Continuity of T

Abstract

Let X be a continuum. By Remark 2.1.5, the image of any subset of X under T is a closed subset of X. Then we may restrict the domain of T to the hyperspace, 2X, of nonempty closed subsets of X. Since 2X has a topology, we may ask if T: 2 X→ 2 X is continuous. The answer to this question is negative, as can be seen from Example 2.1.15. On the other hand, by Theorems 2.1.37 and 2.1.44, T is continuous for locally connected continua and for indecomposable continua, respectively. In this chapter we present results related to the continuity of T and examples of classes of decomposable nonlocally connected metric continua for which T is continuous. In particular, we show that if a continuum X is almost connected im kleinen at each of its points and T is continuous, then X is locally connected.