= Cell-count formula for Euler characteristic
{title2=$\chi(X)=\sum_{i=0}^{d}(-1)^ic_i=\sum_{i=0}^{d}(-1)^ib_i$}
Let $X$ be a finite <CW complex>, with $c_i$ <topological cells> of <dimension> $i$ and none above $d$. Its <Euler characteristic> is the displayed alternating sum. For a finite <simplicial complex>, $c_i$ counts its $i$-dimensional <simplexes>, each counted once rather than once for each containing <simplex>. In <dimension> two this becomes $V-E+F$. A <topological cell>'s interior is counted; identified parts of its boundary are counted in their own lower-dimensional <topological cells>.<Algebraic Topology by Allen Hatcher>[Algebraic Topology, §2.2, p. 146, definition of Euler characteristic and Theorem 2.44]{ref}
The <cellular homology theorem> identifies <cellular homology> with <singular homology>, so the count equals the alternating rational <Betti numbers>. These are the free ranks of <integral homology>; <torsion subgroups> do not contribute to them. Over any <field>, the alternating <homology> dimensions still give the same <Euler characteristic>, although individual <Betti numbers> can depend on the <field>.
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