From: zeno@athena.mit.edu (Richard Duffy) Newsgroups: sci.math Subject: Re: Is Card(R)=Card(R^2)? Date: 23 Aug 92 01:17:46 GMT In article <16g754INNsmk@function.mps.ohio-state.edu> Gerald Edgar writes: >>How about the unit line minus the end points, and the interior of >>the unit square? > >A continuous bijection from the open interval onto the open square >is also not possible. The reason for it is a bit harder, this time. >But still not beyond a first course in point-set topology. > I'm curious to know whether the method Prof. Edgar has in mind is simpler or just radically different than the following: Given a continuous bijection f: I --> I x I where I is an open interval. I is a countable union of closed subintervals K_n , hence I x I is the countable union of the f(K_n). But I x I is open in R x R, hence has the Baire property that it is not a countable union of nowhere-dense subsets. [If you want to avoid the detail of proving that, just use I = R in the first place, via a homeomorphism, and the fact that R x R is a complete metric space.] So some particular f(K_n) is not nowhere-dense; being also closed (since it is compact, as the continuous image of the compact set K_n ), f(K_n) therefore contains an open ball B, which in turn contains a closed sub-ball C . Now the inverse image f^{-1}(C) is a closed subset of K_n and thus compact, so the restriction of f to this set is a homeomorphism onto C , i.e. f^{-1} restricted to C is continuous. But then f^{-1}(C) must be a connected subset of K_n , so it's a closed interval J. (We've essentially reduced the problem to the earlier one). Now just remove three points from C -- you still have a connected set, but its image under f^{-1}, which is J \ {three points} since f is bijective, can't possibly be connected. [Note that a closed interval with *two* points removed could still be connected.] So we've contradicted f out of existence. -- >> Richard Duffy -----------\___ ((lambda (x) (list x (list (quote quote) x))) Internet: zeno@athena.mit.edu \____ (quote (lambda (x) Bitnet: zeno%athena@MITVMA \______(list x (list (quote quote) x))))) Voicenet: +1 617 253 4045 \------------------ ============================================================================== From: wft@math.canterbury.ac.nz (Bill Taylor) Newsgroups: sci.math Subject: [0,1] homeomorphic to most of [0,1]^2 Date: 23 Aug 92 01:51:21 GMT There was a flurry of concern in sci.math a week or two ago, as to whether or not there was a bi-continuous bijection (i.e. a homeomorphism) between [0,1] and [0,1]^2 . Several posters wrote in with their conviction that such *did* exist, probably confusing vaguely remembered classic examples such as Cantor's non-continuous bijection, and Peano's continuous many-1 mapping. There were some excellent replies, ranging from the esoteric, to pointing out the obvious such as that [0,1] could be disconnected by removing a single point, whereas [0,1]^2 couldn't, so they could not possibly be homeomorphic. Hopefully they have killed off this potential piece of mathematical folklore. During the course of the discussion, I mentioned in passing that...... > there *is* a homeomorphism from the unit line to "most of" the unit square. > i.e. so that the image in the unit square has measure arbitrarily close to 1 . An "almost-homeomorphism", one might say, from [0,1] to [0,1]^2 . I have been asked (by email) to substantiate this claim. Being unable to find a reference for it, I have reconstructed and asciied-up a standard example as it was shown to me long ago. I thought I would post it here, in case there should be some others who would like to see such an example. Though it sounds amazing at first, the existence of an almost-homeomorphism will come as no surprise to topologists, who know that there is almost no connection between topology and measure theory. For instance there are dense sets of measure zero (the rationals), and nowhere-dense sets of measure 1-e , in the unit interval, for arbitrarily small positive e . These last could be called "thick Cantor sets". A thick Cantor set is constructed by removing central intervals from [0,1] and its subsequent sub-intervals, NOT all of length 1/3 of their intervals, as in the standard Cantor set, but of proportional lengths e, (e^2)/2, (e^3)/4 etc, so that the measure of the remnant is > 1-e-e^2-e^3-... > 1-2e . The intervals removed can be open, leaving a closed Cantor-like set; or closed, leaving the "quasi-interior" (i.e. no internal end-points) of one; having the same measure. The almost-homeomorphism I display, will have as its range, a subset of [0,1]^2 consisting (mostly) of the cross-product with itself, of one of these quasi-interior-of-thick-Cantor-sets. So this