Euler characteristic of a space minus a point Let $X$ be a topological space and $*$ be the base point of $X$.
How does $\chi(X-*)$ relate to $\chi(X)$ do we have 
$\chi(X-*)=\chi(X)-\chi(*)=\chi(X)-1$?

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.
