Manifold implies nondegenerate

Statement
Every manifold, or more generally, every Hausdorff locally Euclidean space is nondegenerate. In other words, the inclusion of any point in the manifold is a cofibration.

Proof outline
The proof uses two facts:


 * The inclusion of the center plus boundary in a disc, is a cofibration. This is an application of the three sides lemma.
 * Any disc contained in a Euclidean neighbourhood of the point, is a closed subset of the whole manifold. For this we use the fact that the disc itself is a compact space, and the manifold is Hausdorff.

We combine the above two facts and use the gluing lemma for closed subsets to get a homotopy on the whole manifold.

A further abstraction
A way of abstracting this is the notion of a compactly nondegenerate space. We use the fact that Euclidean space is compactly nondegenerate, to conclude that manifolds are also compactly nondegenerate.