Urysohn Lemma is
A topological space $(X, \mathcal{T})$ is normal if and only if for every
pair of disjoint nonempty closed subsets $C, D \subseteq X$ there is a continuous function $f : X \rightarrow [0, 1]$
such that $f(x) = 0$ for all $x \in C$ and $f(x) = 1$ for all $x \in D$.
are there some limits for the topology of $[0,1]$? if the topology of $[0,1]$ is trivial topology $\{\emptyset,X\}$, i think such $f$ always exists?
so my question is that does Urysohn Lemma means for any topology of $[0,1]$ there is always such $f$ exists?
Best Answer
No, it is not valid for any topology. The topology used here is the standard topology, the one induced from the standard topology on $\Bbb R$.