[FOM] Characterization of the real numbers
JoeShipman at aol.com
Sun Feb 6 23:59:48 EST 2005
That's a really nice theorem, which is just what I was looking for; just 2 questions:
1) what's the simplest way to formally define "order continuous" for a function from X^2 to X where X is an ordered space?
2) How do you prove this?
THEOREM. Let X be a linear ordering without endpoints. Then X is order
isomorphic to the real line if and only if
i) X has the least upper bound property;
ii) there is an order continuous F:X^2 into X such that for all x,y, x < y
implies x < F(x,y) < y.
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