Monotone injection of an ordinal into $[0,1]$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:34:05Z http://mathoverflow.net/feeds/question/41598 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41598/monotone-injection-of-an-ordinal-into-0-1 Monotone injection of an ordinal into $[0,1]$ Andreas Thom 2010-10-09T14:33:05Z 2010-10-12T13:11:06Z <p>This is related to my <a href="http://mathoverflow.net/questions/41597/transfinite-induction-a-theorem-of-pedersen-and-chains-of-subalgebras-of-bh" rel="nofollow">recent question</a> and would provide a natural positive answer to Question 2. I am sure this must be known to experts.</p> <blockquote> <p><strong>Question:</strong> Is there a monotone injection $(\omega_1,&lt;) \to ([0,1],&lt;)$ ?</p> </blockquote> http://mathoverflow.net/questions/41598/monotone-injection-of-an-ordinal-into-0-1/41601#41601 Answer by Alessandro Sisto for Monotone injection of an ordinal into $[0,1]$ Alessandro Sisto 2010-10-09T14:44:44Z 2010-10-09T14:44:44Z <p>No, because you could use it to construct an injective map <code>$\omega_1\to\mathbb{Q}$</code>, mapping <code>$\alpha&lt;\omega_1$</code> to some rational number between <code>$\alpha$</code> and <code>`$\alpha+1$</code>.</p> http://mathoverflow.net/questions/41598/monotone-injection-of-an-ordinal-into-0-1/41603#41603 Answer by Péter Komjáth for Monotone injection of an ordinal into $[0,1]$ Péter Komjáth 2010-10-09T15:30:51Z 2010-10-09T15:30:51Z <p>Let me add a slightly different argument to Alessandro's quick and clever solution. Assume that $f:(\omega_1,&lt;)\to([0,1],&lt;)$ is an order preserving map. Let $X$ be the range of $f$. Set $a=\sup(X)$, the least upper bound of $X$. Now $X\cap [0,a]$ is uncountable while $X\cap [0,a-\frac{1}{n}]$ is countable for $n=1,2,\dots,$ an impossibility. Question: where did I use that $f$ is o.p.? </p> http://mathoverflow.net/questions/41598/monotone-injection-of-an-ordinal-into-0-1/41901#41901 Answer by Dave Marker for Monotone injection of an ordinal into $[0,1]$ Dave Marker 2010-10-12T12:46:32Z 2010-10-12T13:11:06Z <p>There is a far reaching generalization of this due to Friedman and Shelah. Suppose $X$ is a Borel set in a Polish space and <code>$&lt;$</code> is a linear order of $X$ that is a Borel subset of $X\times X$. Then there is no order preserving map from $\omega_1$ into <code>$(X,&lt;)$</code>.</p> <p>The Friedman-Shelah result follows from a later structure theorem of Harrington and Shelah who proved that for any such Borel linear order $(X,&lt;)$ there is a Borel measurable order preserving map into ${\bf R}^\alpha$ for some countable ordinal $\alpha$ where ${\bf R}^\alpha$ is ordered lexicographically. The arguments for $[0,1]$ above can be generalized to show that ${\bf R}^\alpha$ has no $\omega_1$-chains. </p>