arc-structure

By an arc-structure on an arbitrary space X we understand a function $A : X \times X \to C(X)$ such that for every two distinct points x and y in X the set A(x,y) is an arc from x to y and that the following metric-like axioms are satisfied for every points x, y and z in X:

(1)

A (x,x) = {x};

(2)

A (x,y) = A (y,x);

(3)

$A (x,z) \subset A(x,y) \cup A (y,z)$,
with equality prevailing whenever $y \in A (x,z)$.

We put (X,A) to denote that the space X is equipped with an arc-structure A (see [42, p. 546]). Note that if there exists an arc-structure on a continuum, then the continuum is arcwise connected.