aa r X i v : . [ m a t h . M G ] M a r AN INTERPOLATION OF METRICS AND SPACES OFMETRICS
YOSHITO ISHIKI
Abstract.
We provide an interpolation theorem of a family ofmetrics defined on closed subsets of metrizable spaces. As an ap-plication, we observe that various sets of all metrics with propertiesappeared in metric geometry are dense intersections of countableopen subsets in spaces of metrics on metrizable spaces. For in-stance, our study is applicable to the set of all non-doubling metricsand the set of all non-uniformly disconnected metrics. Intoroduction
For a metrizable space X , we denote by M( X ) the set of all metricson X which generates the same topology as the original one on X .In 1930, Hausdorff [13] proved the extension theorem stating that forevery metrizable space X , for every closed subset A of X and for every d ∈ M ( A ), there exists D ∈ M ( X ) such that D | A = d (see Theorem2.1). We define a function D X : M( X ) × M( X ) → [0 , ∞ ] by D X ( d, e ) = sup ( x,y ) ∈ X | d ( x, y ) − e ( x, y ) | . The function D X is a metric on M( X ) valued in [0 , ∞ ]. Throughout thispaper, we consider that M( X ) is always equipped with the topologygenerated by D X . We generalize the Hausdorff extension theorem toan interpolation theorem of metrics with an approximation by D X .A family { S i } i ∈ I of subsets of a topological space X is said to be discrete if for every x ∈ X there exists a neighborhood of x intersectingat most single member of { S i } i ∈ I .One of our main results is the following interpolation theorem: Theorem 1.1.
Let X be a metrizable space, and let { A i } i ∈ I be a dis-crete family of closed subsets of X . Then for every metric d ∈ M( X ) ,and for every family { e i } i ∈ I of metrics with e i ∈ M( A i ) , there exists ametric m ∈ M( X ) satisfying the following: (1) for every i ∈ I we have m | A i = e i ; (2) D X ( m, d ) = sup i ∈ I D A i ( e A i , d | A i ) . Date : March 31, 2020.2010
Mathematics Subject Classification.
Primary 54E35; Secondary 54E52.
Key words and phrases.
Metric space, Baire space. Space of metrics.The author was supported by JSPS KAKENHI Grant Number 18J21300.
Moreover, if X is completely metrizable, and if each e i ∈ M( A i ) is acomplete metric, then we can choose m ∈ M( X ) as a complete metricon X . A central idea of the proof of Theorem 1.1 is a correspondense be-tween a metric on a metrizable space and a topological embedding froma metrizable space into a Banach space. A metric d on a metrizablespace X induces a topological embedding from X into a Banach spacesuch as the Kuratowski embedding (see Theorem 2.8). Conversely, atopological embedding F from a metrizable space X into a Banachspace V with norm k · k V induces a metric m ∈ M( X ) on X defined by m ( x, y ) = k F ( x ) − F ( y ) k V . In the proof of Theorem 1.1, we utilize thiscorrespondence to translate the statement of Theorem 1.1 into an inter-polation problem on topological embeddings into a Banach space. Wethen resolve such a problem by using the Michael continuous selectiontheorem (see Theorem 2.2), and by using a similar method to Kura-towski [18] (see also [14] and [1]) of converting a continuous functioninto a topological embedding by extending a codomain.Theorem 1.1 enables us to investigate dense G δ subsets in the topol-ogy of the space (M( X ) , D X ) for a metrizable space X . To describe oursecond result precisely, we define a class of geometric properties thatunify various properties appeared in metric geometry. Remark . For every metrizable space X , and for every closed sub-set A of X , Nguyen Van Khue and Nguyen To Nhu [20] constructeda Lipschitz metric extensor from (M( A ) , D A ) into (M( X ) , D X ), and amonotone continuous metric extensor from M( A ) into M( X ); moreover,if X is completely metrizable, then each of these metric extensors mapsany complete metric in M( A ) into a complete metric in M( X ). To ob-tain such metric extensors, they used the Dugundji extension theoremconcerning locally convex topological linear spaces.Let P ∗ ( N ) be the set of all non-empty subsets of N . For a topologicalspace T , we denote by F ( T ) the set of all closed subsets of T . For asubset W ∈ P ∗ ( N ), and for a set S , we denote by Seq( W, S ) the set ofall finite injective sequences { a i } ni =1 in S with n ∈ W . Definition 1.1.
