A first countable linearly Lindelöf not Lindelöf topological space
Recall that a space is Lindelöf iff every open cover has a countable subcover. It is linearly Lindelöf iff every increasing open cover has a countable subcover. Under what circumstances must a linearly Lindelöf space be Lindelöf? In particular, must a first countable linearly Lindelöf space be Lindelöf? It is known that a first countable linearly Lindelöf space \(X\) has size \(\leq \mathfrak c\), and that a linearly Lindelöf space with netweight \(< \aleph_{\omega}\) is Lindelöf. Thus, if \(X\) is first countable linearly Lindelöf but not Lindelöf then \(\mathfrak c > \aleph_{\omega}\) and \(nw(X) \geq \aleph_{\omega}\). The main result of this paper is to show that under \(MA + \aleph_{\omega} < \mathfrak c\) there is a first countable linearly Lindelöf space which is not Lindelöf. The space is not difficult to describe, so we describe it here. Begin with the usual topology on the Cantor set \((C, \tau)\), together with a descending chain mod finite \(\{A_{\alpha}: \alpha < \aleph_{\omega}\} \subset \wp(\omega)\). Define \(\tau_n\) to be the topology on \(C\) generated by \(\tau\) together with all sets of the form \(U_S = \{x \in C: \exists \alpha < \aleph_n \;S =^* A_{\alpha}\) and \(\forall n \in S \;x(n) = 0\}\). Define \(C_n = \bigcup \{U_S: \exists \alpha < \aleph_n \;S =^* A_{\alpha}\}\) and \(\tilde C_n =\) the \(\tau\)-closure of \(C_n\). Then the desired space is \(P = \bigcup_{k < \omega} \prod_{i = 0}^{k}\tilde C_i \times \prod_{i = k+1}^{\infty}C_i\). Assuming the existence of \(\{A_{\alpha}: \alpha < \aleph_{\omega}\}\), \(P\) is first countable and not Lindelöf. That it is linearly Lindelöf depends on more of \(MA + \aleph_{\omega} < \mathfrak c\), but, as the author points out, the full strength of the hypothesis is not used. Another result is that, again under \(MA\), there is a first countable Michael space which refines the usual topology on the Cantor set. This example is obtained by tweaking an example of \textit{K. Alster} [Proc. Am. Math. Soc. 110, No.~2, 543--547 (1990; Zbl 0723.54026)] which was not first countable.
scientific article; zbMATH DE number 5956069