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I recently came across the formula that the Euler characteristic is equal to $$\sum\limits_{i=-N}^N (-1)^i\dim C_i(X)$$

For this to make sense, $C_{-1}(X)$ would have to exist. What would that be? What is a $-1$-simplex? A $1$-simplex with an odd permutation of vertices?

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    What is $X$? What is $N$? Please give the complete(!) context. – Martin Brandenburg Nov 24 '14 at 14:51
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    If $X$ is a simplicial complex (resp. topological space) and $C_{\bullet} (X)$ is the simplicial (resp. singular) complex, then one typically defines $C_n (X) = 0$ for $n < 0$. – Zhen Lin Nov 24 '14 at 16:50

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If $X$ is a topological space, it is standard to put $C_i(X)=0$ whenever $i<0$.

Mister Benjamin Dover
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