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I have to show that;

infinite product of interiors is subset of interior of infinite products

or I have to give counterexample.

I know even equality is true for finite products in box or product topology and infinite products in box topology. Also I know the equality is false for infinite products in product topology but I do not know this inclusion is true or not.

Any help will be appreciated.

  • If I understood right your question, then one of the answers in this one includes a counterexample. – amrsa Apr 08 '17 at 21:50
  • Oh yes. (int(0,1))^w is not empty in product topology. So that it cannot be the subset of empty set. Im I right? @amrsa –  Apr 08 '17 at 22:40
  • Put another way, the boundary points of the infinite product are a subset of the infinite product of the boundary points. – Jacob Wakem Apr 09 '17 at 00:22
  • @GulcinB, yes I think i got it right. Basically, the example of Brian Scott is an instance of the following: if you have an infinite family of topological spaces, take an open set in each one, and if infinitely many of these open sets are not the full space (in that coordinate) then the product of those open sets is not open, and its interior is empty, an immediate consequence of the definition of product topology. – amrsa Apr 09 '17 at 09:03
  • okey I got that. if I get infinite product as (0,1)^w its interior will be empty set. However, is it enough for me to end this question? May I say (int(0,1))^w is not empty so that it cannot be a subset of right hand side? @amrsa –  Apr 09 '17 at 17:42
  • @GulcinB Of course $(\mathrm{int}(0,1))^{\omega} \neq \varnothing$. That's why I claimed that that answer has a counterexample. – amrsa Apr 09 '17 at 17:56
  • @amrsa Okey. thank you very much. –  Apr 09 '17 at 18:27

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Since interior is monotone increasing it is sufficient to show interior(Infinite product of interiors) is a subset of interior(interior(infinite product). But this is just interior(infinite product of interiors) is a subset of interior(infinite product). But since interior is monotone increasing this is true.

Jacob Wakem
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