fixed point

spaces/S000150/properties/P000089.md

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+---
+space: S000150
+property: P000089
+value: false
+---
+
+Observe that any increasing bijection
+$X\to [a,b]\cap\mathbb Q$, where $0\leq a<b\leq 1$,
+is a continuous self-map of $X$.
+
+Fix $\theta \in (0,1)\setminus \mathbb Q$ and two monotone sequences of irrational numbers
+$1>a_0>\ldots>a_k>a_{k+1}> \theta$
+and $0<b_0<\ldots <b_k<b_{k+1}<\theta$
+such that $a_k\to \theta$ and $b_k\to \theta$.
+Pick also $\tilde a \in (a_1,a_0)\cap\mathbb Q$
+and $\tilde b\in (b_0,b_1)\cap\mathbb Q$.
+
+We construct an increasing bijection $f:X\to [\tilde b,\tilde a]\cap\mathbb Q$ by chooseing arbitrary bijections between the rational intervals
+$[0,b_0)\to[\tilde b, b_1)$, $(a_0,1]\to(a_1,\tilde a]$ and $(b_k,b_{k+1})\to(b_{k+1},b_{k+2})$,
+$(a_{k+2},a_{k+1})\to(a_{k+1},a_k)$ for $k\geq 0$.
+It is evident that union of those bijections
+has no fixed point. 
+

spaces/S000151/properties/P000089.md

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+---
+space: S000151
+property: P000089
+value: false
+---
+
+See the proof that {S150|P89}.
+