Discussion on Lechicki and Spakowski's counterexample

TL;DR

This paper clarifies the role of Lechicki and Spakowski's counterexample in the context of the continuity of intersections of correspondences, emphasizing the importance of boundedness in infinite-dimensional spaces.

Contribution

It provides a detailed and rigorous analysis of the counterexample, clarifying misconceptions and highlighting its significance in economics and optimization theory.

Findings

Counterexample confirms the necessity of boundedness for continuity preservation.

Detailed properties of the correspondences are rigorously computed.

Misinterpretations of the counterexample are addressed and corrected.

Abstract

It is well-known that intersection of continuous correspondences can lost the continuity property. Lechicki and Spakowski's theorem says that intersection of H-lsc functions remains H-lsc if the intersection is a bounded subset of a normed space and its interior is nonempty. Lechicki and Spakowski pointed to the importance of the boundedness assumption in the case of infinite dimensional range giving a counterexample. Even though the counterexample works properly and is one of the most cited patterns of discontinuity, it has no detailed discussion in the literature of economics and optimization theory. What is more, some misleading interpretation of this very important counterexample can be observed. Our technical note clarifies the exact role of Lechicki and Spakowski's counterexample, computing each of the important properties of the correspondences rigorously.