Universal spaces for almost 𝑛-dimensionality

Abry, Mohammad; Dijkstra, Jan

doi:10.1090/S0002-9939-07-08971-X

Universal spaces for almost $n$-dimensionality
HTML articles powered by AMS MathViewer

by Mohammad Abry and Jan J. Dijkstra PDF

Proc. Amer. Math. Soc. 135 (2007), 2623-2628 Request permission

Abstract:

We find universal functions for the class of lower semi-continuous (LSC) functions with at most $n$-dimensional domain. In an earlier paper we proved that a space is almost $n$-dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most $n$-dimensional domain. We conclude that the class of almost $n$-dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdős space.

References

Mohammad Abry and Jan J. Dijkstra, On topological Kadec norms, Math. Ann. 332 (2005), no. 4, 759–765. MR 2179775, DOI 10.1007/s00208-005-0651-5
Jan J. Dijkstra, A criterion for Erdős spaces, Proc. Edinb. Math. Soc. (2) 48 (2005), no. 3, 595–601. MR 2171187, DOI 10.1017/S0013091504000823
Jan J. Dijkstra and Jan van Mill, Homeomorphism groups of manifolds and Erdős space, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 29–38. MR 2048429, DOI 10.1090/S1079-6762-04-00127-1
J. J. Dijkstra and J. van Mill, Erdős space and homeomorphism groups of manifolds, Canad. J. Math., in press.
J. J. Dijkstra and J. van Mill, Characterizing complete Erdős space, preprint.
Jan J. Dijkstra, Jan Van Mill, and Juris Steprāns, Complete Erdős space is unstable, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 465–473. MR 2092071, DOI 10.1017/S0305004104007996
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
Paul Erdös, The dimension of the rational points in Hilbert space, Ann. of Math. (2) 41 (1940), 734–736. MR 3191, DOI 10.2307/1968851
W. Hurewicz, Sur la dimension des produits cartesiens, Ann. of Math. (2) 36 (1935), no. 1, 194–197 (French). MR 1503218, DOI 10.2307/1968674
Kazuhiro Kawamura, Lex G. Oversteegen, and E. D. Tymchatyn, On homogeneous totally disconnected $1$-dimensional spaces, Fund. Math. 150 (1996), no. 2, 97–112. MR 1391294, DOI 10.4064/fm-150-2-97-112
M. Lavrentieff, Contribution à la théorie des ensembles homéomorphes, Fund. Math. 6 (1924), 149–160.
Michael Levin and Roman Pol, A metric condition which implies dimension $\leq 1$, Proc. Amer. Math. Soc. 125 (1997), no. 1, 269–273. MR 1389528, DOI 10.1090/S0002-9939-97-03856-2
M. Levin and E. D. Tymchatyn, On the dimension of almost $n$-dimensional spaces, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2793–2795. MR 1600109, DOI 10.1090/S0002-9939-99-04846-7
J. van Mill and R. Pol, Baire $1$ functions which are not countable unions of continuous functions, Acta Math. Hungar. 66 (1995), no. 4, 289–300. MR 1314008, DOI 10.1007/BF01876046
Lex G. Oversteegen and E. D. Tymchatyn, On the dimension of certain totally disconnected spaces, Proc. Amer. Math. Soc. 122 (1994), no. 3, 885–891. MR 1273515, DOI 10.1090/S0002-9939-1994-1273515-1