[FOM] Characterization of Linear Continua
Jacques Carette
carette at mcmaster.ca
Wed Feb 4 13:55:00 EST 2004
That terminology is standard - see p.31 of "Topology: A First Course" by
James Munkres (1975 edition).
See p.152-154 of the same book for related theorems.
Exercise 13. on p.159 either provides a counter-example OR illustrates a
subtlety in the assumptions of the claim below - I do not currently have the
time to investigate this further, but I hope these references help.
Jacques
-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Harvey Friedman
Sent: February 4, 2004 11:07 AM
To: fom
Subject: [FOM] Characterization of Linear Continua
This will be put into the numbered postings (hopefully with further
results/questions) if it survives claims by others...
#################################
A linear continuum is a linear ordering which
i) has at least two points;
ii) is dense (between any two points there is a third);
iii) every nonempty subset which is bounded above has a least upper bound;
iv) every nonempty subset which is bounded below has a greatest lower bound.
WARNING: This terminology, linear continuum, may not be standard.
Note that a linear continuum may or may not have endpoints.
Let X be a linear continuum. It is obvious that we have a robust notion of
continuous functions f:X into X, using open intervals surrounding points. It
is also obvious that we have a robust notion of continuous f:X dot X into X.
THEROEM. Let X,Y be linear continua both satisfying the following condition.
There is a continuous (everywhere defined) binary function f such that for
all x < y, x < f(x,y) < y. Then X,Y are isomorphic, if we remove the
endpoints (if any). In fact, X,Y with endpoints removed are isomorphic to
the real line.
We can view this as a
*derivation of separability*.
Harvey Friedman
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