= Solution
<Urysohn's lemma> states that, in a <normal space>, disjoint <closed> <subsets> $A,B$ can be separated by a <continuous function> $u:X\to[0,1]$ with $u=0$ on $A$ and $u=1$ on $B$.
First use the shrinking consequence of the <separation> property of a <normal space>: if a <closed set> $F$ lies in an <open set> $G$, there is an <open> $W$ with $F\subseteq W\subseteq\overline W\subseteq G$. To prove it, separate the disjoint <closed sets> $F$ and $X\setminus G$ by disjoint <open> <neighborhoods> $W,V$. Then $\overline W\subseteq X\setminus V\subseteq G$.
Let $D$ be the dyadic rationals in $[0,1]$. <Set> $U_1=X\setminus B$, and shrink around $A$ to choose $U_0$ with $A\subseteq U_0\subseteq\overline{U_0}\subseteq U_1$. At each subsequent dyadic stage, insert between each adjacent pair $s<t$ an <open set> $U_{(s+t)/2}$ such that
$$
\overline{U_s}\subseteq U_{(s+t)/2}\subseteq\overline{U_{(s+t)/2}}\subseteq U_t.
$$
The shrinking lemma applies because the first <set> is <closed> and contained in the last <open set>. Induction gives $\overline{U_s}\subseteq U_t$ for all dyadic $s<t$. Part (a) now constructs a <continuous> $u$. Each point of $A$ lies in $U_0$, so has value zero. Each point of $B$ lies in none of the $U_t$, so has value one. This proves the lemma.
For the embedding, let $I=C(X,[0,1])$, a <set> of <continuous functions>, and define the evaluation map
$$
e:X\to\mathbb R^I,\qquad e(x)=(u(x))_{u\in I}.
$$
Every coordinate is <continuous>, so $e$ is <continuous> in the <product topology>. The <Hausdorff> assumption makes singletons <closed>. For distinct points, <Urysohn's lemma> applied to their singletons gives a coordinate with different values, so $e$ is <injective>.
To check the inverse <topology>, take $x\in O$ with $O$ <open>. Apply the lemma to $\{x\}$ and $X\setminus O$, choosing a coordinate $u$ with $u(x)=1$ and $u=0$ outside $O$. Then
$$
x\in u^{-1}((1/2,3/2))\subseteq O.
$$
This is the preimage of an <open> coordinate cylinder under $e$. Thus every original <neighborhood> is detected by coordinate <neighborhoods>, and $e^{-1}:e[X]\to X$ is <continuous>. \b[The evaluation map is a <topological embedding>.]
Back to article page
