Negligibility and Deficiency

78 2 Fundamental Results on Infinite-Dimensional Manifolds

n

2⁻⁽ⁿ⁻¹⁾ (−∞,(n−1)−2⁻⁽ⁿ⁻¹⁾]×R

{n−1} ×[−2⁻ⁿ,∞)

[n−2⁻ⁿ,∞)× {0} 2⁻ⁿ

n−1

Fig. 2.7. The ambient invertible isotopyφ⁽ⁿ⁾

2.4 Negligibility and Deficiency 79 It should be remarked that 2⁻ⁿ⁺¹γn(x) is the distance of (x(2), . . . , x(n))∈ Rⁿ⁻¹from the complement of [−2⁻ⁿ⁺¹,2⁻ⁿ⁺¹]ⁿ⁻¹ (the 2⁻ⁿ⁺¹-neighborhood of0∈Rⁿ⁻¹) with respect to the norm∥y∥∞= max{|y(1)|, . . . ,|y(n−1)|}.¹³ For each n ∈ N, we denote hn = ψ₁⁽ⁿ⁾· · ·ψ₁⁽¹⁾ : R^N → R^N. For the sake of convenience, let h0 = idR^N. If there exists a homeomorphism h∞ : R^N\ {0} → R^N such that each x ∈ R^N\ {0} has a neighborhood U in R^N\ {0} such that h∞|U = ψ⁽ⁿ⁺¹⁾_t hn|U = hn|U for a sufficiently large n ∈ N and any t ∈ I, and each y ∈ R^N has a neighborhood V such that h⁻¹_∞|V = h⁻¹_n (ψ_t⁽ⁿ⁺¹⁾)⁻¹|V = h⁻¹_n |V for a sufficiently large n ∈ N and any t∈I, then the desired homeomorphismζ can be defined as follows:

ζ(x, t) =

{(ψ⁽ⁿ⁾₂₋₂nthn−1(x), t) if 2⁻ⁿ 6t62⁻⁽ⁿ⁻¹⁾ (h∞(x),0) if t= 0.

To see the existence ofh∞, it suffices to prove the following two facts:

(1) Eachx∈R^N\ {0}has a neighborhoodU inR^N\ {0}andn∈Nsuch that ψ_t^(j)|hn(U) = id forj > n;

(2) Eachy∈R^Nhas a neighborhoodV inR^Nandn∈Nsuch thath⁻¹_n (V)⊂ R^N\ {0}and (ψ^(j)_t )⁻¹|V = id forj > n.

(1): For each x ∈ R^N\ {0}, let m = min{i ∈ N | x(i) ̸= 0}. In other words, x(m) is the first nonzero coordinate ofx. We write y=hm−1(x) and z=hm(x) =ψ₁^(m)(y).

Whenm= 1, y =h0(x) =x andz =ψ⁽¹⁾₁ (x). Sincex(1) ̸= 0, it follows that (z(1), z(2))̸∈[2⁻¹,∞)× {0}, that is,z(1)<2⁻¹ orz(2)̸= 0.

Whenm >1, observe that

hm−1(x) =y= (y(1),0, . . . ,0, y(m), x(m+ 1), x(m+ 2), . . .) and hm−1(x)̸=hm−1(0,0, . . . ,0, x(m+ 1), x(m+ 2), . . .)

= (m−1,0, . . . ,0, x(m+ 1), x(m+ 2), . . .).

Thus, we have y(1)̸=m−1 ory(m)̸= 0.

In the casey(1)̸=m−1, since (z(1), z(m+ 1))̸∈[m−2^−m,∞)× {0}, it follows that z(1)< m−2^−m orz(m+ 1)̸= 0.

Ify(m)̸= 0, then z(m) =y(m)̸= 0. Anyway, we have

z(1)< m−2^−m or max{|z(2)|, . . . ,|z(m+ 1)|} ̸= 0.

Whenz(1)< m−2^−m,V ={z^′∈R^N|z^′(1)< m−2^−m}is a neighborhood ofzinR^Nandψ^(j)_t |V = id for everyj > m. Then,U =h⁻¹_m(V) is the desired neighborhood ofxinR^N.

13This norm of Rⁿ⁻¹ can be replaced with ∥y∥² = ( ∑n−1

i=1 y(i)²)1/2

or ∥y∥¹ =

∑n−1

i=1 |y(i)|. In this case,γnis defined byγn(x) = max{

0,1−2ⁿ⁻¹( ∑n

2x(i)²)1/2} orγn(x) = max{

0,1−2ⁿ⁻¹∑n 2|x(i)|}

.

