A note on continuous mappings and the property of J. L. Kelley

Abstract:

In this paper, it is proved that if $X$ is a continuum and $\omega$ is any Whitney map for $C(X)$, then the following are equivalent: (1) $X$ has property [K]. (2) There exists a (continuous) mapping $F:X \times I \times [0,\omega (X)] \to C(X)$ such that $F(\{ x\} \times I \times \{ t\} ) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}$ for each $x \in X$ and $t \in [0,\omega (X)]$, where $I = [0,1]$. (3) For each $t \in [0,\omega (X)]$, there is an onto map $f:X \times I \to {\omega ^{ - 1}}(t)$ such that $f(\{ x\} \times I) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}$ for each $x \in X$. Some corollaries are obtained also.