Some upper bounds for density of function spaces (Q1019138)

Let \(X, Y\) be topological spaces and let \({\alpha}\) be a hereditarily closed, compact network on \(X\). Let \(C_{\alpha}(X,Y)\) denote the set of all continuous mappings from \(X\) into \(Y\) endowed with the topology having subbase \(\{[A,V]: A \in {\alpha}\), \(V \subseteq Y\) is open\} where \([A,V]= \{f\in C(X,Y): f(A)\subseteq V\}\). If \(\alpha\) consists of all compact subsets of \(X\) then the corresponding function space is denoted by \(C_{k}(X,Y)\). A topological space \(Y\) is equiconnected if there is a continuous map \( \Psi:Y\times Y\times[0,1] \rightarrow Y\) such that \(\Psi(p,p,t)=p\), \(\Psi (p,q,0)=p\), \(\Psi(p,q,1)=q\) for every \(p,q\in Y\) and \(t\in[0,1]\). The map \(\Psi\) is called an equiconnecting function. A subset \(C\) of an equiconnected space \(Y\) is \(\Psi\)-convex if \(\Psi(C\times C\times[0,1])\subseteq C\). The \(i\)-weight \(iw(X)\) of a topological space \(X\) is the least of the cardinals \(w(Y)\) for Tychonoff spaces \(Y\) which are continuous one-to-one images of \(X\). As usual, \(d(X)\) denotes the density of \(X\). The authors show that if \(X\) is a Tychonoff space and \(Y\) is an equiconnected Hausdorff space with equiconnecting function \(\Psi\) such that \(Y\) has a base consisting of \(\Psi\)-convex sets then \(d(C_{\alpha} (X,Y))\leq iw(X)\cdot d(Y)\). For \(\alpha\), \(X\) and \(Y\) satisfying some additional properties, various strengthenings of this result are obtained. The authors leave open the problem whether the condition that \(Y\) has a base consisting of \(\Psi\)-convex sets can be removed or not. On the other hand, it is shown that for each infinite cardinal \(\kappa\) there is a pathwise connected space \(Y\) such that the \(\pi\)-weight of \(Y\) is \(\kappa\) but the Souslin number of the space \(C_{k} ([0,1],Y)\) is \(2^{\kappa}\). Thus in the above mentioned result, the assumption that \(Y\) is equiconnected cannot be weakened to \(Y\) is pathwise connected.