Compactly nondegenerate space

Symbol-free definition
A topological space is termed compactly nondegenerate if every point is contained in an open set, whose closure is compact, and such that the inclusion of the union of the point with the boundary of the open set, inside the closure, is a cofibration.

Facts

 * If, in a Hausdorff space, every point is contained in a compactly nondegenerate open set, then the whole space is compactly nondegenerate.
 * Euclidean space is compactly nondegenerate. This gives a proof that manifolds are nondegenerate, in fact they are compactly nondegenerate.