Group topologies coarser than the Isbell topology |
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Authors: | Szymon Dolecki Francis JordanFrédéric Mynard |
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Institution: | a Université de Bourgogne, Institut de Mathématiques de Bourgogne, Dijon, France b Department of Mathematics and Computer Science, Queensborough Community College, Queens, NY, USA c Georgia Southern University, Statesboro, GA, USA |
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Abstract: | The Isbell, compact-open and point-open topologies on the set C(X,R) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α(X) of compact families of open subsets of a topological space X. Those α(X) for which addition is jointly continuous at the zero function in Cα(X,R) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α(X) for which Cα(X,R) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, that Cα(X,R) can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology. |
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Keywords: | Function space Hyperspace Topological group Consonance Infraconsonance Isbell topology |
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