Definition
A Hausdorff space X is called equi-locally convex (ELCX) if there exists an open neighborhood U of the diagonal , a mapping and an open cover of X such that:
A subset W in X is called convex if
and
.
The cover
is called a convex open cover of X.
An ELCX space is called coparacompact if any open cover has a locally finite convex refinement.
Reference
Vladimir P. Okhezin, On the fixed-point theory for non-compact maps and spaces.I, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, Volume 5, 1995, 83-100.