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By an arc-structure on an arbitrary space X we
understand a function
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)
-
,
with equality prevailing whenever
.
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.
Janusz J. Charatonik and Pavel Pyrih
2000-09-21