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The extraresolvability of some function spaces. (English) Zbl 0927.54004

Summary: A space \(X\) is said to be extraresolvable if \(X\) contains a family \({\mathcal D}\) of dense subsets such that the intersection of every two elements of \({\mathcal D}\) is nowhere dense and \(|{\mathcal D}| >\Delta(X)\), where \(\Delta(X)= \min\{| U|:U\) is a nonempty open subset of \(X\}\) is the dispersion character of \(X\). The authors study the extraresolvability of some function spaces \(C_p(X)\) equipped with the pointwise convergence topology. They show that \(C_p(X)\) is not extraresolvable provided that \(X\) satisfies one of the following conditions: \(X\) is metric; \(nw(X)=\omega\); \(X\) is normal, \(e(X)=nw(X)\) and either \(e(X)\) is attained or \(cf(e(X))\) is countable. Hence, \(C_p(\mathbb{R})\) and \(C_p(\mathbb{Q})\) are not extraresolvable. The authors establish the equivalences \(2^\omega<2^{\omega_1}\) iff \(C_p([0,\omega_1))\) is extraresolvable; and, under GCH, for every infinite cardinal \(\kappa\), the space \(C_p([0,\kappa))\) is extraresolvable iff \(cf(\kappa)>\omega\), where \([0, \kappa)\) has the order topology. They also prove that if \(\kappa^{<cf (\kappa)}=\kappa\) and \(cf(\kappa) >\omega\), then \(C_p(\{0,1\}^\kappa)\) is extraresolvable; and that \(C_p(\beta (\kappa))\) is extraresolvable, for every infinite cardinal \(\kappa\) with the discrete topology. It is shown that \(C_p([0, \beta_{\omega_1}))\) is extraresolvable, where \(\beta_{\omega_1}\) is the beth cardinal corresponding to \(\omega_1\). Under GCH, for a compact space \(X\), they have that \(cf(w(X))>\omega\) iff \(C_p(X)\) is extraresolvable. They prove that \(2^\omega< 2^{\omega_1}\) is equivalent to the statement “\(C_p(\{0,1\}^{\omega_1})\) is strongly extraresolvable”.

MSC:

54A35 Consistency and independence results in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E35 Consistency and independence results

Keywords:

extraresolvable
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