On normal dense subspaces of iterated function spaces (Q2217225)

This paper deals with spaces of the type \(C_{p,n}(X)\) where \(C_p(X) := C_{p,1}(X)\) is the space of continuous real-valued functions on a Tychonoff space \(X\) with the topology of pointwise convergence, and \(C_{p,n+1}(X) = C_p(C_{p,n}(X))\) for \(n \in \mathbb{N}\). The author resumes his study on \(C_p (X)\) made in his previous article [\textit{D. P. Baturov}, Math. Bull. 43, No. 4, 52--55 (1988; Zbl 0662.54010); translation from Vestn. Mosk. Univ., Ser. I 1988, No. 4, 63--65 (1988)], where he proved that ``if a Tychonoff space \(X\) is the union of \(\aleph_1\)-many compact \(\aleph_0\)-monolithic subspaces with countable tightness and \(Y\) is a normal dense subspace of \(C_p(X)\), then \(Y\) is collectionwise normal.'' The main results in this article are the following: \begin{itemize} \item[(1)] If a Tychonoff space \(X\) has a point-countable network of cardinality \(\leq \aleph_1\) and \(Y\) is a normal dense subspace of \(C_p(X)\), then \(Y\) is collectionwise normal. \item[(2)] If a Tychonoff space \(X\) has a point-countable network of cardinality \(\leq \aleph_1\), then \(C_p(C_p(X))\) also has a point-countable network of cardinality \(\leq \aleph_1\). \end{itemize} Some consequences of these theorems are the following: \begin{itemize} \item[(3)] Let \(X\) be a space, \(X = \bigcup_{\alpha \in \omega_1}X_\alpha\), where \(X_\alpha\) is \(\aleph_0\)-monolithic, \(t(X_\alpha) \leq \aleph_0\), and \(d(X_\alpha) \leq \aleph_1\) for every \(\alpha\). If \(n\) is odd and \(Y\) is a normal dense subspace of \(C_{p,n}(X)\), then \(Y\) is collectionwise normal. \item[(4)] If \(X\) is a Lindelöf space, \(w(X) \leq \aleph_1\), \(n\) is even, and \(Y\) is a normal dense subspace of \(C_{p,n}(X)\), then \(Y\) is collectionwise normal. \end{itemize} Definitions of terms that appear in this review can be found in [\textit{V. Arkhangel'skii}, Topological function spaces. Dordrecht: Kluwer Academic (1992)] and [\textit{R. Engelking}, General topology. Berlin: Heldermann Verlag (1989)].