I would like to know what's the Euler characteristic of the hyperboloid of one sheet. I know that $2-2g$ is the Euler characteristic where g is the number of "holes". Using this fact, Euler characteristic of the hyperboloid is -2. Am I right?
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$\begingroup$No. The "fact" you mention is not stated in a rigorous way; there is no definition of 'hole'. (I really dislike this phrasing because of confusions like this.) The precise statement is that if $\Sigma_g$ is the compact surface without boundary of genus $g$, then $\chi(\Sigma_g) = 2-2g$.
The hyperboloid of one sheet is not compact, so it does not fit into this statement. It deformation retracts onto a circle, and the Euler characteristic is a homotopy invariant, so $\chi(H) = \chi(S^1) = 0$.
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