Stability and Deficiency

66 2 Fundamental Results on Infinite-Dimensional Manifolds

Without assuming the completeness ofX = (X, d),X can be embedded isometrically into some normed linear space L with densX = densL as a closed set. In this case, we have X×Π_ℓ^f

1L ≈Π_ℓ^f

1L. Thus, the general case also holds. ⊓⊔

As a corollary, we have the following:

Corollary 2.2.12 For an arbitrary infinite set Γ,ℓ1(Γ)^N≈ℓ1(Γ).

Proof. Sinceℓ1(Γ)^Nis a completely metrizable AR, we have ℓ1(Γ)^N≈ℓ1(Γ)^N×ℓ1(Γ)≈ℓ1(Γ).

Recall that the metrizable cone C(X) over a metrizable space X is the space {0} ∪ (X ×(0,1]) with the topology generated by open sets in the product space X×(0,1] and sets {0} ∪(X×(0, ε)), ε∈ (0,1). If X is compact, thenC(X) is homeomorphic to the quotient space (X×I)/(X×{0}), which is the usual cone. Themetrizable open coneoverX is the subspace {0} ∪(X×(0,1)) of C(X) and denoted byC^o(X). It should be noted that if X is completely metrizable, then so are C(X) andC^o(X). Moreover, the coneC(X) and the open coneC^o(X) over an ANRX are ARs by Corollary 1.8.14.

By using the metrizable open cone, we can easily prove the following the-orem, which is also called the Toru´nczyk Factor Theorem:

Theorem 2.2.13 (Toru´nczyk) For every completely metrizable ANR X withw(X)6τ,X×ℓ1(Γ)is anℓ1(Γ)-manifold. In fact,X×ℓ1(Γ)is home-omorphic to an open set inℓ1(Γ).

Proof. As mentioned above, the open coneC^o(X) is a completely metrizable AR and clearly w(C^o(X))6τ. Then, C^o(X)×ℓ1(Γ)≈ℓ1(Γ) by the above Theorem 2.2.11. Sinceℓ1(Γ)×(0,1)≈ℓ1(Γ),X×ℓ1(Γ) is homeomorphic to the open set (0,1)×X ×ℓ1(Γ) in C^o(X)×ℓ1(Γ). Thus,X ×ℓ1(Γ) can be embedded inℓ1(Γ) as an open set, hence it is anℓ1(Γ)-manifold. ⊓⊔

2.3 Stability and Deficiency 67 we prove that every E-manifold isE-stable, and observe some properties of E-deficient sets.

First, we introduce the reduced products, which will also be used in the following sections. Let X and Y be spaces and Abe a closed set in X. The product ofX andY reduced overAis the space

(X×Y)A=(

(X\A)×Y)

∪A,

with the topology generated by open sets in the product space (X\A)×Y

and sets (

(U \A)×Y)

∪(U∩A),

where U is open in X. Then, the projection pr_X : X ×Y → X is factored through (X ×Y)A into two natural maps qA : X ×Y → (X ×Y)A and pA: (X×Y)A→X defined by

{qA|(X\A)×Y = id, qA|A×Y = pr_A and

{pA|(X\A)×Y = pr_X\A, pA|A= id.

Thus, we have the following commutative diagram:

X×Y ^prX //

qA

&&

LL LL LL LL

LL X

(X×Y)A pA

::u

uu uu uu uu u

The topology of (X×Y)A is the coarsest topology such that the map pA is continuous and the product space (X\A)×Y is its subspace. By the Bing Metrization Theorem 1.3.4, ifX andY are metrizable then (X×Y)Ais also metrizable. In caseY is compact, (X×Y)Ais homeomorphic to the adjunction spaceA∪pr_A(X×Y), where pr_A:A×Y →Ais the projection. It should be also remarked that (I×X){0} is naturally homeomorphic to the (metrizable) coneC(X), so we sometimes regard C(X) = (I×X){0}.

To prove the stability of E-manifolds, our main tool is the coordinate switching pseudo-isotopy. For simplicity, we use the following notational con-vention:

• Given a spaceX, letX^∗ denoteX×E^NorX×E_f^Ndepending onE≈E^N orE ≈E_f^N, and let p0:X^∗→X andpn :X^∗ →X×Eⁿ,n∈N, be the projections. Additionally, letX0=X× {0} ⊂X^∗.

