LAUSR

Given a sequence $${\mathcal{U} =\{U_n: n \in \omega\}}$$U={Un:n∈ω} of non-empty open subsets of a space X, a set $${\{x_n : n \in \omega\}}$${xn:n∈ω} is a selection of $${\mathcal{U}}$$U if $${x_n \in U_n}$$xn∈Un for every $${n \in \omega}$$n∈ω. We show that a space X is uncountable if and only if every sequence of non-empty open subsets of Cp(X) has a closed discrete selection. The same statement is not true for $${C_p(X,[0,1])}$$Cp(X,[0,1]) so we study when the above selection property (which we call discrete selectivity) holds in $${C_p(X,[0,1])}$$Cp(X,[0,1]). We prove, among other things, that $${C_p(X, [0,1])}$$Cp(X,[0,1]) is discretely selective if X is an uncountable Lindelöf $${\Sigma}$$Σ-space. We also give a characterization, in terms of the topology of X, of discrete selectivity of $${C_p(X,[0,1])}$$Cp(X,[0,1]) if X is an $${\omega}$$ω-monolithic space of countable tightness.