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.

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.

Return to “Mathematics GRE Forum: The GRE Subject Test in Mathematics”

Users browsing this forum: lippyrogue, MathCat and 1 guest