#### 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
*C*_{p}(*X*) equipped with the pointwise
convergence topology. We show that
*C*_{p}(*X*)
is not extraresolvable provided that *X* satisfies one of the
following conditions: *X* is metric; *n**w*(*X*)
= ω; *X* is
normal; *e*(*X*) = *n**w*(*X*) and either
*e*(*X*) is attained or *c**f*(*e*(*X*))
is countable. Hence,
*C*_{p}(**R**)
and *C*_{p}(**Q**)
are not extraresolvable. We establish the equivalences
2^{ω} < 2^{ω1} iff
*C*_{p}([0,ω_{1})) is
extraresolvable; and, under GCH, for every infinite cardinal
κ, the space
*C*_{p}([0,κ))
is extraresolvable iff *c**f*(κ) >
ω,
where [0,κ) has the
order topology. We also prove that if
κ^{cf(κ)} = κ
and *c**f*(κ) > ω,
then *C*_{p}({0,1}^{κ})
is extraresolvable; and that
*C*_{p}(β(κ))
is extraresolvable, for every infinite cardinal
κ with the discrete
topology. It is shown that
*C*_{p}([0,β_{ω1}))
is extraresolvable, where
β_{ω1}
is the beth cardinal corresponding to
ω_{1}.
Under GCH, for a compact space *X*, we have that
*c**f*(*w*(*X*)) >
ω iff
*C*_{p}(*X*) is extraresolvable. We proved
that 2^{ω} < 2^{ω1}
is equivalent to the statement
"*C*_{p}({0,1}^{ω1})
is strongly extraresolvable".

**1991 Mathematics Subject Classification.**
54A35, 03E35, 54A25.

**Key words and phrases.** Extraresolvable,
κ-resolvable.

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