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 
$\chi(X-*)=\chi(X)-\chi(*)=\chi(X)-1$?

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.
