Homological codimension of a subspace

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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 $A$.