Euler characteristic of a space minus a point

From the Mayer-Vietoris sequence for (singular) homology it follows that if $U,V\subset X$ are open subsets and $X=U\cup V$, $$\chi(X) = \chi(U) + \chi(V) - \chi(U\cap V).$$ In your example you can take $U=X\backslash \{x\}$ and $V$ a small neighbourhood of $x$. Then you'll have $$\chi(U) = \chi(X) - \chi(V) + \chi(U\cap V).$$ In general this numbers can vary a lot (think on a point in a space not having a contractible neighbourhood, or the vertex of the cone $CY$ of some space $Y$). In the particular case when $X$ is a $n$-manifold, and therefore, locally homeomorphic to $\mathbb{R}^n$, taking $U=X\smallsetminus \{x\}$ and $V$ a small neighbourhood of $x$ homeomorphic to a ball we get that $U\cap V$ is homeomorphic to a ball minus a point, hence homotopy equivalent to $S^{n-1}$, so we get $$\chi(U) = \chi(X) - 1 + 0$$ if $n$ is even and $$\chi(U) = \chi(X) - 1 + 2$$ if $n$ is odd.

This additivity property is true for the compactly supported Euler characteristic $\chi_c$, that is, the alternating sums of the dimensions of the cohomology groups with compact support. More generally one has $$\chi_c(X) = \chi_c(X \setminus Z) + \chi_c(Z)$$ for a closed subset $Z$ of $X$. In this sense the compactly supported Euler characteristic is nicer, but there are other drawbacks, like that the compactly supported cohomology groups is not a homotopy invariant.

On a manifold $M$ of even dimension $2n$, the compactly supported and the ordinary Euler characteristic coincide by Poincaré duality, interpreted as the assertion that there is a perfect pairing between $H^i(M)$ and $H^{2n-i}_c(M)$. This generalizes Poincaré duality on a closed manifold, since on a closed manifold the ordinary cohomology and the cohomology with compact support coincide. This also tells you that on a manifold of odd dimension, the two Euler characteristics only differ by a sign.

To learn about this and much more, see e.g. the book by Bott and Tu.

Let X be a para-compact Hausdorff space. Consider the compactly supported Euler characteristic $e_c(X) = \sum_{i} (-1)^i \dim \ \mathrm{H}_c^i(X,\mathbf{Q})$. Let $U\subset X$ be open with complement $Z$. Since you have a Mayer-Vietoris sequence for compactly supported cohomology groups, we have $$ e_c(X) = e_c(U) + e_c(Z).$$ I don't know how the non-compactly supported Euler characteristic behaves.