Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
9  
how are you getting that a triangle has euler characteristic 1? –  Eric O. Korman Aug 14 '12 at 23:21
    
It depends on your definition of a triangle. How many faces does your triangle have? –  M Turgeon Aug 14 '12 at 23:21
add comment

1 Answer

If by “triangle” you mean “three points and the segments connecting each pair of them”, then the Euler characteristic of it is $0$ (it has three edges and three vertices).

If by “triangle” you mean the same as above, plus the part of plane enclosed by the segment, then it is certainly not homeomorphic to $S^1$, but to a two-dimensional disk $D^2$, and it has Euler characteristic $1$, just like the disk.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.