Compact sets in TVS - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:48:14Z http://mathoverflow.net/feeds/question/67589 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67589/compact-sets-in-tvs Compact sets in TVS Johan Jorg 2011-06-12T17:11:05Z 2011-06-12T19:38:56Z <p>Let $K$ be a compact subset of a Hausdorff topological vector space. Is it true that $\bigcap_{n\in \mathbb{N}}\frac{1}{n}K$ is either empty of or is a set consisting of the origin only?</p> http://mathoverflow.net/questions/67589/compact-sets-in-tvs/67591#67591 Answer by Todd Trimble for Compact sets in TVS Todd Trimble 2011-06-12T17:35:12Z 2011-06-12T17:35:12Z <p>I'll assume this is a TVS over $\mathbb{R}$ or $\mathbb{C}$, and not over some other local field. </p> <p>If $x$ is nonzero and belongs to $\frac1{n}K$ for every $n \geq 1$, then $nx \in K$ for each $n$. If $L$ is the line through $x$, then the subspace topology coincides with the standard topology on the ground field by the TVS axioms, and $L \cap K$ is a compact subset of the line which is unbounded since it contains all the $nx$. This gives a contradiction. </p>