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Countably infinite products of sequential topologies. (English) Zbl 0991.54028
First countability is preserved by countable productivity. But the productivity of weakly first countability is bad. A property \({\mathcal P}\) is said to be almost countably productive provided that \(\prod_{k=1}^n X_k\in{\mathcal P}\) for every \(n\in \omega\) implies that \(\prod_{k=1}^\infty X_k\in{\mathcal P}\). Clearly, if a property is preserved by limits of inverse sequences, then it is almost countably productive. In this paper, the authors investigate the almost countable productivity of some properties related to convergent sequences by the limits of inverse sequences of topological spaces. The main result is that the limit of an inverse sequence of sequentially compact sequential topologies is sequential of order not greater than the supremum of the sequential orders of the topologies of the sequence \(+1\); if moreover the topologies are \(\alpha_3\), then the order is equal to that supremum. This implies almost countable productivity of the considered properties.
Reviewer: Shou Lin (Fujian)
54D55 Sequential spaces
54B10 Product spaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)