Re: Connectedness question

Michiel de Bondt wrote: 
Ryan Reich wrote:


some deep topology

I think a course in topology is not required to transfer what is on stake here. Both in Q and R, there is a concept of distance: the distance between numbers a and b is |a - b|, the absolute difference of
both numbers. More generally, other sets can have a concept of
distance: the distance between a and b is denoted as d(a, b) usually. A
set is complete if the following holds: if a_1, a_2, a_3, ... is a
Cauchy sequence (google for definition), then it converges to a limit
within the set. Now 1/0!, 1/0!+1/1!, 1/0!+1/1!+1/2!, 1/0!+1/1!+1/2!,
... is a Cauchy sequence that converges to e = exp(1). But e is not a
rational number. So Q is not complete. R is complete, and in fact, R is
the smallest complete set that contains Q, and therefore called the completion of Q. 

Please give more context when you quote; I didn't write the three words
you cited and I have no idea what you purport to be responding to.  In
addition, although it IS true that both Q and R have metrizable topologies
(and the hyperreals do not), that is not actually related to my
explanation and, if you connect what I was trying to explain with what
"none" was asking, is actually completely the opposite of what I said.

--
Ryan Reich
ryanr@xxxxxxxxxxxx
.

Relevant Pages

  • Why Topology? (was irrationality 0.201201020100201000201...)
    ... by equating it with a "metric space", ... Much of the motivation for point set topology comes from ... Note that the definition above actually says that the *distance* ... geodesics in string space. ...
    (sci.math)
  • Why Topology? (was irrationality 0.201201020100201000201...)
    ... by equating it with a "metric space", ... Much of the motivation for point set topology comes from ... Note that the definition above actually says that the *distance* ... geodesics in string space. ...
    (sci.cognitive)
  • Why Topology? (was irrationality 0.201201020100201000201...)
    ... by equating it with a "metric space", ... Much of the motivation for point set topology comes from ... Note that the definition above actually says that the *distance* ... geodesics in string space. ...
    (sci.physics)
  • Re: Connectedness question
    ... I think a course in topology is not required to transfer what is on stake here. ... Both in Q and R, there is a concept of distance: the distance between numbers a and b is |a - b|, the absolute difference of both numbers. ... Cauchy sequence that converges to e = exp. ...
    (sci.math)
  • Re: homeomorphism vs isomorphism
    ... But we don't have no distance in topology:) ... As far as I know all structure there is in topology is as much as is ... needed for the notion of continuity, which is open sets. ... homeomorphism is an invertible mapping continuous in both directions. ...
    (sci.math)