Nonseparably connected metric spaces
by
Michal Ryszard Wójcik
Institute of Mathematics and Computer Science, Wroclaw University of Technology
Coauthors: Michal Morayne

DEFINITIONS:

A topological space is said to be separably connected if any two of its points can be contained in a separable connected subset. In the literature, there are four connected metric spaces which fail to be separably connected. Moreover, two of them do not contain any nontrivial connected separable subsets; such spaces will be called nonseparably connected.

MAIN RESULT:

We construct another example of a nonseparably connected metric space as a dense connected graph of a function from the real line into a nonseparable normed space.

MOTIVATION:

Suppose a continuous real-valued function on a connected space has a local extremum at every point without knowing whether it is a maximum or a minimum. Does it have to be constant? In general no, but yes for separably connected metric spaces.

Date received: June 14, 2008