# Thread: Show that [0,1] is uncountable

1. ## 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?

2. Hello.

I was thinking about letting $X$ 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, $(0,1)$.

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

I was hoping that this would establish some kind of bijection between $X$ and $[0,1]$, 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.