Subject: Re: question From: "Carlos Prieto (113)" Date: Mon, 13 Dec 2004 13:03:47 -0600 (CST) To: Don Davis Here an Answer to Adel's question. Prop. Under the given hypothesis, X = lim X(j) is non empty. Moreover, the canonical map X -> X(j) for each j is surjective. Pf. If his index set is a totally ordered set, one has the following: For any (fixed) i and any x_i in X(i), take any other j. If j>i, take x_j in X(j) such that the surjective map X(j) -> X(i) maps x_j to x_i; if j X(j). Then the element (x_j) determined by these elements in the product of the X(j) lies indeed in the inverse limit. Thus it is nonempty and surjects onto X(i). If the index set is only partially ordered, one may do the previous for any given element in X(i) and for indexes j that are not related to i one may use the same technique to construct further entries of an element in the product that lie in the inverse limit. Up to set theoretical subtelties (that I do not know - choice axiom?) I think, this way one can exhaust any index (together with the totally ordered set containing it). Regards Carlos Prieto On Thu, 9 Dec 2004, Don Davis wrote: >> Subject: Re:Question >> From: adel george >> Date: Wed, 8 Dec 2004 06:15:18 -0800 (PST) >> >> From:Dr.George,Adel A. >> >> I have the following question,please post it: >> Let X(i) be an inverse system of contractible >> topological spaces where for j>i the map X(j)--->X(i) >> is a continuous surjection with contractible fibers.I >> wish to show that X(= the inverse limit of the X(i)) >> is nonempty? >> >> Are there some set theoretic conditions that >> insure that an inverse limit is non-empty other than >> the familiar 2 conditions stated in Bourbaki "Set >> Theory"? >> >> Thank you. >> -- =================== PROF. CARLOS PRIETO Instituto de Matemáticas, UNAM 04510 México, DF, MÉXICO cprieto@math.unam.mx Tel. (++52-55) 5622-4489,-4520 Fax (++52-55) 5616 0348 =======================