From: rusin@washington.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: bijection = homeomorphism Date: 16 Feb 1995 16:54:04 GMT In article <3hvfq4$d53@newstand.syr.edu>, Thomas R. Scavo wrote: >>> Under what conditions is a continuous bijection a homeomorphism? > >... I want to be able to say "If a >bijection f: X -> Y is continuous and ..., then its inverse is >continuous." ... >Isn't it true that X compact will suffice? No. If you have a bijection between the points in two spaces, you might as well think of f as being a continuous map between two topological spaces built on the same underlying set, that is, view f as f : (X, T1) --> (X, T2) where T1 and T2 are two topologies and f is the identity map. When is f continuous? If and only if T1 contains T2. When is its inverse continuous? If and only if T2 contains T1. So your question asks, under what conditions can I assert that two given topologies on a space are equal, knowing the first contains the second? (Equivalently, in what family of conditions is the topology T1 minimal?) Having T1 compact is not enough -- just take T2 = { {}, X } (the coarse topology). Having T1 compact and T2 Hausdorff _does_ turn out to be sufficient, that is, a compact cannot properly contain any Hausdorff topologies. I find this to be an entertaining perspective: it sort of says compact topologies have to have relatively few open sets in them. Turned the other way ("a Hausdorff topology cannot be properly contained in a compact topology") it says Hausdorff topologies have to have kind of a lot of open sets. Taken together ("no two compact, Hausdorff topologies are comparable") this makes it clear that compact+Hausdorff is quite a restrictive condition, and thus one from which many nice results can be expected to follow (indeed, Bourbaki _defines_ compact to include the Hausdorff axiom). In view of this it is a pleasant accident of Nature that such familiar spaces as [0,1] are indeed compact and Hausdorff. dave ============================================================================== Date: Mon, 20 Feb 1995 13:58:49 +1000 To: rusin@washington.math.niu.edu (Dave Rusin) From: ibokor@metz.une.edu.au (ibokor) Subject: Re: Re: bijection = homeomorphism Subject: Re: bijection = homeomorphism From: Dave Rusin, rusin@washington.math.niu.edu Date: 16 Feb 1995 16:54:04 GMT In article <3hvvvc$dhp@watson.math.niu.edu> Dave Rusin, rusin@washington.math.niu.edu writes: >In article <3hvfq4$d53@newstand.syr.edu>, >Thomas R. Scavo wrote: >>>> Under what conditions is a continuous bijection a homeomorphism? >> >>... I want to be able to say "If a >>bijection f: X -> Y is continuous and ..., then its inverse is >>continuous." ... >>Isn't it true that X compact will suffice? > >No. > >If you have a bijection between the points in two spaces, you might as >well think of f as being a continuous map between two topological >spaces built on the same underlying set, that is, view f as > f : (X, T1) --> (X, T2) >where T1 and T2 are two topologies and f is the identity map. >When is f continuous? If and only if T1 contains T2. When is its >inverse continuous? If and only if T2 contains T1. So your question >asks, under what conditions can I assert that two given topologies on a >space are equal, knowing the first contains the second? (Equivalently, >in what family of conditions is the topology T1 minimal?) > >Having T1 compact is not enough -- just take T2 = { {}, X } (the >coarse topology). > >Having T1 compact and T2 Hausdorff _does_ turn out to be sufficient, >that is, a compact cannot properly contain any Hausdorff topologies. >I find this to be an entertaining perspective: it sort of says compact >topologies have to have relatively few open sets in them. Turned >the other way ("a Hausdorff topology cannot be properly contained in >a compact topology") it says Hausdorff topologies have to have kind of >a lot of open sets. Taken together ("no two compact, Hausdorff topologies >are comparable") this makes it clear that compact+Hausdorff is quite a >restrictive condition, and thus one from which many nice results can >be expected to follow (indeed, Bourbaki _defines_ compact to include >the Hausdorff axiom). In view of this it is a pleasant accident of >Nature that such familiar spaces as [0,1] are indeed compact and Hausdorff. > >dave Strange! What about the compactification of a non-compact space? How does that fit into this scheme? ============================================================================== Date: Mon, 20 Feb 95 09:20:15 CST From: rusin (Dave Rusin) To: ibokor@metz.une.edu.au Subject: Re: Re: bijection = homeomorphism >>Having T1 compact and T2 Hausdorff _does_ turn out to be sufficient, >>that is, a compact cannot properly contain any Hausdorff topologies. >>I find this to be an entertaining perspective: it sort of says compact >>topologies have to have relatively few open sets in them. Turned >>the other way ("a Hausdorff topology cannot be properly contained in >>a compact topology") it says Hausdorff topologies have to have kind of >>a lot of open sets. Taken together ("no two compact, Hausdorff topologies >>are comparable") this makes it clear that compact+Hausdorff is quite a >>restrictive condition, and thus one from which many nice results can >>be expected to follow (indeed, Bourbaki _defines_ compact to include >>the Hausdorff axiom). In view of this it is a pleasant accident of >>Nature that such familiar spaces as [0,1] are indeed compact and Hausdorff. >> >>dave > >Strange! What about the compactification of a non-compact space? >How does that fit into this scheme? It's interesting that you would think of that. A "compactification" of a space X usually means a larger space Z containing X such that (1) Z is compact, and (2) the topology of Z agrees with that of X on the subset X of Z. (Is that clear? You're not supposed to take X={0,1} with the coarse topology and Z={0,1,2} with the discrete topology, since then when you view X as a subset of Z, it will have different open sets than the topological space X did in the first place.) The key point here is that to make the compactification you add _points_, not _open sets_. In the discussion you quoted, what stays fixed is the collection of points; what is varying is the collections of open sets on that set of points. It follows as a part of that discussion that if (X,T) is a topological space which is compact, then so is (X, T') where T' is any smaller (i.e. "coarser") topology on X; thus it would make perfect sense to define the "Ibokor compactification" of any topological space (X,T) to be the following space: the collection of points X is not changed, but the topology is the largest ("finest", or "strongest") one in T in which X is compact -- the point of all the previous discussion being that this toplogy is well defined. Undoubtedly there are non-trivial topologies on spaces X for which this Ibokor compactification would turn out to be the extreme case (X, coarse). I don't even know, for example, what the I.c. of the open interval (0,1) would be. For the record: there are different possible compactifications in the usual sense, too. There is the one-point compactification X* = X union one more point, with a certain topology. There is also the Stone-Cech compactification, which is quite large but of particular importance in functional analysis. But these all meet the description of Z I posed earlier. I don't know what Bourbaki has to say about these -- is that what you meant? dave