Countable dense homogeneity of definable spaces
Authors:
Michael Hrusák and Beatriz Zamora Avilés
Journal:
Proc. Amer. Math. Soc. 133 (2005), 34293435
MSC (2000):
Primary 54E52, 54H05, 03E15
Published electronically:
May 2, 2005
MathSciNet review:
2161169
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zerodimensional Borel CDH spaces. We also show that for a Borel the following are equivalent: (1) is in , (2) is CDH and (3) is homeomorphic to or to . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr ns and Zhou, by showing that is not CDH.
 [BB]
Stewart
Baldwin and Robert
E. Beaudoin, Countable dense homogeneous spaces under
Martin’s axiom, Israel J. Math. 65 (1989),
no. 2, 153–164. MR 998668
(90f:54010), http://dx.doi.org/10.1007/BF02764858
 [Be]
Ralph
Bennett, Countable dense homogeneous spaces, Fund. Math.
74 (1972), no. 3, 189–194. MR 0301711
(46 #866)
 [Br]
L. E. J. Brouwer, Some Remarks on the coherence type , Proc. Akad. Amsterdam 15 (1912), 12561263.
 [DP]
Alan
Dow and Elliott
Pearl, Homogeneity in powers of
zerodimensional firstcountable spaces, Proc.
Amer. Math. Soc. 125 (1997), no. 8, 2503–2510. MR 1416083
(97j:54008), http://dx.doi.org/10.1090/S0002993997039981
 [Fi]
Ben
Fitzpatrick Jr., A note on countable dense homogeneity, Fund.
Math. 75 (1972), no. 1, 33–34. MR 0301696
(46 #852)
 [FL]
Ben
Fitzpatrick Jr. and Norma
F. Lauer, Densely homogeneous spaces. I, Houston J. Math.
13 (1987), no. 1, 19–25. MR 884229
(88d:54041)
 [FZ1]
Ben
Fitzpatrick Jr. and Hao
Xuan Zhou, Densely homogeneous spaces. II, Houston J. Math.
14 (1988), no. 1, 57–68. MR 959223
(89k:54080)
 [FZ2]
Ben
Fitzpatrick Jr. and Hao
Xuan Zhou, Countable dense homogeneity and the Baire property,
Topology Appl. 43 (1992), no. 1, 1–14. MR 1141367
(93b:54030), http://dx.doi.org/10.1016/01668641(92)90148S
 [FZ3]
Ben
Fitzpatrick Jr. and Hao
Xuan Zhou, Some open problems in densely homogeneous spaces,
Open problems in topology, NorthHolland, Amsterdam, 1990,
pp. 251–259. MR
1078651
 [Fo]
M.
K. Fort Jr., Homogeneity of infinite products of manifolds with
boundary, Pacific J. Math. 12 (1962), 879–884.
MR
0145499 (26 #3030)
 [Fr]
M. Fréchet, Les dimensions d'unensemble abstrait, Math. Ann 68 (1910), 145168.
 [HS]
Michael
Hrušák and Juris
Steprāns, Cardinal invariants related to sequential
separability, Sūrikaisekikenkyūsho
Kōkyūroku 1202 (2001), 66–74.
Axiomatic set theory (Japanese) (Kyoto, 2000). MR
1855551
 [Ka]
Akihiro
Kanamori, The higher infinite, Perspectives in Mathematical
Logic, SpringerVerlag, Berlin, 1994. Large cardinals in set theory from
their beginnings. MR 1321144
(96k:03125)
 [Ke]
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, SpringerVerlag, New York, 1995. MR 1321597
(96e:03057)
 [KLW]
A.
S. Kechris, A.
Louveau, and W.
H. Woodin, The structure of 𝜎ideals of
compact sets, Trans. Amer. Math. Soc.
301 (1987), no. 1,
263–288. MR
879573 (88f:03042), http://dx.doi.org/10.1090/S00029947198708795739
 [Ku]
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, NorthHolland Publishing Co., Amsterdam, 1983.
