**
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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