On product bases
John Morgan · Pacific Journal of Mathematics · 1982
II measure in a σ-finite measure space.EXAMPLE ID.The family of all perfect sets in a complete separable metric space with no isolated points.It is the inclusion of this example which led to the cardinality restriction in Axiom 2.With respect to a given category base (X, ^) we define the generalized Baire category concepts for subsets of X. DEFINITION.A set S is singular if every region contains a subregion which is disjoint from S. A countable union of singular sets is called meager set.A set which is not a meager set is called an abundant set.An abundant set whose complement is a meager set is called a comeager set.NOTATION.The family of all sets which are meager with respect to a given category base (X, ^) will be denoted by THEOREM 1.1.The intersection of any two regions either contains a region or it is a singular set.EXAMPLE IE.If ^ is the family of all complements of finite subsets of an uncountable set then the singular sets, meager sets, and abundant sets coincide with the finite, countable, and uncountable sets, respectively.EXAMPLE IF.If (X, ^) is a topology then the singular, meager, and abundant sets are the nowhere dense sets, sets of the first category, and the sets of the second category, respectively.EXAMPLE lG.Let ^ denote the family of all measurable sets of positive measure in a σ-finite measure space (X, Szf, μ) and let μ denote the completion of μ.The families of singular sets and meager sets with respect to ^ are identical and coincide with the sets of /^-measure zero.The abundant sets coincide with the sets of positive μ-outer measure.EXAMPLE lH.If ^ is the family of all perfect sets in a complete separable metric space with no isolated points then the families of singular sets and of meager sets are identical and coincide with the sets having Marczewski's property (s° 2 ) (see [12]).THEOREM 1.2.The family of all singular sets forms an ideal and the family of all meager sets forms a σ-ideal.