What is a co-dimension? I'm looking for a simple explanation (without complex formula) what a co-dimension is. When does objects have a co-dimension of 0 and when > 0?
Context: A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis (see abstract)
 A: For a subobject $A$ of $X$ such that both $A$ and $X$ have finite dimension, the codimension, which is short for complementary dimension, of $A$ relative to $X$ is $\dim X - \dim A$.  
A: The codimension of a subspace $W$ of a vector space $V$ is the dimension of the space of cosets of $W$ in $V$.  A coset is a set of the form $v+W=\{v+w : w\in W\}$.  One can add two cosets or multiply a coset by a scalar, so the set of cosets is a vector space in its own right.
A: The term arises when you are considering some type of mathematical object for which dimension makes sense, and you have a subobject of a larger object. For example:


*

*A subspace $U$ of a (finite dimensional) vector space $V$.

*A submanifold $N$ of a manifold $M$.


The codimension of the small object is the dimension of the larger object minus that of the small object. So in the examples, $\operatorname{codim}{U}=\dim{V}-\dim{U}$ (which is equal to $\dim{V/U}$ as mentioned in the comment - and this extends the definition to allow $V$ to be infinite dimensional), and $\operatorname{codim}{N}=\dim{M}-\dim{N}$.
Note that the codimension of the small object is not intrinsic, but depends on the choice of large object; if you have vector spaces $U,V,W$ with $U$ a subspace of $V$ and $V$ a subspace of $W$, then the codimension of $U$ in $V$ is different from the codimension of $U$ in $W$ (providing $V\ne W$!).
The case that a subobject has codimension $0$ is special, as this is the case where its dimension is as large as possible, given that it is a subobject of the larger object. So $U$ has codimension $0$ in $V$ only when $U=V$, and $N$ has codimension $0$ in $M$ only when $N$ contains an open subset of $M$.
