Topology: Clopen basis of a space

  • Thread starter talolard
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  • #1
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Homework Statement



So, I'm going through a proof and it is shamelessly asserted that the collection of clopen sets of [tex] {0,1}^{\mathbb{N}} [/tex] is a countable basis. Can anyone reasure me of this, point me in the direction of proving it.
Thanks
Tal
 

Answers and Replies

  • #2
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I assume you mean that [tex]\{0, 1\}[/tex] is to be given the discrete topology and then [tex]\{0, 1\}^\mathbb{N}[/tex] gets the product topology based on that.

Remember that the basic open sets of the product topology on a product [tex]\textstyle\prod_\lambda X_\lambda[/tex] are the the sets [tex]\textstyle\prod_\lambda U_\lambda[/tex] where each [tex]U_\lambda[/tex] is open in [tex]X_\lambda[/tex] and only finitely many of the [tex]U_\lambda[/tex] differ from [tex]X_\lambda[/tex].

How many finite subsets does [tex]\mathbb{N}[/tex] have?
 
  • #3
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Got it.
Thanks.
 

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