An introduction to independence proofs; Reprint of the 1980 original. MR 756630
(85e:03003)
 [La]
L.
Brian Lawrence, Homogeneity in powers of subspaces of
the real line, Trans. Amer. Math. Soc.
350 (1998), no. 8,
3055–3064. MR 1458308
(98k:54061), http://dx.doi.org/10.1090/S000299479802100X
 [Le]
S.
Levi, On Baire cosmic spaces, General topology and its
relations to modern analysis and algebra, V (Prague, 1981) Sigma Ser.
Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 450–454.
MR 698438
(84d:54052)
 [Ma]
M. V. Matveev, Cardinal and a theorem of Pelczynski, (preprint).
 [vM1]
Jan
van Mill, Strong local homogeneity does not
imply countable dense homogeneity, Proc. Amer.
Math. Soc. 84 (1982), no. 1, 143–148. MR 633296
(83e:54033), http://dx.doi.org/10.1090/S00029939198206332960
 [vM2]
Jan
van Mill, The infinitedimensional topology of function
spaces, NorthHolland Mathematical Library, vol. 64,
NorthHolland Publishing Co., Amsterdam, 2001. MR 1851014
(2002h:57031)
 [Mi]
Arnold
W. Miller, Descriptive set theory and forcing, Lecture Notes
in Logic, vol. 4, SpringerVerlag, Berlin, 1995. How to prove theorems
about Borel sets the hard way. MR 1439251
(98g:03119)
 [Sa]
W.
L. Saltsman, Concerning the existence of a connected, countable
dense homogeneous subset of the plane which is not strongly locally
homogeneous, Topology Proc. 16 (1991), 137–176.
MR
1206461 (94c:54008)
 [SZ]
J.
Steprāns and H.
X. Zhou, Some results on CDH spaces. I, Topology Appl.
28 (1988), no. 2, 147–154. Special issue on
settheoretic topology. MR 932979
(89c:54070), http://dx.doi.org/10.1016/01668641(88)900065
 [Zh]
Hao
Xuan Zhou, Two applications of set theory to homogeneity,
Questions Answers Gen. Topology 6 (1988), no. 1,
49–56. MR
940441 (89c:54010)
 [BB]
 S. Baldwin and R. E. Beaudoin, Countable dense homogeneous spaces under Martin's axiom, Israel J. Math. 65 (1989), 153164. MR 0998668 (90f:54010)
 [Be]
 R. B. Bennett, Countable dense homogeneous spaces, Fund. Math 74 (1972), 189194.MR 0301711 (46:866)
 [Br]
 L. E. J. Brouwer, Some Remarks on the coherence type , Proc. Akad. Amsterdam 15 (1912), 12561263.
 [DP]
 A. Dow and E. Pearl, Homogeneity in Powers of zerodimensional, firstcountable spaces, Proc. AMS 125 (1997), 25032510. MR 1416083 (97j:54008)
 [Fi]
 B. Fitzpatrick Jr., A note on countable dense homogeneity, Fund. Math. 75 (1972), 34. MR 0301696 (46:852)
 [FL]
 B. Fitzpatrick, Jr. and N. F. Lauer, Densely homogeneous spaces. I, Houston J. Math. 13 (1987), 1925. MR 0884229 (88d:54041)
 [FZ1]
 B. Fitzpatrick Jr. and H.X. Zhou, Densely homogeneous spaces II, Houston J. Math. 14 (1988), 5768. MR 0959223 (89k:54080)
 [FZ2]
 B. Fitzpatrick Jr. and H.X. Zhou, Countable dense homogeneity and the Baire property, Topology and its Applications 43 (1992), 114.MR 1141367 (93b:54030)
 [FZ3]
 B. Fitzpatrick Jr. and H.X. Zhou, Some Open Problems in Densely Homogeneous Spaces, in Open Problems in Topology (ed. J. van Mill and M. Reed), 1984, pp. 251259, NorthHolland, Amsterdam. MR 1078651
 [Fo]
 M. Fort, Homogeneity of infinite products of manifolds with boundary, Pacific J. Math 12 (1962), 879884. MR 0145499 (26:3030)
 [Fr]
