Notes on the strong Pytkeev property in paratopological groups

Abstract

In this paper, we discuss certain strong Pytkeev property in topological spaces, and prove the following results:

(1)\, a Hausdorff topological space with the strong Pytkeev property is sequential if it is a $k$-space, which gives an affirmative answer to the question proposed by F.C. Lin, A. Ravsky and J. Zhang \cite[Question 4.3]{linf};

(2)\, a regular $\kappa$-Fr\'{e}chet-Urysohn paratopological group with countable $sp^*$-character is submetrizable, and if it still has the property ($\circledast$), then it is first-countable, which gives a partial answer to the question proposed by Z.Y. Cai, P.Q. Ye, S. Lin and B. Zhao \cite[Question 2.8]{Cai}.