Abstract
A subset G of Cp(X) - the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence - is a generator (resp., (0,≠0)-generator, or (0,1)-generator) if whenever a point x is not in a closed set C of X, there is a g in G such that g(x)∉g(C)‾ (resp., g(x)≠0 and g(C)⊆{0}, or g(x)=1 and g(C)⊆{0}).The motivation behind the questions: Which spaces have a first countable(0,≠0)-generator containing the zero function,0? Are they precisely the separable spaces? Which spaces have a second countable(0,≠0)-generator containing0? Are they precisely the cosmic spaces? is elucidated, and partial solutions presented.
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