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fixed point. continuos function from a triangle to a triangle.

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Let $T$ be a subset of $\mathbb{R}^2$ such that $T$ is a triangle.
Let $f:T \rightarrow T$ be a continuous surjective function.
Prove that $f$ has a fixed point.

I wish I had something which can be called an "attempt"... but I don't
My friend asked me this and he found this question in a graph theory book.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
This follows from the proof of Brouwer's Fixed Point Theorem using Sperner's Lemma. I don't think surjection of f is necessary. This seems to be a fascinating example of how a fact from discrete mathematics is used to prove one of the "fundamental theorems of topology" (Wikipedia).