Classification of selectors for sequences of dense sets of Cp(X)

Abstract

For a Tychonoff space X, Cp(X) is the space of all real-valued continuous functions with the topology of pointwise convergence. A subset A⊂X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following 8 properties for Cp(X) are considered. S1(S,S)⇒Sfin(S,S)⇒S1(S,D)⇒Sfin(S,D)⇑⇑⇑⇑S1(D,S)⇒Sfin(D,S)⇒S1(D,D)⇒Sfin(D,D) For example, a space X satisfies S1(D,S) (resp., Sfin(D,S)) if whenever {Dn:n∈N} is a sequence of dense subsets of X, one can take points xn∈Dn (resp., finite Fn⊂Dn) such that {xn:n∈N} (resp., ⋃{Fn:n∈N}) is sequentially dense in X. Other properties are defined similarly. S1(D,D) (=R-separability) and Sfin(D,D) (=M-separability) for Cp(X) were already investigated by several authors. In this paper, we have gave characterizations for Cp(X) to satisfy other 6 properties above. © 2018