Countable dense homogeneity of definable spaces

Authors:
Michael Hrusák and Beatriz Zamora Avilés

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3429-3435

MSC (2000):
Primary 54E52, 54H05, 03E15

Published electronically:
May 2, 2005

MathSciNet review:
2161169

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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 zero-dimensional 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**, 10.1007/BF02764858**[Be]**Ralph Bennett,*Countable dense homogeneous spaces*, Fund. Math.**74**(1972), no. 3, 189–194. MR**0301711****[Br]**L. E. J. Brouwer,*Some Remarks on the coherence type*, Proc. Akad. Amsterdam**15**(1912), 1256-1263.**[DP]**Alan Dow and Elliott Pearl,*Homogeneity in powers of zero-dimensional first-countable spaces*, Proc. Amer. Math. Soc.**125**(1997), no. 8, 2503–2510. MR**1416083**, 10.1090/S0002-9939-97-03998-1**[Fi]**Ben Fitzpatrick Jr.,*A note on countable dense homogeneity*, Fund. Math.**75**(1972), no. 1, 33–34. MR**0301696****[FL]**Ben Fitzpatrick Jr. and Norma F. Lauer,*Densely homogeneous spaces. I*, Houston J. Math.**13**(1987), no. 1, 19–25. MR**884229****[FZ1]**Ben Fitzpatrick Jr. and Hao Xuan Zhou,*Densely homogeneous spaces. II*, Houston J. Math.**14**(1988), no. 1, 57–68. MR**959223****[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**, 10.1016/0166-8641(92)90148-S**[FZ3]**Ben Fitzpatrick Jr. and Hao Xuan Zhou,*Some open problems in densely homogeneous spaces*, Open problems in topology, North-Holland, 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****[Fr]**M. Fréchet,*Les dimensions d'unensemble abstrait*, Math. Ann**68**(1910), 145-168.**[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, Springer-Verlag, Berlin, 1994. Large cardinals in set theory from their beginnings. MR**1321144****[Ke]**Alexander S. Kechris,*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597****[KLW]**A. S. Kechris, A. Louveau, and W. H. Woodin,*The structure of 𝜎-ideals of compact sets*, Trans. Amer. Math. Soc.**301**(1987), no. 1, 263–288. MR**879573**, 10.1090/S0002-9947-1987-0879573-9**[Ku]**Kenneth Kunen,*Set theory*, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR**756630****[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**, 10.1090/S0002-9947-98-02100-X**[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****[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**, 10.1090/S0002-9939-1982-0633296-0**[vM2]**Jan van Mill,*The infinite-dimensional topology of function spaces*, North-Holland Mathematical Library, vol. 64, North-Holland Publishing Co., Amsterdam, 2001. MR**1851014****[Mi]**Arnold W. Miller,*Descriptive set theory and forcing*, Lecture Notes in Logic, vol. 4, Springer-Verlag, Berlin, 1995. How to prove theorems about Borel sets the hard way. MR**1439251****[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****[SZ]**J. Steprāns and H. X. Zhou,*Some results on CDH spaces. I*, Topology Appl.**28**(1988), no. 2, 147–154. Special issue on set-theoretic topology. MR**932979**, 10.1016/0166-8641(88)90006-5**[Zh]**Hao Xuan Zhou,*Two applications of set theory to homogeneity*, Questions Answers Gen. Topology**6**(1988), no. 1, 49–56. MR**940441**

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Additional Information

**Michael Hrusák**

Affiliation:
Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 61-3, Xangari, C. P. 58089, Morelia, Michoacán, México

Email:
michael@matmor.unam.mx

**Beatriz Zamora Avilés**

Affiliation:
Instituto de Matemáticas, UNAM, Unidad Morelia, A. P. 61-3, Xangari, C. P. 58089, Morelia, Michoacán, México

Email:
bzamora@matmor.unam.mx

DOI:
https://doi.org/10.1090/S0002-9939-05-07858-5

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 IN108802-2 and CONACYT grant 40057-F

Communicated by:
Alan Dow

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.