Results 1 to 2 of 2

Thread: Show that [0,1] is uncountable

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    12

    Show that [0,1] is uncountable

    I am supposed to show that [0,1] is uncountable using Urysohn's Lemma (given a normal space X and two disjoint closed sets A and B of X, there is a continuous map from X to [a,b] that maps all of A to {a} and all of B to {b}). Any ideas on how to proceed?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    Hello.

    I was thinking about letting $\displaystyle X$ be the space consisting of all nonconstant sequences with either 0 or 1 as the elements under the discrete topology. (Note that it is T4). This space definitely has the same cardinality as the continuum, $\displaystyle (0,1)$.

    Let A be the singleton $\displaystyle (1,0,0,0,\ldots)$ and B be the singleton $\displaystyle (0,1,0,0,\ldots)$. By Urysohn's Lemma, we can find a continuous function $\displaystyle f: X\to [0,1]$ with $\displaystyle f(A)=\{0\}$ and $\displaystyle f(B)=\{1\}$.

    I was hoping that this would establish some kind of bijection between $\displaystyle X$ and $\displaystyle [0,1]$, but I'm not entirely sure that it is correct. But maybe I gave you some ideas. (And I suppose that it is better than receiving no responses on your post.)

    Good luck.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. show that M is uncountable
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Nov 24th 2011, 07:34 AM
  2. uncountable set
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Sep 20th 2009, 09:45 AM
  3. [SOLVED] uncountable subset is itself uncountable
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Feb 3rd 2009, 10:30 AM
  4. Replies: 4
    Last Post: Oct 11th 2008, 01:42 PM
  5. N^N is uncountable
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Mar 22nd 2008, 01:34 AM

Search Tags


/mathhelpforum @mathhelpforum