Product of compact sets compact in box topology?

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Discussion Overview

The discussion centers on whether the product of compact sets remains compact in the box topology, contrasting it with the product topology as stated in Tychonoff's theorem. Participants explore potential counterexamples and the implications of their findings.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant states that Tychonoff's theorem indicates that products of compact sets are compact in the product topology and questions if this holds true for the box topology.
  • Another participant proposes the product \(\prod_{n\in \mathbb{N}}{[0,1]}\) as a counterexample to the claim of compactness in the box topology.
  • A participant suggests that a specific open cover, defined by sets \(S_n\) where all indices except one contain empty sets, demonstrates that there is no finite subcover, implying a lack of compactness.
  • Further clarification is provided regarding the nature of the sets \(S_n\) and a challenge is issued to consider covers formed by sets of the form \(\prod_{n\in \mathbb{N}}{A_i}\) with specified intervals.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are competing views regarding the validity of the proposed counterexample and the nature of the open cover.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the nature of the sets involved and the specific conditions under which the counterexample is evaluated. The implications of the box topology versus the product topology remain unresolved.

spicychicken
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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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A counterexample is [itex]\prod_{n\in \mathbb{N}}{[0,1]}[/itex]. Can you show why?
 
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
 
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]
 

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