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 1

Outline 0 Introduction 1 Background 2 Dendrites The closure of C( X, Y ) in Cld ∗ F ( X × � 3 Y ) 4 Proof of the Main Theorem 2

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. 3

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. 4

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 . 5

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 d H 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 . cl Cld ∗ F ( X × Y ) C( X, Y ) is a natural metrizable compactification of C( X, Y ) under the assumption of the main theorem. 6

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 (cl Cld ∗ 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 cl Cld ∗ F ( I × R ) C( I , R ) and cl Cld ∗ F ( I × α R ) C( I , R ) , where α R is the one point compactifica- tion of R . 7

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 )] . 8

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 : D 2 × 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 . 9

(Proof) Fix x 0 ∈ D \ E , and define a homotopy h : D × I → D by h ( x, t ) = γ ( x, x 0 , t ) , where γ : D 2 × I → D as in Proposition 2.2. � 10

Theorem 2.4 A space Y is a 1 -dimensional locally compact AR if and only 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. 11

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 Y ) C( X, � C( X, Y ) = cl Cld ∗ Y ) C( X, Y ) = cl Cld ∗ Y ) . F ( X × � F ( X × � 12

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 ) . 13

Theorem 3.1 For each locally compact, locally connected, paracompact space W with no isolated points and each dendrite D , cl Cld ∗ 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 ) . 14

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. 15

Hence if X is connected, then Y ) C( X, � C( X, Y ) = cl Cld ∗ Y ) C( X, Y ) = cl Cld ∗ Y ) F ( X × � F ( X × � = USCC( X, � Y ) . 16

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 Z σ -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 = id B and d ( h ( a ) , a ) < ǫ for every a ∈ A , where d is an admissible metric for Q. 17

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 ) . 18

(Case I) X is discrete. When X is discrete (so Y is non-compact), Y ) C( X, � (C( X, Y ) , C( X, Y )) = (cl Cld ∗ Y ) , C( X, Y )) F ( X × � = (C( X, � Y ) , C( X, Y )) ≈ ( � Y N , Y N ) . Theorem 4.2 Let D be a dendrite and E 0 be a non-empty closed set of D which consists of end points. Then ( D N , ( D \ E 0 ) N ) ≈ ( Q , s ) . 19

(Case II) X is non-discrete. Lemma 4.3 Let W n be a compact AR and Z n be a homotopy dense G δ subset of W n , n ∈ N . Then ∏ ∏ ( Q × W n , s × Z n ) ≈ ( Q , s ) . n ∈ N n ∈ N Hence it is sufficient to show the case X is connected. First, we consider X is compact. 20

(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- Y ) C( X, � motopy dense in cl Cld ∗ Y ) . F ( X × � Theorem 4.4 For each non-degenerate Peano continuum W and each dendrite D , C( W, D ) is homotopy dense in cl Cld ∗ F ( W × D ) C( W, D ) . 21

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 diam d ˜ f ( σ ) ≤ α diam d f ( σ (0) ) for every σ ∈ K . Then Z is homotopy dense in W . 22

Let W = ( W, d W ) be a Peano continuum with a convex met- ric and D = ( D, d D ) be a dendrite with a convex metric. Define an admissible metric ρ on W × D as follows: ρ (( w 1 , y 1 ) , ( w 2 , y 2 )) = max { d W ( w 1 , w 2 ) , d D ( y 1 , y 2 ) } and denote by ρ H 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 f ( σ ) ≤ 4 diam ρ H f ( σ (0) ) for each σ ∈ K . ( ∗ ) diam ρ H ˜ 23

(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 d X and an admissible Y on X and � convex metric d � Y , respectively, and define an admissible metric ρ on X × � Y as follows: ρ (( x, y ) , ( x ′ , y ′ )) = max { d X ( 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 24

such that ˜ h | B = id B and ρ H (˜ h ( a ) , a ) < ǫ for every a ∈ A , where ρ H is the Hausdorff metric of ρ . 25