Ladder systems on trees

Author:
Zoran Spasojevic

Journal:
Proc. Amer. Math. Soc. **130** (2002), 193-203

MSC (2000):
Primary 03E05

DOI:
https://doi.org/10.1090/S0002-9939-01-05072-9

Published electronically:
July 31, 2001

MathSciNet review:
1626486

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We formulate the notion of uniformization of colorings of ladder systems on subsets of trees. We prove that Suslin trees have this property and also Aronszajn trees in the presence of Martin's Axiom. As an application we show that if a tree has this property, then every countable discrete family of subsets of the tree can be separated by a family of pairwise disjoint open sets. Such trees are then normal and hence countably paracompact. As a dual result for special Aronszajn trees we prove that the weak diamond, , implies that no special Aronszajn tree can be countably paracompact.

**[1]**K. Devlin and S. Shelah, A weak version of which follows from , Israel Journal of Mathematics, Vol. 29, Nos. 2-3 (1978) 239-245. MR**57:9537****[2]**K. Devlin and S. Shelah, Suslin properties and tree topologies, Proceedings of The London Mathematical Society, 39 (1979) 237-252. MR**80m:54031****[3]**K. Devlin and S. Shelah, A note on the normal Moore space conjecture, Canadian Journal of Mathematics, 31 (1979) 241-252. MR**81d:54022****[4]**W. Fleissner, When is a Jone's space normal, Proceedings of the American Mathematical Society, 50 (1975) 375-378. MR**52:15384****[5]**W. Fleissner, Remarks on Suslin Properties and Tree Topologies, Proceedings of the American Mathematical Society, 80 (1980) 320-326. MR**81k:54054****[6]**M. Hanazawa, Note on Countable Paracompactness of Collection-wise Hausdorff tree topologies, Saitama Mathematics Journal, Vol. 2 (1984) 7-20.**[7]**S. Shelah, Whitehead groups may not be free even assuming CH, II, Israel Journal of Mathematics, Vol. 35, No. 4 (1980) 257-285. MR**82h:03055****[8]**S. Watson, Separation in countably metacompact spaces, Transactions of the American Mathematical Society, 290 (1985) 831-842. MR**87b:54016**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
03E05

Retrieve articles in all journals with MSC (2000): 03E05

Additional Information

**Zoran Spasojevic**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

DOI:
https://doi.org/10.1090/S0002-9939-01-05072-9

Received by editor(s):
June 5, 1996

Received by editor(s) in revised form:
May 18, 1998

Published electronically:
July 31, 2001

Communicated by:
Andreas R. Blass

Article copyright:
© Copyright 2001
American Mathematical Society