range will have measure very close to 1. Our first step is to create the set [0,1]^2 from which countably many cross-shaped sections have been removed, but crosses of decreasingly small measure. I believe this remnant set is sometimes called "Sierpinski dust". ~~~~~~~~~~~~~~~ It is totally disconnected, nowhere dense, but of very large measure. I illustrate it here.... ------------------------------------------------------------------------ | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | |======||======| |======||======| |======||======| |======||======| | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | ---------------' `--------------- ---------------' `--------------- | | | | ---------------. .--------------- ---------------. .--------------- | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | |======||======| |======||======| |======||======| |======||======| | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | --------------------------------- --------------------------------- | | | area e removed in this central cross | | | --------------------------------- --------------------------------- | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | |======||======| |======||======| |======||======| |======||======| | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | ---------------' `--------------- ---------------' `--------------- | | | area (e^2)/4 removed here | ---------------. .--------------- ---------------. .--------------- | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || + | | || | | || | | || | | || | |======||======| |======||======| |======||======| |=(e^3)/16==removed | || | | || | | || | | || | | + || + | | + || + | | + || + | | + || etc. | | || | | || | | || | | || | ------------------------------------------------------------------------ We will draw a continuous non-self-intersecting image of [0,1] in the unit square, that covers this Sierpinski dust. First join the 4 large subsquares with straight line segments as shown... ------------------------------------------------------------------- | | | | | | | ***** * | | ****** | | * * | | * * | | * * | | * * | | * * | | * * | | * * | | * * | | * * | | * ********------**** * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | --------------*---------------- ---------------*-------------- | 1 | | | | | | 0 B| | --------------*---------------- ---------------*-------------- | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | A| * | | * ***********------******** * | | * * | | ***** | | ***** | | | | | | | | | | | | | | | | | | | | | | | ------------------------------------------------------------------- The dotted lines show where further connections still have to be made. "0" and "1" label the ends of our image-set. Note that the three solid lines form a fixed part of the final curve; they will not be overwritten later (unlike the Peano example). Points "A" & "B" appear again below. To define the curve inside each quarter, join up as shown for the magnified bottom right quarter... (again, the dotted sections to be filled later) | | | B | -----------------------------*-------------------------------| | | | | | | | | | | | | | | | | | / | | | |/ | | | ************ ********** | | * | /| * | | * | / | * | | * | / | * | | * | | | * | | * | | | * | ------------*------------- | --------------*-------------| | \ | | | A *----------\ \--------------/ | | | \ | | -------------*------------ --------------*-------------| | * | | * | | * | | * | | * | | * | | * | | * | | * | | * | | *************-----*************** | | | | | | | | | | | | | | | | | | | | | -------------------------------------------------------------------| The other three quarters are to have internal joins in a similar way. Then make internal joins in the 16 size-(1/16) subsquares in a similar way to this; and so on in reducing recursive fashion; countably many times. The resulting curve is continuous and non-self-intersecting (i.e. 1-1), though both of these facts need a little proof. The curve is made up of countably many straight line sections, of total (areal) measure zero, and an uncountable number of "connecting" limit points, the Sierpinski dust, of measure close to 1. Q.E.D. ------------------------------------------------------------------------------- Bill Taylor wft@math.canterbury.ac.nz ------------------------------------------------------------------------------- Free will - the result of chaotic amplification of quantum events in the brain. Galaxies - the result of chaotic amplification of quantum events in the big bang. -------------------------------------------------------------------------------- ==============================================================================