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  1. Path: sparky!uunet!caen!zaphod.mps.ohio-state.edu!not-for-mail
  2. From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
  3. Newsgroups: sci.math
  4. Subject: Re: Is Card(R)=Card(R^2)?
  5. Date: 13 Aug 1992 07:54:35 -0400
  6. Organization: The Ohio State University, Dept. of Math.
  7. Lines: 13
  8. Message-ID: <16dihrINNpet@function.mps.ohio-state.edu>
  9. References: <1992Aug12.102140.5231@nntp.hut.fi> <1992Aug13.000928.12631@unidus.rz.uni-duesseldorf.de> <1992Aug13.011522.11161@informix.com>
  10. NNTP-Posting-Host: function.mps.ohio-state.edu
  11.  
  12.  
  13. >I believe that there is even an everywhere-continuous 
  14. >bijective mapping from the unit line to the unit square.
  15.  
  16. No, there is not.  The interval and the square are not homeomorphic.
  17. Such a continuous bijective map would be a homeomorphism,
  18. since the two spaces are compact Hausdorff spaces.
  19.  
  20. -- 
  21.   Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  22.   Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  23.   The Ohio State University      telephone: 614-292-0395 (Office)
  24.   Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
  25.