We define a homotopyθ^E:X^∗×E×I→X^∗as follows: For (x, y1, y2, . . .)∈ X^∗ (x∈X,yi∈E,i∈N) andz∈E,

68 2 Fundamental Results on Infinite-Dimensional Manifolds θ₁^E(x, y1, y2, . . . , z) = (x, z,−y1,−y2, . . .), θ₂^E−1(x, y1, y2, . . . , z) = (x, y1, z,−y2,−y3, . . .),

...

θ^E₂−n(x, y1, y2, . . . , z) = (x, y1, . . . , yn, z,−yn+1,−yn+2, . . .), θ^E₂−(n+1)(x, y1, y2, . . . , z) = (x, y1, . . . , yn, yn+1, z, ,−yn+2, . . .),

...

θ₀^E(x, y1, y2, . . . , z) = (x, y1, y2, . . .) and for 2⁻⁽ⁿ⁺¹⁾6t62⁻ⁿ,

θ^E_t(x, y1, y2, . . . , z) = (x, y1, . . . , yn,

zcosπ(1−2ⁿt) +yn+1sinπ(1−2ⁿt), zsinπ(1−2ⁿt)−yn+1cosπ(1−2ⁿt),

−yn+2,−yn+3, . . .).

Then, it should be remarked that

(♮) θ^E_t(x,0) =xfor eachx∈X0andt∈I.

(♯) t62⁻ⁿ⇒pnθ^E_t =pnpr_X∗.

In addition,θ^E induces a homeomorphism ˜θ^E :X^∗×E×(0,1]→X^∗×(0,1]

defined by

θ˜^E(x, y, z, t) = (θ^E(x, y, z, t), t)

for each (x, y)∈X^∗,z∈E, andt∈(0,1].

This means that (θ^E)⁻¹_t : X^∗ →X^∗×E is also continuous with respect to t∈(0,1]. We callθ^E thecoordinate switching pseudo-isotopyforX^∗.

Given a mapτ :X^∗→I, we define a mapθ_τ^E: (X^∗×E)τ⁻¹(0)→X^∗ by θ^E_τ|τ⁻¹(0) = id and θ_τ^E(w, z) =θ^E_τ(w)(w, z) if τ(w)̸= 0.

Then, by (♮), the following is satisfied:

θ^E_τ(x,0) =x for everyx∈X0\τ⁻¹(0).

The continuity of θ^E_τ at w∈τ⁻¹(0) follows from (♯). Indeed, each neighbor-hood U ofθ^E_τ(w) =win X^∗ containsp⁻¹_n (V), whereV is an open neighbor-hood of pn(w) inX×Eⁿ. Then, the following is an open neighborhood ofw in (X^∗×E)_τ⁻¹₍₀₎:

W =(

(p⁻¹_n (V)∩τ⁻¹((0,2⁻ⁿ)))×E)

∪(p⁻¹_n (V)∩τ⁻¹(0)).

2.3 Stability and Deficiency 69 Sincepnθ^E_τ|τ⁻¹((0,2⁻ⁿ)) =pnpr_X^∗|τ⁻¹((0,2⁻ⁿ)) by (♯), we have

θ^E_τ((p⁻¹_n (V)∩τ⁻¹((0,2⁻ⁿ)))×E)⊂p⁻¹_n (V).

Hence, it follows thatθ_τ^E(W)⊂p⁻¹_n (V)⊂U.

Lemma 2.3.1 Let τ : X^∗ →I be a map. SupposeX^∗\τ⁻¹(0) = ∪

n∈NDn

for some ∅=D0⊂D1⊂ · · · such that (1) τ(w)62⁻ⁿ⁻¹ for eachw∈X^∗\Dn and (2) w∈Dn, pn(w) =pn(w^′)⇒τ(w) =τ(w^′).

Then, θ^E_τ is a homeomorphism.

Proof. For simplicity, letA=τ⁻¹(0). First, observe the following two facts:

(3) τ(θ^E_τ(w, z)) =τ(w) for each (w, z)∈(X^∗\A)×E.

(4) τ(pr_X^∗(θ_τ(w)^E )⁻¹(w)) =τ(w) for each w∈X^∗\A.

