Urysohn's Lemma and the Tietze Extension Theorem

Let $C$ be a closed subset of a metric space ($X, \rho$). Show that the distance to $C$ function $\operatorname {dist}_{C}$ is continuous and $\operatorname {dist}_{C}(x) = 0$ if and only if $x$ belongs to $C$.

Provide an example of a continuous real-valued function on the open interval $(0,1)$ that is not extendable to a continuous function on $\mathbb {R}$. Does this contradict the Tietze Extension Theorem?

Deduce Urysohn's Lemma as a consequence of the Tietze Extension Theorem.

State and prove a version of the Tietze Extension Theorem for functions with values in $\mathbb {R}^{n}$.

Suppose that a topological space $X$ has the property that every continuous, bounded real-valued function on a closed subset has a continuous extension to all of $X$. Show that if $X$ is Tychonoff, then it is normal.

Let ($X, \mathcal{T}$) be a normal topological space and $\mathcal{F}$ the collection of continuous real-valued functions on $X$. Show that $\mathcal{T}$ is the weak topology induced by $\mathcal{F}$.

Show that the function $\rho$ defined in the proof of the Urysohn Metrization Theorem is a metric that defines the same topology as the given topology.

Let $X$ be a normal topological space, $F$ a closed subset of $X$, and $f$ a continuous not necessarily bounded, real-valued function on $F$. Then $f$ has a continuous extension to a real-valued function $\bar{f}$ on all of $X$. Prove this as follows:

Show that a mapping $f$ from a topological space $X$ to a topological space $Y$ is continuous if and only if there is a subbase $\mathcal{S}$ for the topology on $Y$ such that the preimage under $f$ of each set in $\mathcal{S}$ is open in $X$. Use this to show that if $Y$ is a closed, bounded interval $[a, b]$, then $f$ is continuous if and only if for each real number $c \in (a, b)$, the sets $\left\{ x \in X \mid f(x) < c\right\}$ and $\left\{ x \in X \mid f(x) > c\right\}$ are open.

Use the preceding problem to prove Lemma 1.

Lemma 1: Let $X$ be a topological space. For $\Lambda$ a dense subset of the open, bounded interval of real numbers $(a, b)$, let $\left\{ O_{\lambda }\right\}_{\lambda \in \Lambda }$ be a normally ascending collection of open subsets of $X$. Define the function $f: X \rightarrow \mathbb {R}$ by setting $f = b$ on $X \sim \bigcup_{\lambda \in \Lambda } O_{\lambda }$ and otherwise setting $f(x) = \inf \left\{ \lambda \in \Lambda \mid x \in O_{\lambda }\right\}$. Then $f: X \rightarrow \left[a, b\right]$ is continuous.