Katsuhisa Koshino and Katsuro Sakai University of Tsukuba December - - PowerPoint PPT Presentation

ぺアノ空間から１次元局所コンパクトＡＲへの 関数空間のコンパクト化

Katsuhisa Koshino and Katsuro Sakai

University of Tsukuba

December 2012

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Outline Introduction 1 Background 2 Dendrites 3 The closure of C(X, Y ) in Cld∗

F(X ×

Y ) 4 Proof of the Main Theorem

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0 Introduction Spaces are regular and maps are continuous. For spaces X and Y , C(X, Y ) is the space of all maps from X to Y with the compact-open topology, which is generated by the following sets: {f ∈ C(X, Y ) | f(K) ⊂ U}, where K is a compact set in X and U is an open set in Y . When X is locally compact and σ-compact, and Y is metriz- able, C(X, Y ) is metrizable. Let Q = [−1, 1]N be the Hilbert cube and s = (−1, 1)N be the pseudo-interior of Q.

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Main Theorem Let X be an infinite, locally compact, locally connected, separable metrizable space and let Y be a 1-dimensional locally compact AR. If X is non-discrete or Y is non-compact, then C(X, Y ) has a natural compactification C(X, Y ) such that (C(X, Y ), C(X, Y )) ≈ (Q, s). Remark If X is discrete and Y is compact, then C(X, Y ) ≈ Q.

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1 Background Let Cld(X) be the set of all non-empty closed subsets of a space X and Cld∗(X) = Cld(X) ∪ {∅}. For each Z ⊂ X, let Z− = {A ∈ Cld∗(X) | A ∩ Z = ∅} and Z+ = {A ∈ Cld∗(X) | A ⊂ Z}. By Cld∗

F(X), we denote Cld∗(X) with the Fell topology, which

is generated by the following sets: (X \ K)+, U−, where K is a compact set in X and U is an open set in X.

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Cld∗

F(X) is a compact metrizable space if and only if X is a

locally compact separable metrizable space. For each compact metric space X = (X, d), the relative topology on Cld(X) ⊂ Cld∗

F(X) is induced by the Hausdorff

metric dH of d. When X is a locally compact, locally connected space, and Y is a locally compact space, C(X, Y ) can be regarded as a subspace of Cld∗

F(X×Y ), where each f ∈ C(X, Y ) is identified

with the graph of f in X × Y . clCld∗

F(X×Y ) C(X, Y ) is a natural metrizable compactification

• f C(X, Y ) under the assumption of the main theorem.

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Theorem 1.1 [K. Sakai and S. Uehara (1999)

• A. Kogasaka and K. Sakai (2009)]

Let X be an infinite, locally compact, locally connected, separable metrizable space. Then (clCld∗

F(X×R) C(X, R), C(X, R)) ≈ (Q, s),

where R = [−∞, +∞] is the extended real line. Remark C(I, R) is not homotopy dense in clCld∗

F(I×R) C(I, R) and

clCld∗

F(I×αR) C(I, R), where αR is the one point compactifica-

tion of R.

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2 Dendrites Definition (Dendrite) A dendrite is a Peano continuum containing no simple closed curves, equivalently it is a 1-dimensional compact AR. Definition (Convex metric) For a metric space X = (X, d), d is convex if for each x, y ∈ X, there exists z ∈ X such that d(x, z) = d(y, z) = d(x, y)/2. When d is complete, there exists an arc from x to y iso- metric to the segment [0, d(x, y)].

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Proposition 2.1 Every Peano continuum admits a convex metric. Hence every dendrite does so. Proposition 2.2 For each dendrite D, there exists a map γ : D2 × I → D such that for any distinct points x, y ∈ D, γ(x, y, ∗) : I ∋ t → γ(x, y, t) ∈ D is the unique arc from x to y. Proposition 2.3 Let D be a dendrite with E the end points. Then D \ E is homotopy dense in D.

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(Proof) Fix x0 ∈ D \ E, and define a homotopy h : D × I → D by h(x, t) = γ(x, x0, t), where γ : D2 × I → D as in Proposition 2.2.

