(a) Given f:X→ℝN, 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∖C. When ρ is the bounded metric on C(X,ℝN), 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 ℝN.
(c) When C⊆X is compact and given ε>0, define Uε(C)={f | Δ(f|C)<ε} Show Uε(C) is compact.
(d) Show that if N=2m+1, then Uε(C) is dense in C(X,ℝN).
(e) Show there exists a continuous map F:X→ℝN such that F(x)→∞.
(f) Complete the proof.
7. Show that every m-manifold can be imbedded as a closed subspace into ℝ2m+1.
Proof: (a) Given n, let C⊆X be compact such that |f(x)|>n+1 for all x∈X∖C. Then |g(x)|>n for all x∈X∖C.
(b) If f(x)→∞, then define F:X∗→(ℝN)∗ by F(ΩX)=ΩℝN and F(x)=f(x) otherwise. Since X is first-countable, it suffices to show f(xn)→f(x) whenever xn→x. This is evident by continuity of f when x≠ΩX, and it follows from the definition of f(x)→∞ and of one-point compactifications when x=ΩX. And when f is injective, we see F is a homeomorphism whose image is closed in (ℝN)∗, so that f is a homeomorphism onto a closed subspace of ℝN.
(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 Uε(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 C(C,ℝN) by the result proved in Theorem 50.5, which by nature of the bounded metric ρ implies Uε(C) is open in C(X,ℝN).
(d) Let f:X→ℝN and δ>0 be given. By the result in Theorem 50.5, let g:C→ℝN 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 {Ui} be a countable basis for X. First, define a sequence Dn of compact subsets of X such that ∪Dn=X, such as by letting Dn be the union of those basis elements Ui for i<n with compact closure. Let C0=ø. Given compact Cn, by local compactness of X cover Cn by finitely many sets open in X of compact closure, and let Cn+1 be the union of these closures together with Dn; then Cn⊆Int Cn+1 for each n, and ∪Cn=X.
For all n, let Sn=Cn−Int Cn. Let f0:Cn→ℝ be void. Given a function fn:Cn→ℝ such that fn(x)=n for all x∈Sn, let gn+1:Cn+1∖Int Cn→[n,n+1] be such that gn+1(Sn)={n} and gn+1(Sn+1)={n+1}, and define fn+1:Cn+1→ℝ by fn+1(x)=fn(x) if x∈Cn and fn+1(x)=gn+1(x) otherwise. Then fn+1 is continuous by the pasting lemma, and is n+1 on Sn+1.
Since we see fn(x)=fm(x) for all x∈Cn whenever n≤m, define f:X→ℝ by f(x)=fn(x) when x∈Cn. Since every compact subset of X must be contained in Cn for some n (lest {Int Cn} 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∖Cn. Finally, define F:X→ℝN by πi(F(x))=f(x) for all i. It follows that F is continuous and F(x)→∞.
(f) By the Baire property and previous arguments, ∩U1/n(Cn) is dense in C(X,ℝN). Hence, let λ:X→ℝN be such that λ∈∩U1/n(Cn) 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 ℝN.
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. ◻
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