Two comments concerning certain topological spaces (Q1286758)

\textit{D. Janković} and \textit{Ch. Konstadilaki} showed [Quest. Answers Gen. Topology 11, No. 1, 87-92 (1993; Zbl 0810.54022)] that \(\omega^*\) contains \(\beta\omega\) so that some regular closed cover of \(\beta\omega\) in \(\omega^*\) has no finite subcover of \(\beta\omega\). The present paper substitutes \(\omega^*\) by \(\beta\omega\times (\omega_0+1)\) endowed with a special topology (in fact more general spaces than \(\beta\omega\) are considered). Finally, the spaces where every Lindelöf subset is closed are characterized as those having that property locally using closed neighborhoods (an example is given that in Hausdorff spaces one cannot change closed neighborhoods to neighborhoods).