 M. Fréchet, Les dimensions d'unensemble abstrait, Math. Ann 68 (1910), 145168.
 [HS]
 M. Hrusák and J. Steprans, Cardinal invariants related to sequential separability, Surikaisekikenkiusho Kokyuroku 1202 (2001), 6674. MR 1855551
 [Ka]
 A. Kanamori, The Higher Infinite, 1994, SpringerVerlag. MR 1321144 (96k:03125)
 [Ke]
 A. S. Kechris, Classical Descriptive Set Theory, 1995, SpringerVerlag. MR 1321597 (96e:03057)
 [KLW]
 A. S. Kechris, A. Louveau and W. H. Woodin, The Structure of ideals of Compact Sets, Trans. AMS 301 (1987), 263288.MR 0879573 (88f:03042)
 [Ku]
 K. Kunen, Set Theory, An Introduction to Independence Proofs, 1990, North Holland.MR 0756630 (85e:03003)
 [La]
 B. Lawrence, Homogeneity in powers of subspaces of the real line, Trans. AMS 350 (1998), 30553064.MR 1458308 (98k:54061)
 [Le]
 S. Levi, On Baire cosmic spaces, Proceednigs of the Fifth Prague Topological Symposium, 1983, pp. 450451, Heldermann Verlag, Berlin.MR 0698438 (84d:54052)
 [Ma]
 M. V. Matveev, Cardinal and a theorem of Pelczynski, (preprint).
 [vM1]
 J. van Mill, Strong local homogeneity does not imply countable dense homogeneity, Proc. AMS 84 (1982), 143148. MR 0633296 (83e:54033)
 [vM2]
 J. van Mill, The InfiniteDimensional Topology of Function Spaces, 2001, North Holland. MR 1851014 (2002h:57031)
 [Mi]
 A. W. Miller, Descriptive Set Theory and Forcing, 1995, Springer, Lecture Notes in Logic 4. MR 1439251 (98g:03119)
 [Sa]
 W. L. Saltsman, Concerning the existence of a connected, countable dense homogeneous subset of the plane which is not strongly locally homogeneous, Topology Proceedings 16 (1991), 137176. MR 1206461 (94c:54008)
 [SZ]
 J. Steprans, H.X. Zhou, Some Results on CDH Spaces, Topology and its Applications 28 (1988), 147154. MR 0932979 (89c:54070)
 [Zh]
 H.X. Zhou, Two applications of set theory to homogeneity, Questions Answers Gen. Topology 6 (1988), 4956. MR 0940441 (89c:54010)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
54E52,
54H05,
03E15
Retrieve articles in all journals
with MSC (2000):
54E52,
54H05,
03E15
Additional Information
Michael Hrusák
Affiliation:
Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 613, Xangari, C. P. 58089, Morelia, Michoacán, México
Email:
michael@matmor.unam.mx
Beatriz Zamora Avilés
Affiliation:
Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 613, Xangari, C. P. 58089, Morelia, Michoacán, México
Email:
bzamora@matmor.unam.mx
DOI:
http://dx.doi.org/10.1090/S0002993905078585
PII:
S 00029939(05)078585
Keywords:
Countable dense homogeneous,
Borel,
Baire
Received by editor(s):
June 13, 2003
Received by editor(s) in revised form:
June 11, 2004
Published electronically:
May 2, 2005
Additional Notes:
The first author’s research was supported partially by grant GAČR 201/03/0933 and by a PAPIIT grant IN1088022 and CONACYT grant 40057F
Communicated by:
Alan Dow
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
