Rays and the fixed point property in noncompact spaces
A problem is discussed in the paper of whether a noncompact space with a nice local structure contains a ray, i.e., a closed homeomorphic copy of \([0,1)\). After the introduction, \(LC^0\)-spaces containing a ray are considered in the second section. It is shown that if a metrizable \(LC^0\)-space either is noncompact, connected and locally compact, or is not completely metrizable, then it contains a ray. The third section is devoted to convex sets which lack the fixed point property. Answering an old question of \textit{V. L. Klee} jun. [Trans. Am. Math. Soc. 78, 30-45 (1955; Zbl 0064.10505)] the authors show that each noncompact convex subset of a metric linear space contains a ray, and therefore fails the fixed point property. The main result of this section says that if a convex subset of a topological vector space is not totally bounded, then it contains a ray and does not have the fixed point property. In the last section two examples of noncompact convex set with the fixed point property are constructed, showing that the previous result cannot be extended to all totally bounded noncompact convex subsets of topological vector spaces.
scientific article; zbMATH DE number 1102912