= Euler characteristic of a disk and its boundary
{c}
{title2=$\chi(D^2)=1,\quad\chi(S^1)=0$}
A filled <triangle> is a <disk> with three <vertices>, three <edges> and one two-dimensional <topological cell>, so $\chi=3-3+1=1$. Its <simplicial homology> has $(b_0,b_1,b_2)=(1,0,0)$: the filled <triangle> is <contractible>, and the three-edge <chain cycle> is the <chain boundary> of its face. Keeping only the <cycle graph> on those three <vertices> gives $\chi=3-3=0$ and $(b_0,b_1)=(1,1)$, since that <chain cycle> no longer bounds a face. The surrounding plane's exterior is not part of either <simplicial complex>; do not add it to $F$.
\Image[/euler-characteristic-triangle-subdivision.png]
{title=The filled triangle, its midpoint subdivision, and its unfilled boundary}
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