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Theorem 2.4 A space Y is a 1-dimensional locally compact AR if and

• nly if Y has a dendrite compactification

Y such that the remainder Y \ Y is closed and contained in the set of all end points of Y . (Proof) Use the following Curtis’ result. Theorem 2.5 [D.W. Curtis (1980)] Every locally compact, connected, locally connected, metriz- able space Y has a Peano compactification Y such that (∗) for each non-empty connected open set U in Y , the sub- set U ∩ Y is a non-empty connected set.

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From now on let X be an infinite, locally compact, lo- cally connected, separable metrizable space and Y a 1- dimensional locally compact AR, and fix a dendrite com- pactification Y of Y with the remainder Y \ Y closed in Y and consisting end points. By Proposition 2.3, Y is homotopy dense in Y . Proposition 2.6 C(X, Y ) is homotopy dense in C(X, Y ). Let C(X, Y ) = clCld∗

F(X×

Y ) C(X, Y ) = clCld∗

F(X×

Y ) C(X,

Y ).

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3 The closure of C(X, Y ) in Cld∗

F(X ×

Y ) For spaces W and Z, let USCC(W, Z) = { φ : W → Cld(Z) φ is u.s.c. and φ(w) is connected for every w ∈ W. } Identifying each φ ∈ USCC(W, Z) with the graph of φ, we can regard USCC(W, Z) ⊂ Cld∗(W × Z).

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Theorem 3.1 For each locally compact, locally connected, paracompact space W with no isolated points and each dendrite D, clCld∗

F(W×D) C(W, D) = USCC(W, D).

Lemma 3.2 Let W be a locally compact, locally connected space and let Z be a compact connected space. Then USCC(W, Z) is closed in Cld∗

F(W × Z).

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Lemma 3.3 Let W be a paracompact space with no isolated points and let D be a dendrite. Then C(W, D) is dense in USCC(W, D). (Proof) Use the following Michael’s selection theorem. Theorem 3.4 [E. Michael (1959)] Let W be a paracompact space and D a dendrite. For every l.s.c. set-valued function φ : W → Cld(D), if each φ(w) is connected, then φ has a continuous selection.

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Hence if X is connected, then C(X, Y ) = clCld∗

F(X×

Y ) C(X, Y ) = clCld∗

F(X×

Y ) C(X,

Y ) = USCC(X, Y ).

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4 Proof of the Main Theorem Theorem 4.1 [R.D. Anderson] Let M ⊂ Q. The pair (Q, M) is homeomorphic to (Q, Q\s) if and only if M is a cap set in Q, that is, it is a Zσ-set and has the following property: (cap) For each pair A, B of compact sets in Q with B ⊂ A ∩ M and each ǫ > 0, there exists an embedding h : A → M such that h|B = idB and d(h(a), a) < ǫ for every a ∈ A, where d is an admissible metric for Q.

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We show the following: (1) C(X, Y ) ≈ s. (2) C(X, Y ) is homotopy dense in C(X, Y ). (3) C(X, Y ) ≈ Q. (4) C(X, Y ) \ C(X, Y ) is a cap set in C(X, Y ).

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(Case I) X is discrete. When X is discrete (so Y is non-compact), (C(X, Y ), C(X, Y )) = (clCld∗

F(X×

Y ) C(X,

Y ), C(X, Y )) = (C(X, Y ), C(X, Y )) ≈ ( Y N, Y N). Theorem 4.2 Let D be a dendrite and E0 be a non-empty closed set of D which consists of end points. Then (DN, (D \ E0)N) ≈ (Q, s).

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(Case II) X is non-discrete. Lemma 4.3 Let Wn be a compact AR and Zn be a homotopy dense Gδ subset of Wn, n ∈ N. Then (Q × ∏

n∈N

Wn, s × ∏

n∈N

Zn) ≈ (Q, s). Hence it is sufficient to show the case X is connected. First, we consider X is compact.

