Pontryagin duality for spaces of continuous functions
Let \(X\) be a completely regular Hausdorff space and \(C_p(X)\) denote the space of continuous real valued functions on \(X\) with the topology of pointwise convergence. We know that if \(X\) is not discrete, then \(C_p(X)\) is not reflexive as a topological linear space. The authors regard \(C_p(X)\) instead as a topological abelian group \(G\), take the dual group of characters with the compact open topology, and say that \(G\) is \(P\)-reflexive in case the natural homomorphism from \(G\) to its second dual space is a topological isomorphism. They give an example where \(X\) is not discrete and \(C_p(X)\) is \(P\)-reflexive, and they show that if \(X\) is metrizable, then \(C_c(X)\) (compact open topology instead of pointwise convergence) is \(P\)-reflexive. They show, among other things, that if \(X\) is an uncountable discrete space together with a point at infinity whose neighborhoods are complements of countable subsets, then the question of whether or not \(C_p(X)\) is \(P\)-reflexive is undecidable in ZFC.
scientific article; zbMATH DE number 1423626
scientific article