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convex

Given a continuum X$ with an arc-structure A$, a subset Z$ of X$ is said to be convex provided that for each pair of points x$ and y$ of Z$ the arc A(x,y)$ is a subset of Z$. If Z$ is a convex subcontinuum of X$, then A\vert Z \times Z$ is an arc-structure on Z$. We define X$ to be locally convex at a point p \in X$ provided that for each open set U$ containing p$ there is a convex set Z$ such that p \in \mathrm{int}\,Z \subset \mathrm{cl}\,Z \subset U$ (see [Fugate et al. 1981, I.2, p. 548-549]).
next up previous contents index
Next: continuum Up: Definitions Previous: converge 0-regularly
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30