[FOM] Question on Number Line

[Lawrence Stout]

Suppose we take a different tack and claim that a physical string does 
not correspond to a closed interval [a,b] in the reals, but rather to 
an interval in which each interior point has a twin:  if $a < c < b$  
then there is another point $c'$ which is in exactly the same open sets 
as $c$ and has exactly the same strict order relationships.  Only the 
endpoints remain untwinned.  Note that from the topology and the order 
we cannot distinguish $c$ and $c'$.  When you cut such a string at $c$ 
you put $c$ in one half and $c'$ in the other.  Since you cannot tell 
them apart it does not matter which half gets which.  The resulting cut 
strings are homeomorphic to the original string and to eachother.

Lawrence Neff Stout
Professor of Mathematics
Illinois Wesleyan University
Bloomington, IL 61702-2900
http://www.iwu.edu/~lstout