ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXIII, 1 (2004)
p. 127 138

A Generalization of Baire Category in a Continuous Set
V. Berinde
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 {\it "large"} or {\it "small"} sets in linearly ordered continuous sets. In particular, for cardinal number $\kappa$, the continuous ordered set $\Tks$ a subset of the set of dyadic sequences of length $\kappa$ is discussed. We establish that this space, and its Cartesian square is not the union of $\cf (\kappa)$ many nowhere dense sets. Further we provide comparative results between Baire category in $\re$ and ``generalized Baire category" in $\Tks$ as well as some of the significant differences concerning Baire category in $\re$ and $\kappa$-category in $\ds{ \Wks}$. For example we have shown that a residual set in $\ds{ \Wks}$ need not contain a perfect set and that there exist perfect sets of cardinality $|\ds{ {}^{<\kappa} 2_{*}}|$.

AMS Subject classification:  54H25;   Secondary:  03E04.
Keywords:  Generalization of Baire category, ordered sets, $\eta_\alpha$-sets, Dedekind complete sets, continuous sets.

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