Sigma-fragmentability of mappings into \(C_p(K)\)
Let \(T\) be a space with topology \(\tau\) and \(\rho\) a metric on \(T\) which need not be compatible with \(\tau\). We say that \(T\) is fragmented by \(\rho\) down to \(\varepsilon\) for some \(\varepsilon>0\) in case each nonempty subset of \(T\) has a relatively \(\tau\)-open subset with \(\rho\)-diameter \(< \varepsilon\). We say that \(T\) is \(\sigma\)-fragmented by \(\rho\) in case, for every \(\varepsilon >0\), we can write \(T=\bigcup \{T_n:1 \leq n<\infty\}\) where each \(T_n\) is fragmented by \(\rho\) down to \(\varepsilon\). Let \(K\) be a compact Hausdorff space and \(C_p(K)\) the Banach space of continuous real-valued functions on \(K\) with the topology of pointwise convergence, where \(\rho\) denotes the sup norm. The authors study various conditions for subsets \(T\) of \(C_p(K)\) to be \(\sigma\)-fragmented by \(\rho\). One of these conditions is that \(T\) be countably separated, that is, there exists a countable family \(U\) of open subsets of \(C_p(K)\) such that each point of \(T\) can be separated from each point of \(C_p(K) \smallsetminus T\) by some member of \(U\). We say that \(K\) has property \(\Sigma\) in case \(C_p(K)\) is \(\sigma\)-fragmented by \(\rho\). The authors show that if compact spaces \(K\), \(L\) have property \(\Sigma\), then so does \(K\times L\). We say that \(K\) has property \(N^*\) in case, for any continuous mapping \(\varphi\) from a Baire space \(B\) into \(C_p(K)\), the set of points where \(\varphi\) is continuous in the norm topology on \(C_p(K)\) is dense in \(B\). The authors show that if \(C_p(K)\) embeds in \(C_p(L)\) and \(L\) has property \(N^*\), then so does \(K\). A consequence of this result is that \(C_p(\beta N)\) does not embed in \(C_p(L)\) where \(L\) is any power of \([0,1]\).
scientific article; zbMATH DE number 1307926