80 2 Fundamental Results on Infinite-Dimensional Manifolds

When max{|z(2)|, . . . ,|z(m+ 1)|} ̸= 0, choose n>m+ 1 so that 2⁻ⁿ <

max{|z(2)|, . . . ,|z(m+ 1)|}. Letz^∗=hn(x). Sincez^∗(2) =z(2), . . . , z^∗(m+ 1) =z(m+ 1), we have

max{|z^∗(2)|, . . . ,|z^∗(n)|}>max{|z(2)|, . . . ,|z(m+ 1)|}>2⁻ⁿ. So,xhas a neighborhoodU in R^Nsuch that

z^′∈hn(U)⇒max{|z^′(2)|, . . . ,|z^′(n)|}>2⁻ⁿ.

Then,γn(z^′) = 0 for eachz^′∈hn(U). Hence,ψ_t^(j)|hn(U) = id for everyj > n.

(2): For each y ∈ R^N, choose n ∈ N so that y(1) < n−2⁻ⁿ. Then, V ={z ∈ R^N| z(1)< n−2⁻ⁿ} is a neighborhood of y in R^N and hn(0) = (n,0,0, . . .) ̸∈ V, that is, h⁻¹_n (V) ⊂ R^N \ {0}. For each j > n, we have ψ^(j)_t |V = id, that is, (ψ^(j)_t )⁻¹|V = id. ⊓⊔

Using Proposition 2.4.1 above, we prove the following:

Proposition 2.4.2 LetX be anE-stable perfectly normal paracompact space andA be anE-deficient closed set in X. Then, for each open cover U of X, there exists an I-preserving homeomorphism

h:X×I→X×I\A× {0}

such that each ht is U-close to idX. In particular, h0 : X → X \A is a homeomorphism that is U-close toidX.

Proof. For simplicity, we denote F = R^N or R^Nf depending on E ≈ E^N or E ≈ E_f^N. Since E ≈ E ×F by Lemma 2.1.7, X is F-stable and A is F-deficient in X (Corollary 2.3.5). Let ζ : F ×I\ {(0,0)} → F ×I be the I-preserving homeomorphism obtained in Proposition 2.4.1. Then, we have anI-preserving homeomorphism

φ: (X×F×I)\(A× {0} × {0})→X×F×I

defined by φ(x, y, t) = (x, ζmax{t,α(x)}(y), t), where α : X → I is a map such thatα⁻¹(0) =A. Let V be an open star-refinement of U. By Corollary 2.3.4(iv), we have a homeomorphismf :X×F →X such thatf(x,0) =xfor eachx∈Aandf isV-close to pr_X, so pr_Xf⁻¹ isV-close to idX. The desired I-preserving homeomorphismhcan be defined byh(x, t) = (f φ⁻¹_t f⁻¹(x), t).

Indeed, each ht =f φ⁻¹_t f⁻¹ is V-close to pr_Xφ⁻¹_t f⁻¹ = pr_Xf⁻¹ because φ preservesX-coordinates. Then, it follows thathtisU-close to idX. ⊓⊔

A set A in a space X is said to be negligible in X ifX \A ≈X. It is said thatAislocally closedinX if eachx∈Ahas a neighborhoodU in X such thatA∩U is closed inU. It is easy to see thatA is locally closed inX if and only if A=W ∩clXA for some open setW in X. Now, we have the following Negligibility Theorem:

2.4 Negligibility and Deficiency 81 Theorem 2.4.3 (Negligibility) LetX be anE-stable metrizable space. If A is an E-deficient locally closed set in X, then the inclusion X\A⊂X is a near-homeomorphism, hence Ais negligible in X.

