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From: andrews@yale.ARPA (Thomas O. Andrews)
Newsgroups: net.math
Subject: Re: cardinality of the unit square & the unit line
Message-ID: <457@yale.ARPA>
Date: Wed, 1-May-85 13:42:30 EDT
Article-I.D.: yale.457
Posted: Wed May  1 13:42:30 1985
Date-Received: Sat, 4-May-85 05:20:11 EDT
References: <1767@decwrl.UUCP> <72@harvard.ARPA> <54@utastro.UUCP> <536@gloria.UUCP>
Reply-To: andrews@yale-comix.UUCP (Thomas O. Andrews)
Organization: Yale University CS Dept., New Haven CT
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In article <536@gloria.UUCP> colonel@gloria.UUCP (Col. G. L. Sicherman) writes:
>
>Of course not.  If it were one-to-one and continuous, it would be a
>homeomorphism--the line and the square would be topologically the same.
>  .
>  .
>  .
>Col. G. L. Sicherman
>...{rocksvax|decvax}!sunybcs!colonel

  This statement is accurate, but somewhat misleading.  A map is a homeo-
morphism if it is 1-1, continuous, and has a continuous inverse.  In general
1-1 and continuous does not imply that the inverse is continuous.

  For example, the map f:[0,2*pi)-->circle defined by f(a)=(cos a,sin a) is
1-1 and continuous, but certainly not a homeomorphism.

  On the other hand, if f:X-->Y is 1-1 and continuous, and X is compact, then
the inverse of f is also continuous, and hence f is a homeomorphism.  Since
the unit interval is compact, the Col. Sicherman's statement is accurate.
-- 
					      Thomas Andrews

17?  My dear, what worthless, superstitious nonsense!