Manifold implies nondegenerate
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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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
Proof outline
The proof uses two facts:
- Given a function on a disc, and a homotopy starting with that function at the center of the disc, there is a way of extending that homotopy to the whole disc, such that on the boundary of the disc, the homotopy is a constant homotopy starting at the initial function.
- 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.