(a) Given $f : X→ℝ^N$, we say $f(x)→∞$ (as $x→∞$) if for all $n∈ℕ$ there exists a compact subspace $C⊆X$ such that $|f(x)| > n$ whenever $x∈X \setminus C$. When $ρ$ is the bounded metric on $\mathcal{C}(X,ℝ^N)$, show that if $ρ(f,g) < 1$ and $f(x)→∞$, then $g(x)→∞$.
(b) Show that if $f(x)→∞$, then $f$ extends to a continuous mapping of one-point compactifications. Conclude that if $f$ is injective, then $X$ can be imbedded as a closed subspace into $ℝ^N$.
(c) When $C⊆X$ is compact and given $ε > 0$, define $$U_ε(C)=\{f~|~Δ(f|_C) < ε\}$$ Show $U_ε(C)$ is compact.
(d) Show that if $N=2m+1$, then $U_ε(C)$ is dense in $\mathcal{C}(X,ℝ^N)$.
(e) Show there exists a continuous map $F:X→ℝ^N$ such that $F(x)→∞$.
(f) Complete the proof.
7. Show that every $m$-manifold can be imbedded as a closed subspace into $ℝ^{2m+1}$.
Proof: (a) Given $n$, let $C⊆X$ be compact such that $|f(x)| > n+1$ for all $x∈X \setminus C$. Then $|g(x)| > n$ for all $x∈X \setminus C$.
(b) If $f(x)→∞$, then define $F: X^*→(ℝ^N)^*$ by $F(Ω_X)=Ω_{ℝ^N}$ and $F(x)=f(x)$ otherwise. Since $X$ is first-countable, it suffices to show $f(x_n)→f(x)$ whenever $x_n→x$. This is evident by continuity of $f$ when $x≠Ω_X$, and it follows from the definition of $f(x)→∞$ and of one-point compactifications when $x=Ω_X$. And when $f$ is injective, we see $F$ is a homeomorphism whose image is closed in $(ℝ^N)^*$, so that $f$ is a homeomorphism onto a closed subspace of $ℝ^N$.
(c) Note that $X$ is metrizable by the Urysohn metrization theorem, so that for each compact $C⊆X$ we see the image of the restriction of $U_ε(C)$ (technically, it requires specifying it is relative to $C$ rather than $X$, though by the Tietze extension theorem the point is moot) is open in $\mathcal{C}(C,ℝ^N)$ by the result proved in Theorem 50.5, which by nature of the bounded metric $ρ$ implies $U_ε(C)$ is open in $\mathcal{C}(X,ℝ^N)$.
(d) Let $f : X→ℝ^N$ and $δ > 0$ be given. By the result in Theorem 50.5, let $g : C→ℝ^N$ be such that $|f(x)-g(x)| < δ$ for all $x∈C$ and $Δ(g) < ε$. Extend $g-f|_C$ to a continuous map $h : X→[-δ,δ]^N$ by the Tietze extension theorem; then $k = h+f$ is such that $ρ(f,k) < δ$, and since $k|_C=g-f|_C+f|_C=g$, we have $Δ(k) < ε$.
(e) Let $\{U_i\}$ be a countable basis for $X$. First, define a sequence $D_n$ of compact subsets of $X$ such that $∪D_n=X$, such as by letting $D_n$ be the union of those basis elements $U_i$ for $i < n$ with compact closure. Let $C_0=ø$. Given compact $C_n$, by local compactness of $X$ cover $C_n$ by finitely many sets open in $X$ of compact closure, and let $C_{n+1}$ be the union of these closures together with $D_n$; then $C_n ⊆ \text{Int }C_{n+1}$ for each $n$, and $∪C_n=X$.
For all $n$, let $S_n=C_n-\text{Int }C_n$. Let $f_0 : C_n→ℝ$ be void. Given a function $$f_n : C_n→ℝ$$ such that $f_n(x)=n$ for all $x∈S_n$, let $$g_{n+1} : C_{n+1} \setminus \text{Int }C_n → [n,n+1]$$ be such that $g_{n+1}(S_n)=\{n\}$ and $g_{n+1}(S_{n+1})=\{n+1\}$, and define $$f_{n+1} : C_{n+1}→ℝ$$ by $f_{n+1}(x)=f_n(x)$ if $x∈C_n$ and $f_{n+1}(x)=g_{n+1}(x)$ otherwise. Then $f_{n+1}$ is continuous by the pasting lemma, and is $n+1$ on $S_{n+1}$.
Since we see $f_n(x)=f_m(x)$ for all $x∈C_n$ whenever $n≤m$, define $f : X→ℝ$ by $f(x)=f_n(x)$ when $x∈C_n$. Since every compact subset of $X$ must be contained in $C_n$ for some $n$ (lest $\{\text{Int }C_n\}$ be a cover with no finite subcover), and since $X$ is compactly generated, we see $f$ is continuous. Further, $f(x)→∞$ because $f(x) ≥ n$ whenever $x∈X \setminus C_n$. Finally, define $F : X→ℝ^N$ by $π_i(F(x))=f(x)$ for all $i$. It follows that $F$ is continuous and $F(x)→∞$.
(f) By the Baire property and previous arguments, $∩U_{1/n}(C_n)$ is dense in $\mathcal{C}(X,ℝ^N)$. Hence, let $λ : X→ℝ^N$ be such that $λ∈∩U_{1/n}(C_n)$ and $ρ(λ,f) < 1$. It follows that $Δ(λ)=0$ so that $λ$ is injective, hence by (b) $λ$ is an imbedding of $X$ onto a closed subspace of $ℝ^N$.
7. Being regular and locally Euclidean, $m$-manifolds are locally compact, and being Hausdorff and second countable the other qualities necessary to apply the previous theorem follow together with an application of Theorem 50.1.$~\square$
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