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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 [Fugate et al. 1981, p. 546]). Note that if there exists an arc-structure on a continuum, then the continuum is arcwise connected.
next up previous contents index
Next: atomic Up: Definitions Previous: arc-smooth
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30