The Property of Baire

  • John C. Oxtoby
Part of the Graduate Texts in Mathematics book series (GTM, volume 2)


The operation of symmetric difference, defined by
$$A\Delta B = (A \cup B) - (A \cap B) = (A - B) \cup (B - A).$$
is commutative, associative, and satisfies the distributive law A ∩ (B △ C) = (A ∩ B) △ (A ∩ C). Evidently, A △ B ⊂ A ∪ B and \(A\,\Delta \,A = \emptyset\). It is easy to verify that any class of sets that is closed under △ and ∩ is a commutative ring (in the algebraic sense) when these operations are taken to define addition and multiplication, respectively. Such a class is also closed under the operations of union and difference. It is therefore a ring of subsets of its union, as this term was defined in Chapter 3.


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Copyright information

© Springer-Verlag New York 1971

Authors and Affiliations

  • John C. Oxtoby
    • 1
  1. 1.Bryn Mawr CollegeBryn MawrUSA

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