Discrete generators (Q2116381)

A subset \(G\) of the space \(C(X)\) of all real continuous functions defined on a topological space \(X\) is called a generator if for each point \(x\) belonging to the exterior of some closed set \(C\) of \(X\) there exists a function \(g\) in \(G\) such that \(g(x)\) is in the exterior of image \(g(C)\). Every generator is considered as a topological subspace of \(C(X)\) endowed with the topology of pointwise convergence. Spaces possesing a discrete generator are characterised, mainly in terms of cardinal invariants. Interesting counterexamples are constructed and open conjectures are formulated as further problems in this area. This paper is a continuation of earlier investigations of authors, concerning spaces with a compact generators and spaces with a first or second countable generators.