Homological codimension of a subspace

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This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.

Definition

Let X be a topological space and A a subspace. A is said to have homological codimension m in X if A is nonempty, and for every point p \in A, there exists a neighbourhood U of p such that:

H_m(U, U \setminus A) = \mathbb{Z}

and all other homologies are 0.

Facts

If X is a manifold and A is a closed subset which is also a tame submanifold, then A has cohomological dimension in X equal to the difference of dimensions of X and A.