On dense subspaces of countable pseudo character in function spaces

摘要

A set A subset of C-p(X) is uniformly dense in C-p(X) if, for any f is an element of C-p(X) and epsilon > 0, there is g is an element of A such that vertical bar g(x) - f (x)vertical bar < epsilon for all x is an element of X. We prove that C-p(X) has a uniformly dense subspace that condenses onto a second countable space if and only if C-p (X) has cardinality of the continuum. This answers a question of Tkachuk published in 2003. It is also proved that C-p(X) has a dense F-sigma-metrizable subspace if X is metrizable or Eberlein compact. If X is Corson compact or C-p(X) is a Lindelof Sigma-space, then it is possible to find a dense set Y subset of C-p(X) with countable pseudocharacter. (C) 2017 Elsevier B.V. All rights reserved.