PROJECTIVE COVERS OF ORDINAL SUBSPACES

Gleason [G] introduced in 1958 the notion of projective cover for compact Hausdorff spaces. Later, Ponomarev [PI, P2, P3] and Strauss [S] independently obtained more general results. A detailed historical survey is given in [Wi2]. The projective cover, also known as the absolute, of a space X, denoted E(X), is characterized as that extremally dis connected space which is the perfect, irreducible preimage of X. Its existence and uniqueness are established in the references, above, for regular topological spaces. Many properties, particularly those of the compactness gender, are preserved from a regular topological space (here we assume all spaces to be completely regular) to its pro jective cover. Of import is that a space X has paracompact projective cover if and only if X is paracompact. The normality of a space is not necessarily preserved in it