Let Q be an at most countable set, P a topologicalspace. Let F : Q → F ( P ) and G : Q → P ∗ ( N ) be maps. Let Z be aset. Let φ be a correspondence assigning a pair ( q, X ) of q ∈ Q and ametrizable space X to a map φ q,X : Seq( G ( q ) , X ) × Z × M( X ) → P .We say that a sextuple ( Q, P, F, G, Z, φ ) is a transmissible paremeter iffor every metrizable space X , for every q ∈ Q , and for every z ∈ Z thefollowing are satisfied:(TP1) for every a ∈ Seq( G ( q ) , X ) the map φ q,X ( a, z ) : M( X ) → P defined by φ q,X ( a, z )( d ) = φ q,X ( a, z, d ) is continuous;(TP2) for every d ∈ M( X ), if S is a subset of X and a ∈ Seq( G ( q ) , S ),then we have φ q,X ( a, z, d ) = φ q,S ( a, z, d | S ). N INTERPOLATION OF METRICS AND SPACES OF METRICS 3
We introduce a property determined by a transmissible parameter.
Definition 1.2.
Let G = ( Q, P, F, G, Z, φ ) be a transmissible param-eter. We say that a metric space (
X, d ) satisfies the G -transmissibleproperty if there exists q ∈ Q such that for every z ∈ Z and for every a ∈ Seq( G ( q ) , X ) we have φ q,X ( a, z, d ) ∈ F ( q ). We say that ( X, d ) sat-isfies the anti- G -transmissible property if ( X, d ) satisfies the negationof the G -transmissible property; namely, for every q ∈ Q there exist z ∈ Z and a ∈ Seq( G ( q ) , X ) with φ q,X ( a, z, d ) ∈ X \ F ( q ). A propertyon metric spaces is a transmissible property (resp. anti-transmissibleproperty ) if it is equivalent to a G -transmissible property (resp. anti- G -transmissible property) for some transmissible parameter G .The class of transmissible properties contains various properties ap-peared in metric geometry. Example 1.1.
The following properties on metric spaces are transmis-sible properties (see Section 4).(1) the doubling property;(2) the uniform disconnectedness;(3) satisfying the ultratriangle inequality;(4) satisfying the Ptolemy inequality;(5) the Gromov Cycl m (0) condition;(6) the Gromov hyperbolicity.To state our second result, we need a more additional condition fortransmissible properties. Definition 1.3.
Let G = ( Q, P, F, G, Z, φ ) be a transmissible parame-ter. We say that G is singular if for each q ∈ Q and for every ǫ ∈ (0 , ∞ )there exist z ∈ Z and a finite metrizable space ( R, d R ) such that(1) δ d R ( R ) ≤ ǫ , where δ d R ( R ) stands for the diameter of R ;(2) card( R ) ∈ G ( q ), where card stands for the cardinality;(3) φ q,R ( R, z, d R ) ∈ X \ F ( q ).Note that not all transmissible parameters are singular; especially,the Gromov hyperbolicity does not have a singular transmissible pa-rameter (see Proposition 4.21).Due to Theorem 1.1, we obtain the following theorem on dense G δ subsets in spaces of metrics: Theorem 1.2.
Let G be a singular transmissible parameter. For everynon-discrete metrizable space X , the set of all d ∈ M( X ) for which ( X, d ) satisfies the anti- G -transmissible property is dense G δ in M( X ) . Theorem 1.2 can be considered as an analogue of Banach’s famousresult stating that the set of all nowhere differentiable continuous func-tions contains a dense G δ set in a function space. YOSHITO ISHIKI
Remark . Theorem 1.2 holds true for the space CM( X ) of all com-plete metrics in M( X ) (see Theorems 4.7).We can apply Theorem 1.2 to the properties (1)–(5) mentioned inExample 1.1. Therefore we conclude that the set of all metrics notsatisfying these properties are dense G δ in spaces of metrics (see alsoCorollary 4.16). We also conclude that the set of all metrics with richpseudo-cones is dense G δ in spaces of metrics (see Theorem 4.14).Our third result is based on the fact that for second countable locallycompact space X the space M( X ) is a Baire space (see Lemma 5.1).For a property P on metric spaces, we say that a metric space ( X, d )satisfies the local P if every non-empty open metric subspace of X satisfies the property P .As a local version of Theorem 1.2, we obtain the following: Theorem 1.3.
Let X be a second countable, locally compact locallynon-discrete space. Then for every singular transmissible parameter G , the set of all metrics d ∈ M( X ) for which ( X, d ) satisfies the localanti- G -transmissible property is dense G δ in M( X ) . Note that all second countable locally compact spaces are metrizable,which is a consequence of the Urysohn metrization theorem.The organization of this paper is as follows: In Section 2, we reviewthe basic or classical theorems on topological spaces and metric spaces.In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2and show that various properties in metric geometry are transmissibleproperties. In Section 5, we prove Theorem 1.3.
