An example of a connected metric space that is not separably connected
by
Manuel Maestre
Departamento de Análisis Matemático, University of Valencia, Spain
Coauthors: Richard M. Aron (University of Kent, Ohio, USA)

J. Candeal, C. Hervés and E. Induráin introduced in the concept of a topological space being separably connected, with the following definition: A topological space X is said to be separably connected if for every two points x, y in X there exists a connected and separable subset C_{{x, y}} such that x, y in C_{{x, y}}. In their work on utility theory [3], the authors ask whether there is a connected metric space that is not separably connected. They observe that there are examples of connected topological spaces which are not separably connected. However, no example of a connected metric space which is not separably connected is given and, in fact, this issue is explicitly raised in [2]. In this talk, based in a joint work with Richard Aron [1], we give a negative answer to this question, using a construction which is based on the standard construction of a non-measurable set. We will show that there exists a subset of the unit ball of any non-separable Banach space that is connected but not separably connected.

[1] R. Aron and M. Maestre Separable Connectedness: A remark on a paper by J. Candeal, C. Hervés, and E. Induráin. Preprint 1999.

[2] Balbas, A., Estévez, M., Hervés, C., and Verdejo, A. Espacios separablemente conexos, Rev. R. Acad. Cien. Exact. Fis. Nat. (Spain), 92, 1, (1998), 35-40.

[3] J. Candeal, C. Hervés and E. Induráin, Some results on representation and extension of preferences, Journal of Mathematical Economics 29 (1998), 75-81.

Date received: May 20, 1999