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.


Full text (PDF) (free access)
Glasnik Matematicki Home Page