Re: No homeomorphism (0,1) <--> [0,1]

From: José Carlos Santos <jcsantos@xxxxxxxx>
Date: Thu, 08 Sep 2005 17:39:32 +0100

On 08-09-2005 17:34, Michael Stemper wrote:

I'm trying to prove that there isn't a homeomorphism between [0,1]
and (0,1), and I'm stuck. What I know so far is the definitions of
homeomorphism and continuity.


Suppose that there was a homeomorphism f:[0,1] --> ]0,1[. It's
continuous, and so, by Weierstrass' theorem, it has a maximum and a
minimum; let's call them M and m respectively. You know that m and M
belong to ]0,1[ and that f([0,1]) is a subset of [m,M]. Therefore, f is
not surjective.

Best regards,

Jose Carlos Santos
.

Follow-Ups:
- Re: No homeomorphism (0,1) <--> [0,1]
  - From: Michael Stemper

References:
- No homeomorphism (0,1) <--> [0,1]
  - From: Michael Stemper

Prev by Date: No homeomorphism (0,1) <--> [0,1]
Next by Date: Re: help with sets
Previous by thread: No homeomorphism (0,1) <--> [0,1]
Next by thread: Re: No homeomorphism (0,1) <--> [0,1]
Index(es):
- Date
- Thread

Relevant Pages

Re: No homeomorphism (0,1) <--> [0,1]
... and, and I'm stuck. ... What I know so far is the definitions of homeomorphism and continuity. ... Math Tutor in Central New Jersey and Manhattan ...
(sci.math)
Re: No homeomorphism (0,1) <--> [0,1]
... >>and, and I'm stuck. ... >>homeomorphism and continuity. ... connectedness; and that connectedness is a homeomorphic ...
(sci.math)
Re: No homeomorphism (0,1) <--> [0,1]
... >continuous, and so, by Weierstrass' theorem, ... definitions of metric space, continuity, and homeomorphism. ... For anybody playing along at home, this exercise is a post script ...
(sci.math)
Re: No homeomorphism (0,1) <--> [0,1]
... >>>and, and I'm stuck. ... >>>homeomorphism and continuity. ... >connectedness; and that connectedness is a homeomorphic ...
(sci.math)
Re: No homeomorphism (0,1) <--> [0,1]
... >>> and, and I'm stuck. ... >>continuous, and so, by Weierstrass' theorem, ... >definitions of metric space, continuity, and homeomorphism. ...
(sci.math)