Insight on difference between Euler characteristics of 2 manifolds: $\chi(U)-\chi(V)$? For the Euler characteristic,
we have the inclusion-exclusion principle:
$$\chi(U\cup V) = \chi(U)+\chi(V)-\chi(U \cap V),$$
and also the connected sum property:
$$
 \chi(U\#V) = \chi(U)+\chi(V)-\chi(S^n).
$$
However, is there any relation or topological/differential-geometric interpretation for the difference between two Euler characteristics
$$
\chi(U)-\chi(V)?
$$
This is studied in usual set theory, but I lack an intuitive understanding of what the above could mean. For instance, such an understanding could be gotten for the $+$ case by using the first equation above: $\chi(U)+\chi(V)=\chi(U\cup V) + \chi(U \cap V)$. I also haven't found too many resources on the matter. Any advice?

I don't know exactly what niceness hypotheses are required for this, but if $Y$ is, say, a finite CW complex and $X$ is a CW subcomplex of $Y$ then we should have
$$\chi_c(Y) - \chi_c(X) = \chi_c(Y \setminus X)$$
where $\chi_c$ is the compactly supported Euler characteristic, defined using cohomology with compact support $H_c^{\bullet}$. $\chi_c$ is not a homotopy invariant but besides that it behaves nicer in some ways, such as this one. We can equivalently write the above rleation as
$$\chi_c(Y) = \chi_c(X) + \chi_c(Y \setminus X).$$
Note that this is manifestly not true for the ordinary Euler characteristic!
Some $\chi_c$ examples:

*

*$\chi_c(X) = \chi(X)$ if $X$ is compact, since then compactly supported and ordinary cohomology coincide. Hence if $X$ and $Y$ are both compact above then we have $\chi(Y) - \chi(X) = \chi_c(Y \setminus X)$.

*$\chi_c(\mathbb{R}^n) = (-1)^n$ (this is a counterexample to homotopy invariance). This is because $\mathbb{R}$ is $[0, 1]$ minus two points, so $\chi_c(\mathbb{R}) = \chi_c([0, 1]) - 2 = -1$. It's still true that $\chi_c(X \times Y) = \chi_c(X) \times \chi_c(Y)$ so this determines the answer for $\mathbb{R}^n$. This can be used to explain the Euler characteristic of the sphere: we have $\chi(S^n) = \chi(\text{pt}) + \chi_c(\mathbb{R}^n) = 1 + (-1)^n$.

*The wedge of $2$ circles satisfies $\chi(S^1 \vee S^1) = \chi(\text{pt}) + \chi \left( (0, 1) \sqcup (0, 1) \right)$ (the result of deleting the wedge point) which gives $\chi(S^1 \vee S^1) = 1 - 2 = -1$ as expected.

*The torus $T^2$ satisfies $\chi(T^2) = \chi(S^1 \vee S^1) + \chi_c( (0, 1)^2) ) = -1 + 1 = 0$ as expected.

*The above argument generalizes to the following: if $X$ is a finite CW complex with $c_i$ different $i$-cells, then $\chi_c(X) = \sum (-1)^i c_i$ (with no compactness hypotheses).

