We define the compact-open topology on the dual of an abelian topological group. Please describe compact open topology more explicitly in the case where G is equipped with the discrete topology, for example G=Z. Are the sup-norm topology and compact-open topology coincide on dual of Z?
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4The compact subsets of a discrete space are the finite subsets. So in this case the compact-open topology is just the pointwise convergence topology. – egreg Aug 03 '15 at 08:15