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:
- The inclusion of the center plus boundary in a disc, is a cofibration. This is an application of the three sides lemma. Further information: Center plus boundary in disc is cofibration
- 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.