(3): Let (w, z) ∈(X^∗\A)×E. Then, w ∈ Dn\Dn−1 for some n∈ N, whenceτ(w)62⁻ⁿ by (1). Thus, it follows from (♯) that

pn(θ^E_τ(w, z)) =pnθ^E_τ(w)(w, z) =pnpr_X^∗(w, z) =pn(w), which impliesτ(θ^E_τ(w, z)) =τ(w) by (2).

(4): Eachw∈X^∗\Ais contained in someDn\Dn−1. In the same way as the above, we havepnθ^E_τ(w)=pnpr_X^∗. Then,pn(w) =pn(pr_X^∗(θ^E_τ(w))⁻¹(w)), which impliesτ(pr_X^∗(θ_τ(w)^E )⁻¹(w)) =τ(w) by (2).

Now, we define η^E_τ : X^∗ → (X^∗ ×E)A by η_τ^E|A = id and η_τ^E(w) = (θ^E_τ(w))⁻¹(w) forw∈X^∗\A(=X^∗\τ⁻¹(0)). Then, by (4),

w∈X^∗\A ⇒ η_τ^E(w) = (θ^E_τ(w))⁻¹(w)∈(X^∗\A)×E= (X^∗×E)A\A.

Sinceτ(pr_X∗η_τ^E(w)) =τ(w) forw∈X^∗\A, it is easy to see that η_τ^Eθ^E_τ = id andθ_τ^Eη_τ^E= id, namely thatη_τ^E is the inverse ofθ^E_τ.

The continuity ofη_τ^E remains to be proved. It follows from the continuity of (˜θ^E)⁻¹ that η^E_τ is continuous at each point ofX^∗\A. We shall show the continuity of η_τ^E at w∈A. Each neighborhood of w=η^E_τ(w) in (X^∗×E)A

contains (W ∩A)∪((W \A)×E) for some open neighborhood W of w in X^∗. We can findn∈ Nand an open neighborhoodW0 ofpn(w) in X×Eⁿ such that p⁻¹_n (W0) ⊂ W. Then, V = p⁻¹_n (W0)∩τ⁻¹([0,2⁻ⁿ)) is an open neighborhood ofwin X^∗. Moreover,

η_τ^E(V)⊂(W ∩A)∪((W \A)×E).

Indeed,η_τ^E(V ∩A) =V ∩A⊂p⁻¹_n (W0)∩A⊂W∩A. For eachv∈V \A, we writeη_τ^E(v) = (θ^E_τ(v))⁻¹(v) = (v^′, z)∈(X^∗\A)×E. Sinceτ(v^′) =τ(v)<2⁻ⁿ by (4), it follows from (♯) that

70 2 Fundamental Results on Infinite-Dimensional Manifolds

A0 A1

h

W (X^∗×E)A1

(X^∗×E)A0

X^∗ W

A0 W

θ^E_τ1 θ^E_τ0

A1

Fig. 2.5.Homeomorphismh

pn(v^′) =pnθ^E_τ(v′)(v^′, z) =pnθ_τ(v)^E (v^′, z) =pn(v)∈W0, hencev^′ ∈W\A. Thus, we haveη_τ^E(v) = (v^′, z)∈(W \A)×E. ⊓⊔

Lemma 2.3.2 Let X be an E-stable perfectly normal space, A be an E-deficient set, A0 ⊂A1 be closed sets, and W be an open set in X such that cl(A1\A0)⊂W. Then, there is a homeomorphismh: (X×E)A0 →(X×E)A1

such that

(1) h(x,0) = (x,0)for each x∈A\A1, (2) h(x,0) =xfor each x∈A∩(A1\A0), and (3) h|(X\(A0∪W))×E∪A0= id.

Proof. Since X ×E ≈ X and E ≈ E^N or E ≈ E_f^N, we may replace X by X^∗, where A can be regarded as a subset of X0 =X × {0} ⊂X^∗ because A is E-deficient. For j = 0,1, we shall construct maps τj : X^∗ → I and

∅ =D^j₀ ⊂ D₁^j ⊂ · · · so that X^∗\τ_j⁻¹(0) =∪

n∈ND_n^j and those satisfy the conditions of Lemma 2.3.1 and the following additional conditions:

(4) τ0|X^∗\W =τ1|X^∗\W andτ₀⁻¹(0) =A0⊂A1=τ₁⁻¹(0).