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(2) C(X, Y ) is homotopy dense in C(X, Y ). By Proposition 2.6, it remains to prove that C(X, Y ) is ho- motopy dense in clCld∗

F(X×

Y ) C(X,

Y ). Theorem 4.4 For each non-degenerate Peano continuum W and each dendrite D, C(W, D) is homotopy dense in clCld∗

F(W×D) C(W, D).

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Lemma 4.5 [K. Sakai and S. Uehara (1999)] Let W = (W, d) be a compact metric space and Z be a dense subset of W which has the following property: (∗) There exists α ≥ 1 such that for any locally finite simpli- cial complex K, each map f : K(0) → Z extends to a map ˜ f : |K| → Z such that diamd ˜ f(σ) ≤ α diamd f(σ(0)) for every σ ∈ K. Then Z is homotopy dense in W.

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Let W = (W, dW) be a Peano continuum with a convex met- ric and D = (D, dD) be a dendrite with a convex metric. Define an admissible metric ρ on W × D as follows: ρ((w1, y1), (w2, y2)) = max{dW(w1, w2), dD(y1, y2)} and denote by ρH the Hausdorff metric on Cld(W × D) in- duced from it. Lemma 4.6 Let K be a locally finite simplicial complex. If W is non-degenerate, then any map f : K(0) → C(W, D) extends to a map ˜ f : |K| → C(W, D) such that ( ∗ ) diamρH ˜ f(σ) ≤ 4 diamρH f(σ(0)) for each σ ∈ K.

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(4) C(X, Y ) \ C(X, Y ) is a cap set in C(X, Y ). Theorem 4.7 M = USCC(X, Y ) \ C(X, Y ) is a cap set in USCC(X, Y ). (Proof) Take an admissible metric dX and an admissible convex metric d

Y on X and

Y , respectively, and define an admissible metric ρ on X × Y as follows: ρ((x, y), (x′, y′)) = max{dX(x, x′), d

Y (y, y′)}.

By Theorem 4.1, it remains to show the following: (cap) For each compacta A, B ⊂ USCC(X, Y ) with B ⊂ A ∩ M and each ǫ > 0, there exists an embedding ˜ h : A → M

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such that ˜ h|B = idB and ρH(˜ h(a), a) < ǫ for every a ∈ A, where ρH is the Hausdorff metric of ρ.

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Let α : A → I be a map defined by α(a) = min{1, ǫ, ρH(a, B)}/3. Since C(X, Y ) is homotopy dense in USCC(X, Y ), we can construct a map f : A → USCC(X, Y ) such that f|B = idB, f(A \ B) ⊂ C(X, Y ) and ρH(f(a), a) ≤ α(a) for every a ∈ A. In addition, we can find an embedding g : A \ B → C(X, Y ) so that ρH(g(a), f(a)) < α(a) for each a ∈ A \ B because C(X, Y ) ≈ s. Fix x0 ∈ X and define h : A \ B → Cld(X × Y ) by h(a)(x) = { B(g(a)(x0), α(a)) if x = x0, g(a)(x) if x = x0, where B(g(a)(x0), α(a)) is the closed ball.

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Then each h(a) is an u.s.c. set-valued function. Because d

Y is convex, h is continuous and B(g(a)(x0), α(a))

is a non-degenerate subcontinuum of Y . Hence h(A \ B) ⊂ USCC(X, Y ) \ C(X, Y ) = M. Since x0 is not isolated point and g is an injection, h is also an injection. It follows that ρH(h(a), a) ≤ ρH(h(a), g(a)) + ρH(g(a), f(a)) + ρH(f(a), a) < 3α(a) ≤ min{1, ǫ, ρH(a, B)}. Therefore h : A \ B → M can be extended to ˜ h : A → M by ˜ h|B = idB. Since h(A \ B) ∩ B = ∅, ˜ h is the desired embedding.

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Next, we consider X is non-compact. Take the one-point compactification αX = X ∪ {∞} of X, which is a Peano continuum. Then use the result of the compact case.

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