On some generalizations of compactness in spaces \(C_p(X,2)\) and \(C_p(X,\mathbb Z)\) (Q1429077)

The authors discuss duality between properties of a Tychonoff space \(X\) and several generalizations of compactness of spaces \(C_p(X,\mathbb Z)\), \(C_p(X,{\mathbf 2})\) and \(C_p(X,{\mathbf n})\), \(n\in \mathbb N\). The results are compared with the corresponding known facts concerning the space \(C_p(X)\) of continuous real-valued functions on \(X\) with the pointwise topology. Two main results: (1) a zero-dimensional space \(X\) is an Eberlein compactum iff \(C_p(X,\mathbb Z)\) is \(\sigma\)-compact, (2) for a normal zero-dimensional space \(X\), \(C_p(X,{\mathbf 2})\) is \(\sigma\)-compact iff \(X\) is an Eberlein-Grothendieck compactum and the set \(X'\) of non-isolated points of \(X\) is an Eberlein compact space.