Further extending a complete convex metric

Abstract:

A metric $D$ is convex if for every two points $x,z$ there is a third point $y$ such that $D(x,y) + D(y,z) = D(x,z)$. A generalized continuum is a connected, locally compact, metric space. Let ${M_1}$ be a nonempty space with a complete convex metric ${D_1}$ and let ${M_2}$ be a nonempty locally connected generalized continuum. The following condition is shown to be necessary and sufficient for there to exist a complete convex metric for ${M_1} \cup {M_2}$ that extends ${D_1}:{M_1} \cap {M_2}$ is a nonempty subspace of both ${M_1}$ and ${M_2}$ which is closed in ${M_2}$ and whose ${M_2}$ boundary is closed in ${M_1}$.