Using BULL'S concept of an a-paracompact subset, the various axioms of separation and embeddings, we define in this paper some classes of topological spaces: the classes T*(Ti) for each i = 2,3,3a, 4,5,5a and r*(Tr) for each j = 4,5,5a. (Throughout this paper a T , space is a regular To space, a Tap space is a completely regular To space, a T4 space is a normal T I space, a T, space is a hereditarily normal T , space, a TSp space is a perfectly normal To space, a Ti space is a collectionwise normal T , space, a T: space is a hereditarily collectionwise normal T, space and a T& space is a space in which all closed subsets are These clwes allow us to give characterizations of compact, LINDELOB and paracompact spaces in the realm of T, spaces. IR fact: "& TS space is compact if and only if it is a-paracompact in each TSa space in which it is embedded as a closed subset" (Proposition 1.2), "a T4 space is LWDELOB if and only if it is a-paracompact in each T4 space in which is embedded as a closed subset" (Proposition 1.11) and "a collectionwise normal TI space is paracompact if and only if it is a-paracompact in each collectionwise normal TI space in which is embedded as a cloaed subset" (Proposition 1.16). Analogously, we define in Section 2 of this paper the classes II*(Ti) (for each 1 = 2, 3,3a, 4,5, 5a) and n*(q) (for each j = 4,5,5a) related to ! I ~W ~S I I Y ' S class IT*.
(Cf.
[a] II)).
The classes P ( T 1 ) and P ( T j )
C. E. BULL defined in [l] the notion of an a-paracompact subset. A subset E of a topological space X is said to be a-paracompact in X if every covering of E by open subsets of X has a refinement by open subseta of X which is locally finite in X and which covers E. a-paracompact subsets have been studied in [l] and [3]. Using this *) The resulfa in this paper are contained in the author's Doctoral Thesis, written under the dmction of Professor E. Outerelo.