Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
monotonically normal; compactness; linear ordered spaces
monotonically normal; compactness; linear ordered spaces
Summary:
For a compact monotonically normal space X we prove: \, (1) \, $X$ has a dense set of points with a well-ordered neighborhood base (and so $X$ is co-absolute with a compact orderable space); \, (2) \, each point of $X$ has a well-ordered neighborhood $\pi $-base (answering a question of Arhangel'skii); \, (3) \, $X$ is hereditarily paracompact iff $X$ has countable tightness. In the process we introduce weak-tightness, a notion key to the results above and yielding some cardinal function results on monotonically normal spaces.
References:
[Ar1] Arhangel'skii A.V.: Structure and classification of topological spaces and cardinal invariants. Russian Math. Surv. 33 (1977), 33-95. MR 0526012
[Ar2] Arhangel'skii A.V.: Some properties of radial spaces. Math. Zametki 27 (1980), 95-104. MR 0562480
[BR] Balogh Z., Rudin M.E.: Monotone Normality. Topology and Appl. 47 (1992), 115-127. MR 1193194 | Zbl 0769.54022
[Bo1] Borges C.J.R.: On stratifiable spaces. Pacific J. Math. 17 (1966), 1-16. MR 0188982 | Zbl 0175.19802
[Bo2] Borges C.J.R.: Direct sums of stratifiable spaces. Fundamenta Math. 100 (1978), 95-99. MR 0505891 | Zbl 0389.54018
[En] Engelking R.: General Topology. Polish Scientific Publishers, 1977. MR 0500780 | Zbl 0684.54001
[Gr] Gruenhage G.: Generalized metric spaces. Handbook of Set-theoretic Topology, North Holland, 1984, pp.423-502. MR 0776629 | Zbl 0794.54034
[HLZ] Heath R.W., Lutzer D.J., Zenor P.L.: Monotonically normal spaces. Trans. Amer. Math. Soc. 155 (1973), 481-494. MR 0372826 | Zbl 0269.54009
[Ho] Hodel R.: Cardinal Functions I. Handbook of set-theoretic Topology, North Holland, 1984, pp.1-61. MR 0776620 | Zbl 0559.54003
[Ju] Juhasz I.: Cardinal Functions in Topology. Math. Centre Tracts 34, 1971. MR 0340021 | Zbl 0479.54001
[Ni] Nikiel J.: Some problems on continuous images of compact ordered spaces. Quest. & And. in Gen. Top. 4 (1986), 117-128. MR 0917893
[NP] Nyikos P.J., Purisch S.: Monotone normality and paracompactness in scattered spaces. Annals on the New York Acad. Sci. 552, 1989, pp.124-137. MR 1020780 | Zbl 0887.54019
[Os] Ostacewski A.: Monotone normality and $G_\delta $-diagonals in the class of inductively generated spaces. Coll. Math. Soc. Janos Bolyai, Topology 23 (1978), 905-930. MR 0588837
[Ru1] Rudin M.E.: Dowker spaces. Handbook of Set-theoretic Topology, North Holland, 1984, pp.761-780. MR 0776636 | Zbl 0566.54009
[Ru2] Rudin M.E.: Monotone normality and compactness. to appear in Topology and Appl. volume of Ehime Conference Proceedings, 1995. MR 1425938 | Zbl 0874.54004
[Ru3] Rudin M.E.: Compact monotonically normal spaces. invited lecture, 8th Prague Topological Symposium, August, 1996. Zbl 0983.54021
[St] Stephenson R.M., Jr.: Initially $k$-compact and related spaces. Handbook of Set-theoretic Topology, North Holland, 1984, pp.603-602. MR 0776632
[UH] HASH(0x976e968): The University of Houston Problem Book, 6/30/90.
[Wi] Williams S.W.: Trees, Gleason spaces, and co-absolutes of $\beta N$-$N$. Trans. Amer. Math. Soc. 271 (1982), 83-100. MR 0648079
[WZ] Williams S.W., Zhou H.: Strong versions of normality. Proceedings of the New York Conference on Topology and its Applications. Zbl 0797.54011
Search
Partner of
