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|Z x 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 [42, I.2, p. 548-549]).