Then, we would have homeomorphismsθ_τ^E_j : (X^∗×E)Ai→X^∗,j= 0,1, that satisfy the following:

(5) θ^E_τj(x,0) =xfor everyx∈X0\Aj, (6) θ^E_τj|Aj= idAj, and

(7) θ^E_τ0|(X^∗\(A0∪W))×E=θ_τ^E₁|(X^∗\(A0∪W))×E.

Then,h= (θ^E_τ1)⁻¹θ^E_τ0 is the desired homeomorphism. — Fig. 2.5

We denoteA2=A0∪(X^∗\W). Then,X^∗\A2=W \A0. Now, we take Bi ∈ cov(X^∗\Ai), i = 1,2, which consist of basic open sets inX^∗.⁸ Here,

8 I.e., open setsp⁻¹n (U),n∈N, whereU is open inX×Eⁿ.

2.3 Stability and Deficiency 71 B ⊂ X^∗ is said to be n-basic if p⁻¹_n (pn(B)) = B. Note that each n-basic subset ofX^∗ism-basic for everym>n. For eachn∈N, let

B_i^[n]=∪ { B∈ Bi

B isn-basic}

, i= 1,2.

Then, B_i^[n] isn-basic andpn(B_i^[n]) is open inX×Eⁿ. Since X×Eⁿ≈X is perfectly normal, we have a mapki,n:X×Eⁿ→Isuch that

k⁻¹_i,n(0) = (X×Eⁿ)\pn(B_i^[n]) =pn(X^∗\B^[n]_i ).

For eachm∈N, letDⁿ_i,m=k⁻¹_i,n([1/m,1]). Then,pn(B^[n]_i ) =∪

m∈ND_i,mⁿ , each Dⁿ_i,m is closed inX×Eⁿ, andDⁿ_i,m⊂intD_i,m+1ⁿ . For eachn∈N, we define

D^[n]_i =∪ {

p⁻¹_m (D^m_i,n)m6n}

, i= 1,2.

Observe that eachD_i^[n]isn-basic,D^[n]_i ⊂intD^[n+1]_i , and∪

n∈ND^[n]_i =X^∗\Ai. For eachn∈N, let

H_i^[n]=D^[n]_i \intD_i^[n−1], i= 1,2, whereD^[0]_i =∅. Then, it follows that

∪

n∈N

H_i^[n]= ∪

n∈N

D^[n]_i =X^∗\Ai, X^∗\D_i^[n] ⊂Ai∪ ∪

m>n

H_i^[m] and H_i^[n]∩H_i^[n+1]= bdD^[n]_i for eachn∈N.

Moreover,H_i^[m]∩H_i^[n] =∅if|m−n|>1. Sincepn(bdD^[n]_i ) andpn(bdD^[n−1]_i ) are disjoint closed sets in pn(H_i^[n]), we can take maps τ_i^[n] : pn(H_i^[n]) → [2⁻ⁿ⁻¹,2⁻ⁿ],i= 1,2, such that

τ_i^[n](pn(bdD^[n]_i )) = 2⁻ⁿ⁻¹ and τ_i^[n](pn(bdD^[n−1]_i )) = 2⁻ⁿ. We defineτi:X^∗→I,i= 1,2, as follows:

τi(Ai) = 0 and τi|H_i^[n]=τ_i^[n]pn|H_i^[n].

It is easy to see that τi is continuous at each point of X^∗\Ai. To verify the continuity ofτiatx∈Ai, for eachε >0, choosen∈Nso that 2⁻ⁿ< ε. Then, U =X^∗\D^[n]_i is a neighborhood ofxinX^∗. Note thatU\Ai ⊂∪

m>nH_i^[m]. Ify∈U\Ai, then y∈H_i^[m] for some m > n, hence

τi(y) =τ_i^[m]pm(y)62^−m<2⁻ⁿ< ε.

72 2 Fundamental Results on Infinite-Dimensional Manifolds

Therefore,τi(U)⊂[0, ε). It follows from the definitions thatτi and (D_i^[n])n∈N

satisfy the conditions of Lemma 2.3.1. Moreover,τ_i⁻¹(0) =Ai.

