On discretely generated box products (Q306117)

A topological space \(X\) is called discretely generated if for any \(A\subseteq X\) and \(x\in \overline{A}\) there exists a discrete set \(D\subseteq A\) such that \(x\in \overline{D}\).NEWLINENEWLINEIn this paper, two problems 3.19 and 3.3 from [\textit{V. V. Tkachuk} and \textit{R. G. Wilson}, Topology Appl. 159, No. 1, 272--278 (2012; Zbl 1236.54005)], are solved, namely: NEWLINENEWLINEProblem 3.19. Does the space \(\left\{ \xi \right\}\cup \omega \) embed into a box product of real lines when \(\xi \in \beta \omega \backslash \omega \)? NEWLINENEWLINEFor any \(\xi \in \beta \omega \backslash \omega \), this is answered in the negative. Theorem 8. There is no embedding \(\phi :\left\{ \xi \right\}\cup \omega \to \square {{\mathbb R}^{\omega }}\). Corollary 9. For any cardinal \(\kappa\), there is no embedding from \(\left\{ \xi \right\}\cup \omega \) to \(\square {{\mathbb R}^{\kappa}}\). NEWLINENEWLINEProblem 3.3. Is any box product of first countable spaces discretely generated? NEWLINENEWLINEThis is answered positively by assuming that the spaces are regular. Theorem 11. Suppose \(\left\{ {{X}_{t}}:t\in T \right\}\) is a family of regular first countable spaces. Then \({{\square}_{t\in T}}{{X}_{t}}\) is discretely generated.