Proof. LetU ∈cov(X) andd∈Metr(X). Choose an open setW inX so that A⊂W andAis closed inW. Due to Corollary 2.3.4(i), (ii),W isE-stable and AisE-deficient. By Proposition 2.1.16,W has an open coverV that is fitting inXand refinesU. By Proposition 2.4.2, we have a homeomorphismh:W → W \Athat is V-close to id. Then, hcan be extended to a homeomorphism

˜h:X →X\Aby ˜h|X\W = id. The inverse homeomorphism ˜h⁻¹:X\A→X isU-close to the inclusionX\A⊂X. ⊓⊔

A locally closed set in a metrizable space X is Fσ in X, because it is the intersection of a closed set and an open set. Hence, anE-deficient locally closed set inX is a countable union of E-deficient closed sets inX, where it should remarked that a countable union of E-deficient locally closed sets in X can be written as a countable union ofE-deficient closed sets inX. In the case where X is completely metrizable, Theorem 2.4.3 can be strengthened slightly as follows:

Theorem 2.4.4 (Negligibility) Let X be an E-stable completely metriz-able space. IfA is a countable union ofE-deficient closed sets inX, then the inclusion X\A⊂X is a near-homeomorphism, henceA is negligible inX. Proof. LetA=∪

n∈NAn, where eachAnis anE-deficient closed set inX. For eachn∈N, letXn=X\∪n

i=1Ai. Note thatXn is open inX, and is hence completely metrizable. For eachU ∈ cov(X),X has an admissible complete metricd0 such that{Bd0(x,1)|x∈X} ≺ U (cf. 1.3.18(1)).

Since the inclusionX1⊂X is a near-homeomorphism by Theorem 2.4.3, we have a homeomorphism f1 : X → X1 with d0(f1,id) < 2⁻¹ (hence d0(f₁⁻¹,id)<2⁻¹). Choose an admissible complete metricd1 forX1 so that

d1(x, x^′)>max{d0(x, x^′), d0(f₁⁻¹(x), f₁⁻¹(x^′))} for eachx, x^′∈X1. Since A2 ∩X1 is an E-deficient closed set in X1 and X2 = X1\A2, the inclusionX2⊂X1is a near-homeomorphism by Theorem 2.4.3. Then, we have a homeomorphism f2 :X1→X2 with d1(f2,id) <2⁻² (henced1(f₂⁻¹,id) <

2⁻²). By induction, we can obtain homeomorphisms fn : Xn−1 → Xn, and admissible complete metricsdn forXn, such that dn−1(fn,id)<2⁻ⁿ (hence dn−1(f_n⁻¹,id)<2⁻ⁿ) and

dn(x, x^′)>max{dn−1(x, x^′), dn−1(f_n⁻¹(x), f_n⁻¹(x^′))} for eachx, x^′∈Xn, where X0 = X. Then, (fnfn−1· · ·f1)n∈N is a Cauchy sequence in C(X, X) with respect to the sup-metric induced byd0, so it converges to a map

f = lim

n→∞fnfn−1· · ·f1:X →X.

82 2 Fundamental Results on Infinite-Dimensional Manifolds Since d0(f,id) 6∑

n∈N2⁻ⁿ = 1, it follows that f is U-close to id. For each x∈X andi∈N, lety =fi· · ·f1(x)∈Xi. Then, (fnfn−1· · ·fi+1(y))n>i is a Cauchy sequence inXi with respect todi, which implies

f(x) = lim

n>ifnfn−1· · ·fi+1(y)∈Xi. Therefore,f(X)⊂∩

i∈NXi=X\A. It remains to show that f :X →X\A is a homeomorphism.

To construct the inverse of f :X →X\A, observe that for eachn∈N andx∈X\A,

d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹(x), f₁⁻¹· · ·f_n−1⁻¹ (x))

6d1(f₂⁻¹· · ·f_n−1⁻¹ f_n⁻¹(x), f₂⁻¹· · ·f_n−1⁻¹ (x))6· · ·

6dn−2(f_n−1⁻¹ f_n⁻¹(x), f_n−1⁻¹ (x))6dn−1(f_n⁻¹(x), x)<2⁻ⁿ. Hence, (f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹|X\A)n∈Nis a Cauchy sequence in C(X\A, X) with respect to the sup-metric induced byd0, and this converges to a map

g= lim

n→∞f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹:X\A→X.

For each x∈X\Aandε >0, choosen∈Nso that

d0(f g(x), f f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹(x))< ε/2 and d0(f, fnfn−1· · ·f1)< ε/2, which implies thatd0(f g(x), x)< ε. Hence,f g(x) =xfor eachx∈X\A.

Now, we shall show that gf(x) = xfor each x∈ X.¹⁴ For each ε > 0, choose n∈Nso that

d0(g, f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹|X\A)< ε/3 and

d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹|Xm, f₁⁻¹· · ·f_m−1⁻¹ f_m⁻¹)< ε/3 for everym>n.

