Homological codimension of a subspace

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 $$A$$.