An asymmetric Ellis theorem |
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Authors: | S Andima R Kopperman |
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Institution: | a Long Island University - C.W. Post Campus, Department of Mathematics, Brookville, NY, USA b City College of CUNY, Department of Mathematics, New York, NY, USA c University of Wollongong, School of Mathematics and Applied Statistics, NSW 2522, Australia |
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Abstract: | In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff. |
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Keywords: | primary 54H11 secondary 06F30 22A05 54D10 54D45 54D50 54E55 54F05 |
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