By the continuity off₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹ atf(x), we haveδ >0 such that y∈Xn, dn(f(x), y)< δ

⇒d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹f(x), f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹(y))< ε/3.

Choosem>nso that 2^−m< δ. Then, dn(f(x), fmfm−1· · ·f1(x))6

∑∞ i=1

dm+i(fm+i,id)<2^−m< δ, which implies that

14Note thatg is defined on∩

n∈NXn =X \A but the image fnfn−1· · ·f1(X) is not contained inX\A.

2.4 Negligibility and Deficiency 83 d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹f(x), f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹fmfm−1· · ·f1(x))< ε/3.

It should be remarked that

d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹fmfm−1· · ·f1,id)

=d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹|Xm, f₁⁻¹· · ·f_m−1⁻¹ f_m⁻¹)< ε/3.

Then, it follows that

d0(gf(x), x)6d0(gf(x), f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹f(x))

+d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹f(x), f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹fmfm−1· · ·f1(x)) +d0(f₁⁻¹· · ·f_n−1⁻¹ f_n⁻¹fmfm−1· · ·f1(x), x)

< ε/3 +ε/3 +ε/3 =ε.

Therefore,gf(x) =x. Thus,f is a homeomorphism withf⁻¹=g. ⊓⊔ We considerC(X) = (I×X){0} and

C^o(X) = ([0,1)×X){0}≈([0,∞)×X){0}, whereC^o(X) is called theopen coneoverX.

Proposition 2.4.5 E≈C^o(E).

Proof. LetSE be the unit sphere of E. Then, we have homeomorphisms:

E≈E\ {0}≈^f SE×(0,∞)≈SE×R,

where the first homeomorphism is obtained by the Negligibility Theorem 2.4.3 above (or Proposition 2.4.2) and the second homeomorphismf is defined by f(x) = (∥x∥⁻¹x,∥x∥). Note thatE×RisE-stable and{0} ×RisE-deficient in E×R. By the Negligibility Theorem 2.4.3, we have

E×R≈E×R\(

{0} ×(R\ {0}))

= (E\ {0})×R∪ {(0,0)}.

The desired homeomorphism is obtained by the following diagram:

E ≈ E×R ≈ (E\ {0})×R∪ {(0,0)}

≈

 y

 y^g C^o(E)≈C^o(SE×R)≈(

[0,∞)×(SE×R))

{0}, whereg is defined as follows:

g(0,0) =0 and g(x, t) = (

∥x∥+|t|, x

∥x∥, ∥x∥+|t|

∥x∥ ·t )

.

84 2 Fundamental Results on Infinite-Dimensional Manifolds

E R

SE

(x, t)

SE×R

∥x∥+|t|=r

( x

∥x∥, r

∥x∥·t )

rSE×R

= (z, s)

rSE 0

x

∥x∥ s

r

Fig. 2.8. (E\ {0})×R∪ {(0,0)} ≈(

[0,∞)×(SE×R))

{0}

It is easy to see thatg is continuous. The inverse ofg can be defined by g⁻¹(0) = (0,0) and g⁻¹(r, z, s) =

( r²

r+|s|·z, rs r+|s|

) . The continuity ofg⁻¹ at0comes from the following:

r²

r+|s|·z

6r and rs

r+|s|

6r

for (r, z, s)∈(0,∞)×SE×R. The proof is complete. — Fig. 2.8. ⊓⊔

For a closed setX inE, the open coneC^o(X) is closed inC^o(E)≈Eand the coneC(X) = (I×X){0} is closed in ([0,∞)×E){0}≈C^o(E)≈E. Thus, the following holds:

Corollary 2.4.6 For every X ∈ FE,C^o(X)∈ FE andC(X)∈ FE. ⊓⊔ Moreover, we have the following:

Corollary 2.4.7 For every locally finite-dimensional simplicial complex K with cardK⁽⁰⁾6w(E), the polyhedron |K|m with the metric topology can be embedded inE as a closed set, i.e.,|K|m∈ FE.

