On separable sequential spaces and \(\sigma\)-hcp \(k\)-networks
Let \(X\) be a space, and \({\mathcal P}\) a cover of \(X\). \({\mathcal P}\) is said to be a \(k\)-network for \(X\) if \(K\subset U\) with \(K\) compact and \(U\) open, then \(K\subset \bigcup{\mathcal P}'\subset U\) for some finite subset \({\mathcal P}'\) of \({ \mathcal P}\). A space \(X\) is called an \(\aleph\)-space if it is a regular space with a \(\sigma\)-locally finite \(k\)-network. \textit{Chuan Liu} in [Adv. Math., Beijing 24, No. 6, 558-560 (1995; Zbl 0863.54023)] raised the following question: Is a separable regular \(k\)-space with a \(\sigma\)-hereditarily closure-preserving \(k\)-network an \(\aleph\)-space? In this paper the author proves that a separable regular \(k\)-space \(X\) with a \(\sigma\)-hcp \(k\)-network is an \(\aleph\)-space if the sequential order \(so(X)<\omega_1\). A space with a \(\sigma\)-locally finite \(k\)-network is a space with a \(\sigma\)-hereditarily closure-preserving \(k\)-network, and a space with a \(\sigma\)-hereditarily closure-preserving \(k\)-network is a space with \(\sigma\)-compact-finite \(k\)-network. Recently, M. Sakai showed that a separable regular \(k\)-space \(X\) with a \(\sigma\)-compact-finite \(k\)-network is an \(\aleph_0\)-space if \(so(X)<\omega_1\). And Liangxue Peng showed the a regular \(k\)-space \(X\) with a \(\sigma\)-hereditarily closure-preserving \(k\)-network is a meta-Lindelöf space, thus a separable regular \(k\)-space with a \(\sigma\)-hereditarily closure-preserving \(k\)-network is an \(\aleph\)-space. But it is still an open question whether a regular \(k\)-space with a \(\sigma\)-compact-finite \(k\)-network is a meta-Lindelöf space.
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