Topological Properties of Quasilinear Spaces and Set-Valued Maps

This paper presents a comprehensive study on the topological properties within the framework of quasilinear spaces, particularly focusing on their application to set-valued maps. We generalize the established concept of normed quasilinear spaces, primarily as introduced by Aseev in 1986, to the more encompassing notion of topological quasilinear spaces1. Our investigation demonstrates that if S is an arbitrary topological space and Z is a metrizable topological vector space, then the space of minimal semi-continuous compact-valued maps, denoted as M(S,Y), can be mapped into a linear space. This significantly extends previous results where S was restricted to being a Baire space. A key contribution of this work is the detailed analysis of the topological vector space Ms(S, Z), defined by the topology of strong uniform convergence on compact subsets of S. We establish several fundamental properties of this space, including its completeness when S is a locally compact Hausdorff space and Z is complete. Furthermore, we prove that Ms(S, Z) is metrizable if and only if S is hemicompact. The paper also undertakes a comparative study of the topologies τs and τk, demonstrating that τs is finer than τk, with equality holding when S is locally compact. Through rigorous mathematical illustrations and examples, we show that the principle of localization, typically valid for topological vector spaces, may not hold universally for all topologically quasilinear spaces, particularly in the context of singular elements. These findings collectively contribute fresh insights into the topology of quasilinear spaces, multivalued mappings, and set-valued analysis.