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 [42, p. 546]). Note that a dendroid is smooth if and only if it is arc-smooth.