-dimensional?

Yes, the product of compact sets in box topology can be infinite-dimensional. This is because the product of compact sets is defined as the Cartesian product of the individual sets, and there is no limit to the number of sets that can be included in a Cartesian product. Therefore, the product of compact sets in box topology can be infinite-dimensional.

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