For each n ∈ N, let D^[n]₀ =D^[n]₁ ∪D₂^[n]. Define τ0 : X^∗ → I byτ0(x) = max{τ1(x), τ2(x)}. Then, τj and (D_j^[n])n∈N, j = 0,1, satisfy the required conditions. Indeed,τ₀⁻¹(0) =τ₁⁻¹(0)∩τ₂⁻¹(0) =A1∩A2=A0. SinceX^∗\W ⊂ A2 = τ₂⁻¹(0), τ0(x) = τ1(x) for each x ∈ X^∗ \W, that is, τ0|X^∗ \W = τ1|X^∗\W. Thus, those satisfy condition (4). Finally, using the fact that τi

and (D^[n]_i )n∈N,i= 1,2, satisfy the conditions of Lemma 2.3.1, we can easily show that τ0 and (D^[n]₀ )n∈Nalso satisfy the same conditions. ⊓⊔

Remark 2.2 In the above proof, each x ∈ X^∗ \A1 is contained in some D^[n]₁ . Then, there is somem6nsuch that pm(x)∈D_1,n^m ⊂pm(B^[m]₁ ), hence x ∈ B₁^[m]. Thus, x is contained in some m-basic B ∈ B1. Since τ1(x) 6 2^−m, it follows from (♯) that pmθ_tτ^E_1(x)(x, z) = pm(x), so θ^E_tτ1(x)(x, z) ∈ B for every z ∈ E and t ∈ I where θ₀^E = pr_X∗. By choosing B1 to refine a given U ∈ cov(X^∗\A1), the homeomorphism θ_τ^E₁ : (X^∗×E)A1 → X^∗ is U-homotopic to pr_X^∗ by the homotopyθ^E_tτ1,t ∈I. Recall that we have the natural mapqA1 :X^∗×E→(X^∗×E)A1 defined byqA1|(X^∗\A1)×E= id andqA1|A1×E= pr_A1. Then,f =θ_τ^E₁qA1 :X^∗×E→X^∗is a map such that f|A1×E= pr_A1 andf|(X^∗\A1)×Eis a homeomorphism ontoX^∗\A1that isU-homotopic to pr_X^∗_\A1.

In this remark, for an open setU ⊂X, letA1 =X \U. Then, we have the following:

Proposition 2.3.3 LetX be anE-stable perfectly normal space,Abe an E-deficient set in X, and U be an open set in X. Then, for each open cover U of U, there exists a map f : X×E →X such that f(x,0) = xfor each x∈A∩U, f|(X\U)×E = pr_X\U andf|U ×E is a homeomorphism onto U that is U-homotopic topr_U. ⊓⊔

Corollary 2.3.4 Let X be an E-stable perfectly normal space and A be an E-deficient set inX. Then, the following hold:

(i) Every open set inX isE-stable.

(ii) For each open setU inX,A∩U is E-deficient in U. (iii) The projectionpr_X:X×E→X is a near-homeomorphism.

(iv) For any open coverU of X, there is a homeomorphismf :X ×E →X such thatf(x,0) =xfor each x∈Aandf ≃Upr_X. ⊓⊔

Replacing E = (E,0) with a pointed space L = (L,∗), the L-stability and theL-deficiency are similarly defined, that is, a space X is L-stable if X×L≈X, and a subsetAofX isL-deficientif there is a homeomorphism h:X →X×Lsuch thath(A)⊂X× {∗}.

2.3 Stability and Deficiency 73 Corollary 2.3.5 Let X be an E-stable perfectly normal paracompact space, A be an E-deficient set in X, and L = (L,∗) be a pointed space such that E×L≈E. Then, for each open coverU ofX, there exists a homeomorphism h : X ×L → X such that h(x,∗) = x for each x ∈ A and h ≃U pr_X. Consequently, the projectionpr_X:X×L→X is a near-homeomorphism and every E-deficient set inX isL-deficient.

Proof. Letφ: E×L→E be a homeomorphism withφ(0,∗) =0. Since X is paracompact, there exists V ∈ cov(X) such that stV ≺ U. By Corollary 2.3.4(iv) above, we have a homeomorphismf :X×E→Xsuch thatf(x,0) = xfor eachx∈Aand f ≃V pr_X. Then, the desired homeomorphism h:X× L→X can be defined as the composition of the following homeomorphisms:

X×L ^f

−1×idL

//X×E×L ^idX×φ //X×E ^f //X

Indeed, it is easy to see that h(x,∗) =xfor eachx∈Aand

h≃V pr_X(idX×φ)(f⁻¹×idL) = pr_X(f⁻¹×idL) = pr_Xf⁻¹pr_X, where the same notation pr_X stands for three different projections. Since pr_Xf⁻¹≃V f f⁻¹= idX, it follows thath≃stVpr_X, henceh≃Upr_X. ⊓⊔

It is said that X is locally E-stable if each x ∈ X has an E-stable neighborhood inX. Applying Michael’s Theorem 1.3.16 on local property, we prove the following theorem:

Theorem 2.3.6 For every perfectly normal paracompact space X, if X is locallyE-stable, thenX isE-stable and the projectionpr_X:X×E→X is a near-homeomorphism.

