almost chainable

A continuum M is almost chainable if, for each positive number $\varepsilon$, there exists an open cover D of M with mesh $\varepsilon$ and a chain C of elements of D and an end link L of C such that no element of $D \setminus C$ intersects any link of C other than L and every point of M is at a distance less than $\varepsilon$ from some link of C.