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Semicontinuous limits of nets of continuous functions

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Abstract

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space \(X\) that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions \(\mathcal{C }(X)\) is the set of upper semicontinuous functions on \(X\), and (2) the pointwise convergence of a net in \(\mathcal{C }(X)\) to an upper semicontinuous limit automatically ensures convergence in this finer topology.

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Authors and Affiliations

  1. Department of Mathematics, California State University, Los Angeles, Los Angeles, CA, 90032, USA

    Gerald Beer

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  1. Gerald Beer
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Correspondence to Gerald Beer.

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Dedicated to Jonathan Borwein on the occasion of his 60th birthday.

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Beer, G. Semicontinuous limits of nets of continuous functions. Math. Program. 139, 71–79 (2013). https://doi.org/10.1007/s10107-013-0660-1

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  • DOI: https://doi.org/10.1007/s10107-013-0660-1

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