Proof. LetP be the property for open sets in X to be E-stable. The result follows from Michael’s Theorem 1.3.16 on local properties ifP isG-hereditary.

The condition (G-1) is Corollary 2.3.4(i), and (G-3) is obvious. It remains to show (G-2).

(G-2): For anyE-stable open setsU1andU2inX, letU =U1∪U2. SinceU is normal, we can choose open setsV1andV2inU so that clUV1∩clUV2=∅, U\U2⊂V1, andU\U1⊂V2. Applying Lemma 2.3.2 toA0=∅,A1= clUV1, W = U1\clUV2, and X = U1, we have a homeomorphismh1 : U1×E → (U1×E)clUV1 such thath1|(U1∩clUV2)×E = id. Then,h1 can be extended to a homeomorphism ¯h1:U×E→(U×E)clUV1 by ¯h1|(U\U1)×E= id.

Next, applying Lemma 2.3.2 to A0 = U2∩clUV1, A1 = U2, and W = X =U2, we have a homeomorphism h2 : (U2×E)U2∩clUV1 → U2 such that h2|U2∩clUV1 = id. Then, h2 can be extended to a homeomorphism ¯h2 : (U ×E)clUV1 → U by ¯h2|U \U2 = id. Thus, we have a homeomorphism

¯h2h¯1:U×E→U, that is,U isE-stable. — Fig. 2.6 ⊓⊔

74 2 Fundamental Results on Infinite-Dimensional Manifolds

U1 V2

U1 U2

V1 V2

U

¯h1 ¯h2

U2

V1

U2∩clUV1

(U1∩clUV2)×E

Fig. 2.6.Homeomorphisms ¯h1 and ¯h2

Note that everyE-manifold is perfectly normal and paracompact because it is metrizable. Since each open set in E is E-stable by Corollary 2.3.4(i), every E-manifold is locally E-stable. Thus, as a corollary of Theorem 2.3.6 above, we have the following Stability Theorem:

Theorem 2.3.7 (Stability) EveryE-manifoldM isE-stable and the pro-jection pr_M :M×E→M is a near-homeomorphism. ⊓⊔

It is said that a subsetAof anE-stable space X islocally E-deficient inX if eacha∈Ahas a neighborhoodU inX such thatA∩U isE-deficient in U. By making small changes in the proof of Theorem 2.3.6, we can prove the following:

Proposition 2.3.8 LetX be anE-stable perfectly normal paracompact space.

Then, every locallyE-deficient closed setA inX isE-deficient in X. As the result, if a closed setA⊂X isE-deficient in an open setU inX, thenA is E-deficient in X.

Proof. Let P be the property for open sets U in M such that A∩U is E-deficient inU. This property implies thatU isE-stable. Actually, an open set U with A∩U =∅ has property P ifU is E-stable. Then,X hasP locally.⁹ In the proof of Theorem 2.3.6, applying Corollary 2.3.4(ii) instead of (i) gives a homeomorphismh:X×E→X such thath(x,0) =xfor eachx∈A. ⊓⊔

Because [−1,1]²≈B², there exists an isotopyφ: [−1,1]²×I→[−1,1]² such that φ0= id,φt(0,0) = (0,0) for each t∈I, and φ1(x, y) = (−y, x) for each (x, y)∈[−1,1]². We define the isotopyφ^Q:Q²×I→Q² as follows:

φ^Q_t(x, y) =ψ(

(φt(x(i), y(i)))i∈N

),

whereψ:(

[−1,1]²)^N

→Q²is the homeomorphism defined by ψ(

(x(i), y(i))i∈N

)=(

(x(i))i∈N,(y(i))i∈N

).

9 WhenA is not closed inM, it is unknown whether a point a∈clA\Ahas an open neighborhoodU such thatA∩U isE-deficient inU.

