Why is definable compact equivalent to bounded and closed for sets with o-minimal structures? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:37:57Z http://mathoverflow.net/feeds/question/66536 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66536/why-is-definable-compact-equivalent-to-bounded-and-closed-for-sets-with-o-minimal Why is definable compact equivalent to bounded and closed for sets with o-minimal structures? profiles116990727936402512408 2011-05-31T11:57:23Z 2011-12-25T16:54:23Z <p>We have $M$ an o-minimal structure. $X \in M^n$ with the induced topology. I'm reading an article which shows that $X \in M^n$ is definable compact is equivalent to $X$ being bounded and closed.</p> <p>Definable compactness of $X$ means that any $M$-definable curve in $X$ is completable. (a curve in $X$ is a $M$-definable continuous embedding $f: (a,b) \rightarrow X$). It is said to be completable if $\lim_{x\rightarrow a^{+}}f(x)$ and $\lim_{x\rightarrow b^{-}}f(x)$ exists.)</p> <p>When it shows that any definably compact subset $X \in M^n$ is bounded I've got the feeling that its proof is very complicated. I might be doing something wrong but I've got the feeling that this proof can be done much more easily.</p> <p>Here is how it goes: it first shows that definable compactness is preserved under projection on the k first coordinates.</p> <p>Then it proceeds by induction. let's assume that any definably compact subset $X \in M^n$ is bounded. Let $X \in M^{n+1}$, then as $p(X)$ is definably compact (by preservation of definable compactness under projection), it is by induction bounded ($p$ is the projection onto the $n$ first coordinates). So the first $n$ coordinates of X are bounded.</p> <p>Now why can't we also say the projection of X onto his last n coordinates is definably compact (by preservation of definable compactness under projection), it is by induction bounded. So the last n coordinates of X are bounded.</p> <p>So X is bounded.</p> <p>(for the proof of "Let $S \in M^n$ be definably compact and let $p : M^n \rightarrow M^k$ be a projection map. Then p(S) is definably compact." is the following: By induction, it suffices to show that if $S \in M^{n+1}$ is definably compact then p(S), where $p: M^{n+1} \rightarrow M^n$ denotes projection onto the first n coordinates, is definably compact as well. For a contradiction, assume not. Then there is a definable continuous embedding $f: (a, b) \rightarrow p(S)$ such that,say, f does not have a right-hand limit point in $p(S)$. By o-minimality, for every $a \in p(S)$, the set <code>$S_a = \{b \in M : (a, b) \in S\}$</code> is the union of finitely many intervals. Since S is definably compact, $S_a$ is closed and bounded. Let m(a) be the least element of $S_a$. We now define $g: (a, b) \rightarrow S$ by $g(x) = (f(x),m(f(x))$. It follows that g does not have a right-hand limit point in S since f does not have one in $p(S)$, a contradiction.)</p>