If \(K\) is Gul'ko compact, then every iterated function space \(C_{p,n}(K)\) has a uniformly dense subspace of countable pseudocharacter (Q1799780)

In this work all topological spaces under consideration are assumed to be Tychonoff. For any spaces \(X\) and \(Y\), \(C_p(X,Y)\) denotes the space of continuous maps from \(X\) to \(Y\) endowed with the topology of pointwise convergence. When \(Y=\mathbb{R}\), the function space \(C_p(X,\mathbb{R})\) is denoted by \(C_p(X)\). The function space \(C(X)\) endowed with the uniform convergence topology is denoted by \(C_u(X)\). Iterated function spaces are defined as follows: \(C_{p,0}(X) =X\) and \(C_{p,n+1}(X)=C_p( C_{p,n}(X)) \) for each \(n\in \mathbb{N}\). A set \(A\subset C_p(X)\) is called uniformly dense in \(C_p(X)\) if it is dense in \(C_u(X)\). The authors prove that \(C_p(X)\) has a dense subspace of countable \(i\)-weight if and only if \(d(C_p(X))\leq \mathfrak{c}\). In this work it is also proved that \(C_p(X)\) has a dense \(F_\sigma\)-discrete subspace of cardinality \(\kappa =d(C_p(X))\), whenever \(X\) is a Corson compact space. A compact space is Corson compact if it embeds in \(\{x\in \mathbb{R}^A:x^{-1}(\mathbb{R}\setminus \{ 0\} ) \mathrm{~is ~ countable} \}\subset \mathbb{R}^A\), for some set \(A\). A space \(X\) is called Lindelöf \(\Sigma\) if there exists a space \(Y\) which maps continuously onto \(X\) and perfectly onto a second countable space. A compact space \(X\) is called Gul'ko compact if \(C_p(X)\) is a Lindelöf \(\Sigma\)-space. The main result in this paper is Theorem 3.12 which answers a question of the first author [Topology Appl. 221, 59--68 (2017; Zbl 1382.54006)]. It was proved in Theorem 3.12 that if \(X\) is Gul'ko compact, then \(C_{p,n}(X)\) has a uniformly dense subspace of countable pseudocharacter for all \(n\in \mathbb{N}\). It is worth mentioning that this result is new even for \(C_p(K)\) given that \(K\) is Eberlein compact. A list of open questions is included in the last section of this paper.