12.12.2012 5:24, William Elliot wrote: > On Tue, 11 Dec 2012, Kaba wrote: > >> Let X be a locally compact Hausdorff space. Is every open set of X >> precompact (compact closure)? > > Yes. How would you prove it?
Related, let X be a Hausdorff space. Royden (Real analysis) defines E subset X to be _bounded_ if it is contained in a compact set. It seems to me that precompact and bounded are equivalent properties.
Assume E is precompact. Then cl(E) is a compact set which contains E. Therefore E is bounded. Assume E is bounded. Then there is a compact set K such that E subset K. Since X is Hausdorff, K is closed. Therefore cl(E) subset K. Since cl(E) is a closed subset of a compact set K, cl(E) is compact. Therefore E is precompact.