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C^*$-smooth

Let X$ be a continuum. Define C^*: C(X) \to C(C(X))$ by C^*(A) = C(A)$. It is known that for any continuum X$ the function C^*$ is upper semi-continuous, [Nadler 1978, Theorem 15.2, p. 514], and it is continuous on a dense G_\delta$ subset of C(X)$, [Nadler 1978, Corollary 15.3, p. 515]. A continuum X$ is said to be C^*$-smooth at A \in C(X)$ provided that the function C^*$ is continuous at A$. A continuum X$ is said to be C^*$-smooth provided that the function C^*$ is continuous on C(X)$, i.e., at each A \in C(X)$ (see [Nadler 1978, Definition 5.15, p. 517]).
next up previous contents index
Next: cyclic element Up: Definitions Previous: cut point
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30