© 1996, Topology Atlas

A size map is a variation on the idea of a Whitney map, but less restrictive. Let $X$ be a continuum and let $C(X)$ denote the hyperspace of subcontinua of $X$ [1, p.1]. A Whitney map for $C(X)$ is a continuous function $m : C(X) --> [0, +\infty)$ such that $m(\{x\}) = 0$ and if $A$ is a proper subset of $B$ then $m(A) < m(B)$. Much work has been done in the study of Whitney levels (point inverses) of Whitney maps (see [1]). More generally, a continuous function $s : C(X) --> [0, +\infty)$ is a size map provided that $s(\{x\})=0$ and $s(A) <= s(B)$ whenever $A$ is a subset of $B$ [2, p. 243]. For example, the diameter map is a size map which is not in general a Whitney map. Point inverses of size maps are called size levels. Whitney levels for arcs are arcs (or degenerate). In [2] the size levels for any size function on the hyperspace of an arc have been characterized as follows.

Theorem. Assume $Z$ is a continuum and consider the following three conditions:

1.1. $Z$ is a planar Absolute Retract.

1.2. Cut points of $Z$ have component number two.

1.3. Any true cyclic element of $Z$ contains at most two cut points of $Z$.

Then any size level for an arc satisfies 1.1-1.3 and conversely, if $Z$ satisfies 1.1-1.3, then $Z$ is a diameter level for some arc.

References

[1] S. B. Nadler, Jr., Hyperspaces of Sets, Marcel Dekker, New York 1978.

[2] S. B. Nadler, Jr. and Thelma West, Size levels for arcs, Fundamenta Mathematicae 141 (1992), 243-255.