On Frechet-Urysohn property

Abstract

In this article, it is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is a sequential a_{4}-space if and only if it is a strongly Fr'echet-Urysohn space, which deduces that if the quotient space G/H is a weakly first-countable space if and only if it is metrizable. Then, it is shown that every Fr'echet-Urysohn Hausdorff paratopological gyrogroup having the property (**) is a strong a_{4}-space, which deduces that every Fr'echet-Urysohn Hausdorff paratopological gyrogroup having the property (**) is a strongly Fr'echet-Urysohn space. Moreover, it is shown that if a Hausdorff paratopological gyrogroup having the property (**) is a sequential a_{4}-space, then it is a strongly Fr'echet-Urysohn space. Finally, we investigate the Fr'echet-Urysohn Hausdorff paratopological gyrogroup with an w^w-base and show that every Fr'echet-Urysohn Hausdorff paratopological gyrogroup having the property (**) with an w^w-base is first-countable.

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