Exercise 11, § 4

(a) Let $X$ be a completely regular space, $Y$ a closed subspace of $X$; the mapping $\varphi : u \mapsto u|Y$ of $C^b(X; \mathbf{R})$ into $C^b(Y; \mathbf{R})$ is a continuous linear mapping of norm $\leq 1$. If $X$ is normal, $\varphi$ is a strict surjective morphism; moreover, if $X_Y$ denotes the quotient space of $X$ obtained by identifying all the points of $Y$ (space which is normal by virtue of IX, p. 105, exerc. 19), the kernel of $\varphi$ identifies (with its norm) with the subspace of $C^b(X_Y; \mathbf{R})$ formed by the functions which vanish at the point $\omega$, canonical image of $Y$ in $X_Y$.

(b) Suppose that $X$ is metrisable and that the boundary of $Y$ in $X$ is of countable type. Now prove that $\varphi : u \mapsto u|Y$ is a strict right invertible morphism (III, p. 47, prop. 3). (Let $d$ be a distance compatible with the topology of $X$; let $(a_n)$ be a sequence of points of $Fr(Y)$, dense in $Fr(Y)$, and let $V_{n,m}$ be the set of the $x \in X$ such that $d(x, a_n) < 1/m$ and $d(x, Y) > 1/2m$; form a family of continuous mappings $f_{n,m}$ of $X$ into $(0, 1)$, such that the support of $f_{n,m}$ is contained in $V_{n,m}$ and that one has $\sum_{n,m} f_{n,m}(x) = 1$ in $\mathscr{C} Y$, the family $(f_{n,m})$ being uniformly summable in a suitable neighbourhood of every point of $\mathscr{C} Y$. Show then that for every function $v \in C^b(Y; \mathbf{R})$, the function $u$ equal to $v$ in $Y$, to $\sum_{m,n} v(a_n) f_{n,m}(x)$ for every $x \in \mathscr{C} Y$, belongs to $C^b(X; \mathbf{R})$ and is such that $\varphi(u) = v$.\footnote{This result does not extend to the case where $X$ is the non-metrisable compact space $\beta N$, $Y$ the closed set $\beta N - N$ (EVT, IV, § 5, exerc. 5 c))}