Proof. First, suppose that dimK < ∞. Then, by induction on dimK, we shall show that |K|m ∈ FE. Since E contains a discrete set D with cardD=w(E) by Lemma 2.1.8, we have the case dimK = 0. Assuming the (n−1)-dimensional case, we prove the case dimK =n. For each v ∈K⁽⁰⁾,

2.4 Negligibility and Deficiency 85

|Lk(v, K)|m ∈ FE and |St(v, K)|m is homeomorphic to the metrizable cone C(|Lk(v, K)|m), hence |St(v, K)|m ∈ FE by Corollary 2.4.6 above. In the same way as Theorem 2.1.10, we can apply Michael’s Theorem on local prop-erties (Corollary 1.3.17) to obtain|K|m∈ FE.

When K is locally dimensional, it follows from the above finite-dimensional case that |St(v, K)|m ∈ FE for each v ∈ K⁽⁰⁾. Again, we can apply Michael’s Theorem (Corollary 1.3.17) to obtain|K|m∈ FE. ⊓⊔ Corollary 2.4.8 E×[0,1)≈E.

Proof. First, note that E ×[0,1) is E-stable. Then, by Lemma 2.3.2 and Proposition 2.4.5, we have

E×[0,1)≈((E×[0,1))×E)E×{0}=E×([0,1)×E){0}

=E×C^o(E)≈E×E≈E.

A subsetAin a space X is said to be collaredin X if there is an open embeddingk : A×[0,1)→ X such that k(x,0) = xfor all x∈A. Such an embeddingkis called acollarofAinX. The following can be easily observed:

Fact Every collared set in X is locally closed inX.

Using collars, we can characterize theE-deficiency as follows:

Theorem 2.4.9 Let X be an E-stable perfectly normal paracompact space with A ⊂X. Suppose that E is a normed linear space. Then, the following are equivalent:

(a) AisE-deficient in X;

(b) The closureclAis contained in some collared set in X;

(c) There is a homeomorphismh:X→X×[0,1) such thath(A)⊂X× {0}.

Proof. The implication (c)⇒(b) is trivial, because h(clA) = clh(A)⊂X× {0} in the condition (c).

(a)⇒(c): SinceE×[0,1)≈E(Corollary 2.4.8), we have a homeomorphism h:X →X×[0,1) such thath(A)⊂X× {0}by Corollary 2.3.5.

(b)⇒(a): LetCbe a collared set inX with clA⊂C. Then,C×[0,1) is homeomorphic to an open set inX, hence it isE-stable by Corollary 2.3.4(i).

It suffices to show thatC× {0}isE-deficient inC×[0,1). Indeed, this implies that clA×{0}isE-deficient inC×[0,1). Since every locallyE-deficient closed set in X is E-deficient (Proposition 2.3.8), clA is E-deficient in X, which implies (a). By Lemma 2.3.2, we have

(C×[0,1), C× {0})

≈(

((C×[0,1))×E)C×{0}, C× {0})

=(

C×([0,1)×E){0}, C× {0})

86 2 Fundamental Results on Infinite-Dimensional Manifolds

On the other hand, ([0,1)×E){0} =C^o(E)≈E≈[0,1)×E by Proposition 2.4.5 and Corollary 2.4.8. Then, it follows from the homogeneity ofE that

(C×[0,1), C× {0})

≈(

C×[0,1)×E, C× {0} × {0}) , which implies thatC× {0}isE-deficient inC×[0,1). ⊓⊔ Corollary 2.4.10 C(E)≈E andE×I≈E.

Proof. By Proposition 2.4.5 and Corollary 2.4.8,C(E) =C^o(E)∪(0,1]×E is anE-manifold, hence it isE-stable (Theorem 2.3.7). Since a collared set in anE-stable space is negligible (Theorems 2.4.9 and 2.4.3), we have

C(E)≈C(E)\E× {1}=C^o(E)≈E

by Proposition 2.4.5. On the other hand, E ×I is also an E-manifold by Corollary 2.4.8. Similarly, we haveE×I≈E×[0,1)≈E. ⊓⊔

We denoteQf = [−1,1]^N_f ⊂[−1,1]^N=Q.

Corollary 2.4.11 Depending on E≈E^N orE≈E_f^N, E×Q≈E or E×Qf ≈E.

Remark 2.4 Since E×I ≈ E by Corollary 2.4.10, Theorem 2.4.9 is valid even if [0,1) in (c) can be replaced with I. Moreover, under the assumption of Theorem 2.4.9,Ais E-deficient inX if and only ifA isQ-deficient (resp.

Qf-deficient) inX in the case that E^N ≈ E (resp. E_f^N ≈E) by Corollaries 2.3.5 and 2.4.11.

