Subject: Re: response and question From: John N Francis Date: Mon, 9 May 2005 13:49:33 -0400 (EDT) Andre's question was whether for any one map f there exists such a compactification. His question was NOT whether there exists such a compactification that simultaneously works for ALL maps. Thus, the question is still open (and still interesting). >> Subject: Re: related question >> From: adamp@mimuw.edu.pl (Adam Przezdziecki) >> Date: Mon, 9 May 2005 06:45:16 +0200 (CEST) >> >> The answer is no again. If the collection of "nice" maps is big enough to >> separate closed subsets >> (that is for each A and B closed in D there is a "nice" map f such that >> the closures of f(A) and f(B) are disjoint in X; this is the case for >> example when X contains an interval and smooth functions are "nice") >> then your extension property implies that the compactification B has to be >> homeomorphic to the Stone-\vCech compactification of D which has >> cardinality 2^\continuum, in particular (B,D) cannot be homeomorphic to >> (\bar D,D). >> >> Best Regards, >> Adam Przezdziecki >> >> >> Subject: related question >> >> From: Andre Henriques >> >> Date: Thu, 5 May 2005 23:11:28 -0400 (EDT) >> >> >> >> Here's a harder question >> >> related to that of Johannes Huebschmann >> >> whose answer I'd be interested to know: >> >> >> >> Let D be the open n-disk and \bar D be the closed n-disk. >> >> Let f be a 'nice' map from D to some 'nice' compact space X. >> >> Is there a compactification B of D such that f extends to B and >> >> such that the pair (B,D) is homeomorphic to the pair (\bar D,D)? >> >> >> >> Feel free to substitute 'nice' by any notion that you find convenient. >> >> Andre Henriques