In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring.
We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U).
In the last chapter, we introduce a new ideal of subsets of Polish spaces consisting of what we call residually null sets. From this ideal, we introduce an algebra consisting of what we call R-sets which is consistently a strict extension of the algebra of Baire property sets. We show that the algebra of R-sets is closed under the Alexandrov-Suslin operation and generalize Pettis' Theorem. From this, we provide new automatic continuity results and give a generalization of a result of D. Montgomery which shows that minimal assumptions on the continuity of group operations of an abstract group G with a Polish topology imply that G is actually a Polish group. We also see that many results pertaining to the algebra of Baire property sets generalize to the context of R-sets.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc984214 |
| Date | 05 1900 |
| Creators | Caruvana, Christopher |
| Contributors | Kallman, Robert R., Urbański, Mariusz, Fishman, Lior, 1964- |
| Publisher | University of North Texas |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | v, 124 pages, Text |
| Rights | Public, Caruvana, Christopher, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved. |
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