Some dense sets of the Tychonoff cube \(I^c\) (Q2087780)

This paper presents these two interesting results about the existence of certain countable dense subsets of \(I^{\mathfrak{c}}\). \begin{itemize} \item[(I)] There is a countable dense subset \(D\) of \(I^{\mathfrak{c}}\) with these two properties: \begin{itemize} \item[(\(\star\))] \(D\) contains a discrete nowhere dense subset \(E\) such that \(cl_{ I^{\mathfrak{c}}}E\) is homeomorphic to \(\beta\omega\) and \item[(\(\star\star\))] for every nontrivial convergent sequence \(s \subseteq I\) where \(x_s\) is the limit of the sequence in \(I\), there is \(\alpha \in \mathfrak{c}\) such that for the projection \(\pi_{\alpha}: I^{\mathfrak{c}}\rightarrow I_{\alpha}\) \(\pi_{\alpha}[E] = s\) and \(\pi_{\alpha}[cl_{ I^{\mathfrak{c}}}E\backslash E] = s \cup \{\pi_{\alpha}(x_s)\}\). \end{itemize} \item[(II)] There is a countable dense subset \(D\) of \(I^{\mathfrak{c}}\) with this property: every countable set \(E \subseteq D\) contains a discrete nowhere dense subset \(E'\) with this property: for every infinite compact metrizable space \(X\), there is a countable subset \(S \subseteq \mathfrak{c}\) with the property that for the projection \(\pi_S:I^{\mathfrak{c}} \rightarrow I^{S}\), \({\pi_S}[cl_{I^{\mathfrak{c}} }E']\) is homeomorphic to \(X\). \end{itemize}