Fixed point property of "3-star"

Question

Let $X = (I_1\sqcup I_2 \sqcup I_3)/(0_1 \sim 0_2\sim0_3),(I_i=[0,1]_i)$. I spend much time to trying to prove that any continuous map $X\to X$ have fixed point, but with no results..

Answer 1

If $f$ fixes $0$, we're done. Otherwise, $f$ sends zero into the interior of one of the legs, WLOG into the interior of $I_1$. If $f(I_1)$ intersects $0$, then the two paths $f(I_1)$ and $I_1$ must cross somewhere between $f(0)$ and $0$ by the IVT, so there is a fixed point somewhere in $I_1$. If $f(I_1)$ doesn't intersect $0$, then $f(I_1) \subseteq I_1$, and we're reduced to showing that any continuous map $[0, 1] \to [0, 1]$ has a fixed point. This also follows by IVT, and so there is again a fixed point in $I_1$.