Abstract:Some strong versions of the Fr\'echet-Urysohn property are introduced and studied. We also strengthen the concept of countable tightness and generalize the notions of first-countability and countable base. A construction of a topological space is described which results, in particular, in a Tychonoff countable Fr\'echet-Urysohn space which is not first-countable at any point. It is shown that this space can be represented as the image of a countable metrizable space under a continuous pseudoopen mapping. On the other hand, if a topological group $G$ is an image of a separable metrizable space under a pseudoopen continuous mapping, then $G$ is metrizable (Theorem 5.6). Several other applications of the techniques developed below to the study of pseudoopen mappings and intersections of topologies are given (see Theorem 5.17).
Keywords: first-countable, Fr\'echet-Urysohn, countably compact, closure-sensor, topological group, strong FU-sensor, pseudoopen mapping, side-base, $\omega $-Fr\'echet-Urysohn space
AMS Subject Classification: 54D20 54G20 54J99