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order preserving mapping

For each point p$ of a continuum X$ equipped with an arc-structure A : X \times X \to C(X)$ we define a partial order \le_p$ by letting x \le_p y$ whenever A(p,x) \subset A(p,y)$. Let X$ and Y$ be continua with fixed arc-structures A$ and B$, respectively. We say that a surjective mapping f: X \to Y$ is a \le_p$-mapping provided that x \le_p y$ in X$ implies that f(x) \le_{f(p)} f(y)$ in Y$. If, in addition, Y \subset X$, B = A\vert(Y \times Y)$, and f$ is a retraction, then f$ is called a \le_p$-retraction (or \le_p$-preserving retraction). The concept of a <_p$-mapping is defined in a similar manner (with f(x) \ne f(y)$ implied by x \ne y$). For order preserving mappings see e.g. [Fugate et al. 1981, I.7, p. 553].
next up previous contents index
Next: ordinary point Up: Definitions Previous: order
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30