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The sequentiality is equivalent to the $${\mathcal F}$$-Fréchet-Urysohn property. (English) Zbl 0696.54020
Some relations between sequentiality, $${\mathcal F}$$-Fréchet-Urysohn property and $${\mathcal F}$$-sequentiality are shown. Theorem 1. There exists a filter $${\mathcal F}$$ on $$\omega$$ such that every Hausdorff sequential space is $${\mathcal F}$$-Fréchet-Urysohn. Example 1, [CH]. For every ultrafilter p on $$\omega$$ which is a P-point on $$\omega^*$$ there exists a compact Franklin’s space whose index of p-sequentiality equals 2. Example 2, [CH]. There exists a sequential compact space which is p- Fréchet-Urysohn for no P-point $$p\in \omega^*$$. Theorem 2. Every compact Franklin’s space is p-Fréchet-Urysohn for no P-point $$p\in \omega^*$$.

##### MSC:
 54D55 Sequential spaces 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 54A35 Consistency and independence results in general topology
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