proof-polish

spaces/S000150/properties/P000089.md

@@ -8,16 +8,20 @@ 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. 
+From now on, to simplify the notation,
+every interval is assumed to consist exclusively of rational numbers.
 
+We construct an increasing bijection $f:X\to [\tilde b,\tilde a]$
+by choosing arbitrary order isomorphism between the rational intervals:
+$[0,b_0)\to[\tilde b, b_1)$, $(a_0,1]\to(a_1,\tilde a]$,
+$(b_k,b_{k+1})\to(b_{k+1},b_{k+2})$, and $(a_{k+2},a_{k+1})\to(a_{k+1},a_k)$ for all $k\geq 0$.
+Function $f$ being union of those increasing bijections
+has no fixed point.