The Euler characteristic of $S^1$ should be equal to that of a triangle, the two being homeomorphic to each other. But it is $1$ for the triangle and zero for $S^1$; how does it make sense?
The Euler characteristic of S1: a paradox
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general-topology
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10how are you getting that a triangle has euler characteristic 1? – 2012-08-14
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2It depends on your definition of a triangle. How many faces does your triangle have? – 2012-08-14