Borel Algebra Definition

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Borel Algebra Definition

Postby Hom » Sun Dec 02, 2012 12:08 pm

Quoted from wikipedia, "The Borel algebra on X is the smallest σ-algebra containing all open sets." What exactly is a smallest σ-algebra? Is it like {Ω, ∅}? If it has to contains all open sets, does it make that a powerset? If so, how come it is the smallest? I am so confused.

dexter
Posts: 39
Joined: Sun Oct 07, 2012 6:08 pm

Re: Borel Algebra Definition

Postby dexter » Sun Dec 02, 2012 12:36 pm

Smallest in this context refers to the smallest σ-algebra that contains the topological space X. It can't be X itself of course, because the axioms of σ-algebra automatically account for more elements that just the open sets. One can prove that such a σ-algebra exist easily: Rudin's Real and Complex Analysis is a good resource for this; in particular, page 12 has a pretty detailed treatment.

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: Borel Algebra Definition

Postby Hom » Mon Dec 03, 2012 8:36 am

Thanks.




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