2.3 Stability and Deficiency 75 Then, φ^Q₀ = id,φ^Q_t (0,0) = (0,0) for eacht ∈I, and φ^Q₁(x, y) = (−y, x) for each (x, y)∈Q². Using this isotopy, we can define thecoordinate switching pseudo-isotopy

θ^Q :X×Q^N×Q×I→X×Q^N

in the same way asθ^E:X^∗×E×I→X^∗. By the same arguments as above, we have the following Q-stable versions of Proposition 2.3.3 and Corollary 2.3.4:

Proposition 2.3.9 Let X be a Q-stable perfectly normal space, A be a Q-deficient set in X, and U be an open set in X. Then, for each open cover U of U, there exists a map f : X×Q→ X such that f(x,0) = xfor each x∈A∩U,f|(X\U)×Q= pr_X\U andf|U×Q is a homeomorphism onto U that is U-homotopic topr_U. ⊓⊔

Corollary 2.3.10 Let X be a Q-stable perfectly normal space and A be a Q-deficient set inX. Then, the following hold:

(i) Every open set inX isQ-stable.

(ii) For each open setU inX,A∩U is Q-deficient in U. (iii) The projectionpr_X:X×Q→X is a near-homeomorphism.

(iv) For any open coverU of X, there is a homeomorphism f :X ×Q→X such thatf(x,0) =xfor each x∈Aandf ≃Upr_X. ⊓⊔

SinceQ×Iⁿ ≈Qfor every n∈N, the following can be obtained in the same way as Corollary 2.3.5:

Corollary 2.3.11 For everyQ-stable perfectly normal paracompact spaceX andn∈N, the projectionpr_X :X×Iⁿ→X is a near-homeomorphism. ⊓⊔

It is said thatX islocally Q-stableif eachx∈X has aQ-stable neigh-borhood inX. The following is theQ-manifold version of Theorem 2.3.6:

Theorem 2.3.12 For every perfectly normal paracompact space X, if X is locally Q-stable, thenX is Q-stable and the projectionpr_X :X×Q→X is a near-homeomorphism. ⊓⊔

Thus, we have the followingQ-manifold version of Theorem 2.3.7:

Theorem 2.3.13 (Stability for Q-manifolds) Every Q-manifold M is Q-stable and the projection pr_M : M ×Q→ M is a near-homeomorphism.

Additionally, for each n ∈ N, the projection pr_M : M ×Iⁿ → X is also a near-homeomorphism. ⊓⊔

In the same way, locallyQ-deficient sets and Q-deficient embeddings are defined. Then, the following can also be obtained:

76 2 Fundamental Results on Infinite-Dimensional Manifolds

Proposition 2.3.14 Let X be a perfectly normal paracompact space. Then, every locallyQ-deficient closed set in X is Q-deficient in X. ⊓⊔

An embeddingf :Y →X is said to beE-deficientiff(Y) isE-deficient in X. Recall that FE is the class of spaces which can be embedded inE as closed sets. Applying the Stability Theorem 2.3.7, we can prove the follow-ing theorem, which is also known as the Closed Embeddfollow-ing Approximation Theorem:

Theorem 2.3.15 (Strong Universality of E-manifolds) LetM be an E-manifold and A ∈ FE, with a closed set B ⊂A and a map f : A → M, such thatf|Bis anE-deficient closed embedding. Then, for eachU ∈cov(M), there exists anE-deficient closed embeddingh:A→M such thath|B=f|B andh≃Uf.

Proof. First, note thatE³≈E andM is an ANR. By Corollary 2.3.4(iv), we have a homeomorphismg:M ×E³→M such that

g(f(x),0,0,0) =f(x) for each x∈B and g≃U pr_M.

From the assumption, there is a closed embedding j : A → E. Take d ∈ Metr(A) anddM ∈Metr(M) such thatd, dM 61/2, and definedA∈Metr(A) as follows:

dA(x, y) =d(x, y) +dM(f(x), f(y)) for each x, y∈A.

Then, dM(f(x), f(y)) 6 dA(x, y) 6 1 for each x, y ∈ A. Let k : A → I be the map defined by k(x) =dA(x, B). Let e ̸= 0 ∈ E be fixed. The desired embeddingh:A→M can be defined by

h(x) =g(f(x), k(x)e, k(x)j(x),0).

