Abstract
We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let 
be two separable metric topologies on the same set X. We prove that every point in X has an 
-neighbourhood basis consisting of sets that are 
-closed if and only if there exists a function φ: X→ℝ that is -lower semi-continuous and such that is the weakest topology on X that contains and that makes φ continuous. An immediate corollary is that the class of almost n-dimensional spaces consists precisely of the graphs of lower semi-continuous functions with at most n-dimensional domains.
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References
Abry, M., Dijkstra, J. J.: Universal spaces for almost n-dimensionality. In preparation
Abry, M., Dijkstra, J. J., van Mill, J.: Sums of almost zero-dimensional spaces. Topology Proc. 29, (2005) to appear
Bessaga, C., Pełczyński, A.: Selected Topics in Infinite-Dimensional Topology. PWN, Warsaw, 1975
Davis, W. J., Johnson, W. B.: A renorming of nonreflexive Banach spaces. Proc. Amer. Math. Soc. 37, 486–488 (1973)
Dijkstra, J. J., van Mill, J.: Homeomorphism groups of manifolds and Erdős space. Electron. Res. Announc. Amer. Math. Soc. 10, 29–38 (2004)
Dijkstra, J. J., van Mill, J.: Erdős space and homeomorphism groups of manifolds. preprint
Dijkstra, J. J., van Mill, J.: Characterizing complete Erdős space. preprint
Dijkstra, J. J., van Mill, J.,
, J.: Complete Erdős space is unstable. Math. Proc. Cambridge Philos. Soc. 137, 465–473 (2004)Engelking, R.: Theory of Dimension, Finite and Infinite. Heldermann Verlag, Berlin, 1995
Erdős, P.: The dimension of the rational points in Hilbert space. Ann. of Math. 41, 734–736 (1940)
Kadec, M. I.: On strong and weak convergence. (Russian) Dokl. Akad. Nauk SSSR 122, 13–16 (1958)
Kadec, M. I.: On the connection between weak and strong convergence. (Ukrainian) Dopovidi Akad. Nauk Ukrain. RSR 9, 465–468 (1959)
Kawamura, K., Oversteegen, L. G., Tymchatyn, E. D.: On homogeneous totally disconnected 1-dimensional spaces. Fund. Math. 150, 97–112 (1996)
Kechris, A. S.: Classical Descriptive Set Theory. Springer Verlag, New York, 1995
Levin, M., Pol, R.: A metric condition which implies dimension ≤ 1. Proc. Amer. Math. Soc. 125, 269–273 (1997)
Levin, M., Tymchatyn, E. D.: On the dimension of almost n-dimensional spaces. Proc. Amer. Math. Soc. 127, 2793–2795 (1999)
Mayer, J. C., Mohler, L. K., Oversteegen, L. G., Tymchatyn, E. D.: Characterization of separable metric ℝ-trees. Proc. Amer. Math. Soc. 115, 257–264 (1992)
Oversteegen, L. G., Tymchatyn, E. D.: On the dimension of certain totally disconnected spaces. Proc. Amer. Math. Soc. 122, 885–891 (1994)
, J.: Complete Erdős space is unstable. Math. Proc. Cambridge Philos. Soc. 137, 465–473 (2004)