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arc-smooth

Given a continuum X$ with an arc-structure A$, the pair (X,A)$ (see arc-structure) is said to be arc-smooth at a point v \in X$ provided that the induced function A_v: X \to C(X)$ defined by A_v (x) = A(v,x)$ is continuous. Then the point v$ is called an initial point of (X,A)$. The pair (X,A)$ is said to be arc-smooth provided that there exists a point in X$ at which (X,A)$ is arc-smooth. An arbitrary space X$ is said to be arc-smooth at a point v \in X$ provided that there exists an arc-structure A$ on X$ for which (X,A)$ is arc-smooth at v$. The space X$ is said to be arc-smooth if it is arc-smooth at some point (see [Fugate et al. 1981, p. 546]). Note that a dendroid is smooth if and only if it is arc-smooth.
next up previous contents index
Next: arc-structure Up: Definitions Previous: arc
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30