Recently B. M. P. Nayar has shown that C-compact spaces are characterized by the property that each function on the space with a strongly-subclosed inverse is a closed function. In this paper we study spaces characterized by the following two properties:\smallskip (1) Each

continuous function from the space maps closed subsets onto sequentially

closed subsets.\smallskip (2) Each function on the space with a strongly-subclosed

inverse maps closed subsets onto sequentially closed subsets.

\smallskip\noindent It is shown that each countably compact space 

satisfies (1) and (2), that each space satisying (2) satisfies (1), and

that any space satisfying (1) is sequentially H-closed in the sense of 

Ovsepian. Moreover, it is proved that any sequential topology on a space $X$ satisfying (1) or (2) is minimal in the class of sequential topologies on $X$. However, a space with a minimal sequential topology need not 

satisfy (1). Characterizations of spaces satisfying

(1) and (2) are obtained using filterbases, sequences, a collection of 

subsets containing the family of H-sets of Veli\v cko, and relations satisfying certain graph conditions, and the question of how they are related to some known spaces is investigated.