A Generalization of Baire Category in a Continuous Set
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چکیده
The following discusses a generalization of Baire category in a continuous set. The objective is to provide a meaningful classification of subsets of a continuous set as “large” or “small” sets in linearly ordered continuous sets. In particular, for cardinal number κ, the continuous ordered set 2∗ a subset of the set of dyadic sequences of length κ is discussed. We establish that this space, and its Cartesian square is not the union of cf(κ) many nowhere dense sets. Further we provide comparative results between Baire category in R and “generalized Baire category” in 2∗ as well as some of the significant differences concerning Baire category in R and κ-category in 2∗. For example we have shown that a residual set in 2∗ need not contain a perfect set and that there exist perfect sets of cardinality |2∗|.
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