Indeed,h|B=f|B by definition, andh(A) isE-deficient inM becauseM × E² ≈ M. Since g ≃U pr_M, it follows that h ≃U f. If x ̸= y ∈ A, then k(x) ̸= k(y), k(x)j(x) ̸= k(y)j(y), or k(x) = k(y) = 0. In the third case, x, y∈B, which impliesf(x)̸=f(y). Thus,his injective.

It remains to see thathis closed. Letxn∈A,n∈N, such that (h(xn))n∈N

is convergent in M. Then, it is enough to show that (xn)n∈N is convergent.

The sequences (f(xn))n∈N, (k(xn))n∈N, and (k(xn)j(xn))n∈N are convergent.

Whenk(xn)→t̸= 0,k(xn)>0 for sufficiently largenandk(xn)⁻¹→t⁻¹, so it follows that (j(xn))n∈N is convergent. Since j is a closed embedding, (xn)n∈Nis convergent.

Whenk(xn)→ 0, we can choose yn ∈ B so thatdA(xn, yn) →0, hence dM(f(xn), f(yn)) → 0 by the definition of dA. Then, (f(yn))n∈N is conver-gent. Sincef|Bis a closed embedding, (yn)n∈Nis convergent, and hence so is (xn)n∈N. ⊓⊔

2.3 Stability and Deficiency 77 Note that the class M₀ of compacta (= compact metrizable spaces) is equal to the classFQ of spaces which can be embedded intoQ. By the same proof as Theorem 2.3.15 above, we can prove the following:

Theorem 2.3.16 (Strong Universality of Q-manifolds) Let M be a Q-manifold and A be a compactum, with a closed set B ⊂ A and a map f : A → M, such that f|B is a Q-deficient closed embedding of a closed set B ⊂ A. Then, for each U ∈ cov(M), there exists a Q-deficient closed embedding h:A→M such thath|B=f|B andh≃U f. ⊓⊔

The following is a non-compact version of the above:

Theorem 2.3.17 (Closed Embedding Approximation) Let M be a Q-manifold, andAbe a locally compact separable metrizable space, with a closed set B ⊂ A and a proper map f : A → M, such that f|B is a Q-deficient closed embedding. Then, for each U ∈ cov(M), there exists a Q-deficient closed embeddingh:A→M such that h|B=f|B andh≃^p_Uf.¹⁰

Proof. ReplacingU by a refinement, we can assume thatU is locally finite and every U ∈ U has the compact closure inM. By Corollary 2.3.10(iv), we have a homeomorphism g:M ×Q² →M such thatg(f(x),0,0) = f(x) for each x∈Bandg≃U pr_M. We have an embeddingj:A→Qwithj(A)⊂pr⁻¹₁ (1) (≈ Q),¹¹ where pr₁ : Q → [−1,1] is the projection of the first factor. Let k:A→Ibe a map withk⁻¹(0) =B. The desired embeddingh:A→M can be defined as follows:

h(x) =g(f(x), k(x)j(x),0).

Indeed, by definition,h|B =f|B andh(A) is Q-deficient in M. Let x̸=

x^′∈A. Ifk(x)̸=k(x^′), thenk(x)j(x)̸=k(x^′)j(x^′) because pr₁(k(x)j(x)) =k(x)̸=k(x^′) = pr₁(k(x^′)j(x^′)).

If k(x) = k(x^′) > 0, then k(x)j(x) ̸= k(x^′)j(x^′) because j(x) ̸= j(x^′). If k(x) = k(x^′) = 0, then x, x^′ ∈ B, so h(x) = f(x) ̸=f(x^′) = h(x^′) because f|B is an embedding. In any case,h(x)̸=h(x^′). Thus,his injective.

Sinceg≃Upr_M, it easily follows thath≃U f. Then, becausef is proper, it follows from Proposition 2.1.17 thath≃^p_Uf. In particular,his proper, hence it is closed (cf. Proposition 1.3.1(c)). Therefore, his a closed embedding. ⊓⊔ Remark 2.3 In Theorem 2.3.15, it suffices to assume thatM isE-stable like Proposition 2.3.8. In Theorems 2.3.16 and 2.3.17, it suffices to assume that M isQ-stable like Proposition 2.3.14.

10Recallh≃^pUf means thathis properlyU-homotopic tof.

11This embedding is not closed unless A is compact. An embedding ofA can be obtained by restricting an embedding of the one-point compactification αA = A∪ {∞}.

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⁾

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