On countable connected locally connected almost regular Urysohn spaces |
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Authors: | RG Ori M Rajagopalan |
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Institution: | Department of Mathematics, University of Durban, Westville, Durban 4000, South Africa;Department of Mathematics, University of Toledo, Ohio 43606, USA |
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Abstract: | We construct connected, locally connected, almost regular, countable, Urysohn spaces. This answers a problem of G.X. Ritter. We show that there are 2c such non-homeomorphic spaces. We also show that there are 2c non-homeomorphic spaces which are further rigid. We discuss the group of homeomorphisms of such spaces.The following question was raised by G.X. Ritter: Does there exist a countable connected locally connected Urysohn space which is almost regular? We answer this question in the affirmative and in fact, show that not only are there as many as 2c such spaces but that there are just as many rigid spaces with the same properties. Furthermore we show that every countable Urysohn space is a subspace of such a space. We also prove that every countable group is isomorphic to the group of autohomeomorphisms of some connected locally connected almost regular Urysohn space. Examples are given of groups of order c which can be represented in this manner. |
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Keywords: | 54D15 54G20 54B15 countable space connected locally connected almost regular homeomorphism group ordinals |
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