Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines

Authors:
Yuan-Qing Qiao and Franklin D. Tall

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3929-3936

MSC (2000):
Primary 54F05, 54A35; Secondary 03E05, 03E35

DOI:
https://doi.org/10.1090/S0002-9939-03-06966-1

Published electronically:
July 16, 2003

MathSciNet review:
1999943

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of ``generalized Souslin lines", *i.e.,* linearly ordered spaces in which every collection of disjoint open intervals is -discrete, but which do not have a -discrete dense set. The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace.

**[BDDGNRTW]**Z. Balogh, S. Davis, A. Dow, G. Gruenhage, P. Nyikos, M. E. Rudin, F. D. Tall, and S. Watson, New Classic Problems,*Top. Proc.***15**(1990) 201-220.**[BL]**H. R. Bennett and D. J. Lutzer,*Problems in Perfect Ordered Spaces*, 231-236 in Open Problems in Topology, J. van Mill and G. M. Reed, eds., North-Holland, Amsterdam, 1990. MR**92c:54001****[F]**M. J. Faber,*Metrizability in Generalized Ordered Spaces,*Mathematical Centre Tracts 53, Amsterdam, 1974. MR**54:6097****[J]**A. Jones, On dense non-archimedean spaces of linearly ordered spaces, preprint.**[N]**P. J. Nyikos,*A Survey of Zero-Dimensional Spaces,*87-114 in Proc. Ninth Annual Spring Topology Conf., Memphis State Univ., Memphis, Tenn., 1975, S. Franklin and B. S. Thomas, eds. MR**56:1245****[P]**S. Purisch, The orderability of non-Archimedean spaces,*Topology Appl.***16**(1983) 273-277. MR**85f:54063****[Q1]**Y.-Q. Qiao, On Non-Archimedean Spaces, Ph.D. Thesis, University of Toronto, 1992.**[Q2]**Y.-Q. Qiao, Martin's Axiom does not imply perfectly normal non-Archimedean spaces are metrizable,*Proc. Amer. Math. Soc.***129**(2001) 1179-1188. MR**2001g:54007****[QT]**Y.-Q. Qiao and F. D. Tall, Perfectly normal non-Archimedean spaces in Mitchell models,*Topology Proc.***18**(1993) 231-243. MR**96g:54033****[T]**S. Todorcevic, Some Consequences of ,*Topology Appl.***12**(1981) 187-202. MR**83b:03060****[W]**S. Watson,*Problems I wish I could Solve,*37-76 in ``Open Problems in Topology'', J. van Mill and G. M. Reed, eds., North-Holland, Amsterdam, 1990. MR**92c:54001**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54F05,
54A35,
03E05,
03E35

Retrieve articles in all journals with MSC (2000): 54F05, 54A35, 03E05, 03E35

Additional Information

**Yuan-Qing Qiao**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3

**Franklin D. Tall**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3

DOI:
https://doi.org/10.1090/S0002-9939-03-06966-1

Keywords:
Non-archimedean,
perfect,
metrizable,
tree base,
generalized Souslin line,
generalized Lusin line,
$\sigma$-discrete chain condition

Received by editor(s):
December 10, 1992

Received by editor(s) in revised form:
July 5, 2002

Published electronically:
July 16, 2003

Additional Notes:
The authors acknowledge support from grant A-7354 of the Natural Sciences and Engineering Research Council of Canada

Communicated by:
Andreas Blass

Article copyright:
© Copyright 2003
American Mathematical Society