Spaces embeddable as regular closed subsets into acc spaces and (a)-spaces

It is well known that some classes of spaces such as absolutely countably compact (abbreviated acc) spaces and (a)-spaces are not hereditary with respect to closed, and even regular closed subspaces. In this paper, we investigate conditions for spaces being regular closed embeddable into spaces with certain covering properties, in particular we characterize spaces which can be embedded as regular closed subsets into acc spaces and (a)-spaces. Some examples are presented as applications of the criteria. Two problems raised by Matveev [Topology Appl. 80 (1997) 169-175] are answered.