# Homological codimension of a subspace

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 $A$.