In this paper we further investigate the results given in [7, 8, 9]. In Section 2 we consider space $X$ for which the closure of each countably compact (strongly countably compact, hypercountably compact) subspace of $X$ has countably compact (strongly countably compact, hypercountablycompact) property. In Section 3 we study some notions related to the classical concepts of being a Lindelöf, Menger or a Rothberger space.