Glasnik Matematicki, Vol. 34, No.1 (1999), 23-35.
THE EXTRARESOLVABILITY OF SOME FUNCTION SPACES
O. T. Alas, S. Garcia-Ferreira and A. H. Tomita
Departamento de Matematica, Instituto de Matematica e Estatistica,
Universidade de Sao Paulo, Caixa Postal 66281, CEP 05315-970,
Sao Paulo, Brasil
e-mail: alas@ime.usp.br
Instituto de Matematica, Ciudad Universitaria (UNAM), D.F. 04510
Mexico, Mexico
e-mail: agarcia@servidor.unam.mx
Departamento de Matematica, Instituto de Matematica e Estatistica,
Universidade de Sao Paulo, Caixa Postal 66281, CEP 05315-970,
Sao Paulo, Brasil
e-mail: tomita@ime.usp.br
Abstract. A space X is said to be extraresolvable
if X contains a family
D of dense subsets such
that the intersection of every two elements of
D is nowhere dense and
|D| >
Δ(X),
where Δ(X)
= min{|U| : U is a nonempty open subset of X}
is the dispersion character of X. In this paper, we study
the extraresolvability of some function spaces
Cp(X) equipped with the pointwise
convergence topology. We show that
Cp(X)
is not extraresolvable provided that X satisfies one of the
following conditions: X is metric; nw(X)
= ω; X is
normal; e(X) = nw(X) and either
e(X) is attained or cf(e(X))
is countable. Hence,
Cp(R)
and Cp(Q)
are not extraresolvable. We establish the equivalences
2ω < 2ω1 iff
Cp([0,ω1)) is
extraresolvable; and, under GCH, for every infinite cardinal
κ, the space
Cp([0,κ))
is extraresolvable iff cf(κ) >
ω,
where [0,κ) has the
order topology. We also prove that if
κcf(κ) = κ
and cf(κ) > ω,
then Cp({0,1}κ)
is extraresolvable; and that
Cp(β(κ))
is extraresolvable, for every infinite cardinal
κ with the discrete
topology. It is shown that
Cp([0,βω1))
is extraresolvable, where
βω1
is the beth cardinal corresponding to
ω1.
Under GCH, for a compact space X, we have that
cf(w(X)) >
ω iff
Cp(X) is extraresolvable. We proved
that 2ω < 2ω1
is equivalent to the statement
"Cp({0,1}ω1)
is strongly extraresolvable".
1991 Mathematics Subject Classification.
54A35, 03E35, 54A25.
Key words and phrases. Extraresolvable,
κ-resolvable.
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