As with the model spaceE(Corollary 2.4.10), the Hilbert cubeQis home-omorphic to the coneC(Q). This will be proved in a different way in Section 2.7. Moreover, theQ-deficient closed set in aQ-manifold can be characterized in the same way as in Theorem 2.4.9, whereEand [0,1) are replaced withQ andI. This will be shown in Theorem 2.10.9.

In the rest of this section, we shall prove that every finite union of E-deficient sets is alsoE-deficient. The following is a key lemma:

Lemma 2.4.12 There exists a homeomorphism h:I×E→(I×E){0} such that h|(0,1]× {0}= id andh(0,0) = 0.

Proof. Since E ≈C(E) = (I×E){0} by Corollary 2.4.10 and E is homoge-neous, there is a homeomorphism f :E →(I×E){0} with f(0) = 0. Then, we have a homeomorphism

f^′: (I×E){0}→(I×(I×E){0}){0}

such thatf^′(0) = 0 andf^′|(0,1]×E= id×f. As is easily observed,

2.4 Negligibility and Deficiency 87 I²≈{

x∈I²x(1)6x(2)}

≈(I×I){0}.

We can easily construct a homeomorphism g : I×I→(I×I){0} such that g(0,0) = 0 andg|(0,1]× {0}= id, which induces the homeomorphism

g^′ : ((I×I)×E)I×{0}→((I×I)_{0}×E){0}∪(0,1]×{0}.

Then, g^′|I× {0} =g|I× {0} (i.e., g^′(0,0) = 0 andg^′|(0,1]× {0}= id), and g^′|(I×(0,1])×E= (g|I×(0,1])×idE. We define the desired homeomorphism has the composition of three homeomorphisms in the following diagram:

I×E

id_I×f

h //(I×E){0}

I×(I×E){0} (I×(I×E){0}){0}

f^{′ −1}

OO

((I×I)×E)^I×{0}

g^′ //((I×I){0}×E){0}∪(0,1]×{0}

For each t ∈ (0,1], since f^′h(t,0) = g^′(t, f(0)) = g^′(t,0) = (t,0) and f^′(t,0) = (t, f(0)) = (t,0), it follows that h(t,0) = (t,0). Moreover, f^′h(0,0) =g^′(0, f(0)) =g^′(0,0) = 0. ⊓⊔

Proposition 2.4.13 Let X be an E-stable perfectly normal space. Every fi-nite union ofE-deficient sets inX is also E-deficient.

Proof. It suffices to show that the union of twoE-deficient sets A andB in X is also E-deficient, where we may assume that A and B are closed in X because their closures inX areE-deficient.

SinceI×E≈E by Corollary 2.4.10, we have a homeomorphismf :X→ X×Isuch thatf(A)⊂X× {0} (Corollary 2.3.5). LetC =f⁻¹(X × {0}).

Then,A⊂C andf induces the homeomorphism

f^′: (X×E)C→((X×I)×E)X×{0}=X×(I×E){0}.

Then, f^′|C=f|C andf^′|(X\C)×E= (f|X\C)×idE. By Lemma 2.4.12 above, we have a homeomorphismh:I×E→(I×E){0} such thath(t,0) = (t,0) for everyt∈(0,1] andh(0,0) = 0. Leth^′:X×E→(X×E)C be the homeomorphism defined by the following diagram:

X×E

f×idE

h^′

//(X×E)C

X×I×E _id

X×h //X×(I×E){0}

f^{′ −1}

OO

88 2 Fundamental Results on Infinite-Dimensional Manifolds

For eachx∈C, h^′(x,0) =xbecausef(x) =f^′(x)∈X × {0}. For each x∈ X\C, letf(x) = (x^′, t). Then,t >0. Sincef^′h^′(x,0) = (x^′, h(t,0)) = (x^′, t,0) andf^′(x,0) = (f(x),0) = (x^′, t,0), we have h^′(x,0) = (x,0).

Applying Lemma 2.3.2 (A = B, A0 = C, A1 = X), we have a homeo-morphism g: (X×E)C →X such thatg(x,0) =xfor each x∈B\C and g|C = id. Then, gh^′ : X ×E → X is a homeomorphism. For each x ∈ C, gh^′(x,0) = g(x) = x. For each x ∈ B \C, gh^′(x,0) = g(x,0) = x. Thus, gh^′(x,0) =xfor eachx∈A∪B. ⊓⊔

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