Co-Absolutes with Homeomorphic Dense Subspaces

Williams, Scott W.

Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 857-861

Résumé

Recall that the absolute ∈(X) of a regular space X is the unique (up to a homeomorphism) extremally disconnected space whose image is X under a perfect irreducible map. X and Y are co-absolute whenever ∈(X) and ∈(Y) are homeomorphic. Completely regular spaces X and Y are weakly co-absolute whenever βX and βY are co-absolute. For a survey of this area we suggest [6] and [8].In this paper we proveTHEOREM 1. Suppose, for i ∈ {0, 1};, X(i) is a compact connected linearly ordered space. Then X(0) and X(l) are co-absolute if, and only if, X(0) and X(l) have homeomorphic dense sets.Making use of Theorem 1 and a result from [7] we give Theorem 2, a cardinal generalization ofCOROLLARY 1. Suppose for each i ∈ {0, 1};, X(i) is a Čech-complete space with a Gδ-diagonal.

Détail

DOI : 10.4153/CJM-1981-066-1

Bibliographie

[1] 1. Frolick, Z., Generalizations of the Gh-property of complete metric spaces, Cech. J. Math. 85 (1960), 359–378. Google Scholar

[2] 2. Gates, C., A study of remote points of metric spaces, Ph.D. Dissertation, University of Kansas (1977). Google Scholar

[3] 3. Hager, A. W., Isomorphisms of some completions of C(X), to appear in Topology Proceedings vol. 4. Google Scholar

[4] 4. Lutzer, D. J., A metrization theorem for linearly ordered spaces, Proc. A.M.S. 22 (1969), 557–558. Google Scholar

[5] 5. Maharam, D. and Stone, A. H., Category algebras of complete metric spaces, to appear in Aust. J. Math. Google Scholar

[6] 6. Ponomarev, V. I. and Sapiro, L., Absolutes of topological spaces and their continuous maps, Russ. Math. Surveys 31 (1976), 138–154. Google Scholar

[7] 7. Williams, S. W., Trees, Gleason spaces, and co-absolutes of (βN ∼ N, preprint. Google Scholar

[8] 8. Woods, R. G., A survey of absolutes of topological spaces, to appear. Google Scholar