A space X is C-starcompact if for every open cover U of X, there exists a countably compact subset C of X such that St(C,U) = X. In this paper we investigate the relations between C-starcompact spaces and other related spaces, and also study topological properties of C-starcompact spaces.
A subspace Y of a space X is almost Lindelöf (strongly almost Lindelöf) in X if for every open cover U of X (of Y by open subsets V of X), there exists a countable subset V of U such that Y ⊆ S {V : V ∈ V }. In this paper we investigate the relationships between relatively almost Lindelöf subset and relatively strongly almost Lindelöf subset by giving some examples, and also study various properties of relatively almost Lindelöf subsets and relatively strongly almost Lindelöf subsets.