In this lecture we introduce a new transfinite dimension function for metric spaces which utilizes Henderson's topological D-dimension and ascribes to any metric space either an ordinal number or the symbol Ω. The construction of our function is motivated by that of Urbański's transfinite Hausdorff dimension, tHD. Henderson's dimension is a topological invariant, however, like Hausdorff dimension and tHD the function presented will be invariant under bi-Lipschitz continuous maps and generally not under homeomorphisms. We present some original results on D-dimension and build the general theory for the D-variant of transfinite Hausdorff dimension, \mathrm{t}_D\mathrm{HD}. In particular, we will show for any ordinal number α, existence of a metrizable space which has \mathrm{t}_D\mathrm{HD} greater than or equal to α and less than or equal to \omega_\tau, where τ is the least ordinal which satisfies α < \omega_\tau.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc1944329 |
| Date | 05 1900 |
| Creators | Decker, Bryce |
| Contributors | Urbanski, Mariusz, Fishman, Lior, Brozovic, Douglas |
| Publisher | University of North Texas |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | Text |
| Rights | Public, Decker, Bryce, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved. |
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