On certain generalizations of countably compact spaces and Lindelöf spaces (Q2762666)

The authors investigate some generalizations of countably compact spaces and Lindelöf spaces which were introduced by \textit{C. M. Pareek} [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 24, 169-192 (1990; Zbl 0727.54019)] and studied by \textit{L. D. Kočinac} [Mat. Vesn. 44, No. 1-2, 33-44 (1992; Zbl 0795.54002)]. They point out that some of the results in these papers are false and provide corrected versions of them. Additionally they prove the following results concerning feebly compact spaces (which, for completely regular \(T_1\)-spaces, are known to coincide with pseudocompact spaces):NEWLINENEWLINENEWLINETheorem 1: Every weakly normal feebly compact \(T_1\)-space \(X\) is almost starcompact, i.e. has the property that for every open cover \({\mathcal U}\) of \(X\) there exists an \(A\in[X]^{<\omega}\) such that \(X=\text{cl} St(A,{\mathcal U})\). Theorem 2: Every metacompact almost starcompact space is feebly compact. NEWLINENEWLINENEWLINETheorem 3: Every quasi-regular almost starcompact space is feebly compact. NEWLINENEWLINENEWLINEA topological space \(X\) is called \(\omega\)-starcompact (respectively \(\omega\)-starLindelöf) if for every open cover \({\mathcal U}\) of \(X\) there exists an \(A\in [X]^{<\omega}\) (respectively an \(A\in [X]^{\leq\omega})\) and an \(n< \omega\) such that \(X=St^n(A, {\mathcal U})\).NEWLINENEWLINENEWLINETheorem 4: Every normal \(\omega\)-starcompact space is \(\omega_0\)-compact.NEWLINENEWLINENEWLINETheorem 5: Every \(\aleph_1\)-collectionwise normal \(\omega\)-star Lindelöf space is \(\omega_1\)-compact.