Acknowledgements.
The author would like to thank Professor KoichiNagano for his advice and constant encouragement.2.
Preliminaries
In this paper, the symbol N stands for the set of all positive integers.Let ( X, d ) be a metric space. Let A be a subset of X . We denoteby δ d ( A ) the diameter of A . We denote by B ( x, r ) (resp. U ( x, r )) theclosed (resp. open) ball centered at x with radius r .2.1. The Hausdorff extension theorem.
The following celebratedtheorem was first proven by Hausdorff [13] (cf [14], [1], [3], [2], [24]).
Theorem 2.1.
Let X be a metrizable space, and let A be a closedsubset of X . Then for every d ∈ M( A ) there exists D ∈ M( X ) with D | A = d . Moreover, if X is completely metrizable, and if d ∈ M( A ) isa complete metric on A , then we can choose D ∈ M( X ) as a completemetric on X . N INTERPOLATION OF METRICS AND SPACES OF METRICS 5
The Michael continuous selection theorem.
Let V be a Ba-nach space. We denote by CC ( V ) the set of all non-empty closedconvex subsets of V . For a topological space X we say that a map φ : X → CC ( V ) is lower semi-continuous if for every open subset O of V the set { x ∈ X | φ ( x ) ∩ O = ∅ } is open in X .The following theorem is known as one of the Michael continuousselection theorems proven in [22]. Theorem 2.2.
Let X be a paracompact space, and A a closed subsetsof X . Let V be a Banach space. Let φ : X → CC ( V ) be a lower semi-continuous map. If a continuous map f : A → B satisfies f ( x ) ∈ φ ( x ) for all x ∈ A , then there exists a continuous map F : X → V with F | A = f such that for every x ∈ X we have F ( x ) ∈ φ ( x ) . By using linear structure, we have the following:
Proposition 2.3.
Let V be a Banach space, and x, y ∈ V . Then forevery r ∈ (0 , ∞ ) we have H ( B ( x, r ) , B ( y, r )) = k x − y k V , where H is the Hausdorff distance induced from the norm k · k V of V . Corollary 2.4.
Let X be a topological space, and let V be a Banachspace. Let H : X → V be a continuous map and r ∈ (0 , ∞ ) . Thena map φ : X → CC ( V ) defined by φ ( x ) = B ( H ( x ) , r ) is lower semi-continuous.Proof. For every open subset O of V , and for every point a ∈ X with φ ( a ) ∩ O = ∅ , choose u ∈ φ ( a ) ∩ O and l ∈ (0 , ∞ ) with U ( u, l ) ⊂ O . ByProposition 2.3, we can take δ ∈ (0 , ∞ ) such that for every x ∈ U ( a, δ )we have H ( φ ( x ) , φ ( a )) = k H ( x ) − H ( a ) k V < l. Then we have φ ( x ) ∩ U ( u, l ) = ∅ , and hence φ ( x ) ∩ O = ∅ . Thereforethe set { x ∈ X | φ ( x ) ∩ O = ∅ } is open in X . (cid:3) Paracompact spaces.
The following theorem is known as theStone theorem proven in [23].
Theorem 2.5.
All metrizable spaces are paracompact.
By this theorem, we can apply Theorem 2.2 to all metrizable space.Let X be a topological space. We say that a family { u j } j ∈ J ofcontinuous functions on X valued in [0 ,
1] is a partition of unity of X iffor every x ∈ X we have P j ∈ J u j ( x ) = 1. A partition of unity { u j } j ∈ J of X is locally finite if the family { u − j ((0 , } j ∈ J of open subsets of X is locally finite. For an open covering { U i } i ∈ I of X , a partition of unity { u j } j ∈ J of X is subordinated to { U i } i ∈ I if the family { u − j ((0 , } j ∈ J isa refinement of { U i } i ∈ I . YOSHITO ISHIKI
The following proposition guarantees the existence of a locally finitepartition of unity on paracompact spaces. The proof can be seen in[21, Proposition 2].
Proposition 2.6.
Every open covering of a paracompact space has alocally finite partition of unity subordinated to it.
Corollary 2.7.
Let X be a paracompact space, and let { A i } i ∈ I be adiscrete family of closed subsets of X . Then there exists a locally finitepartition of unity { u i } i ∈ I of X with the same index set I such that forevery i ∈ I and for every x ∈ A i we have u i ( x ) = 1 .Proof. Since X is paracompact, we can take a discrete family { U i } i ∈ I of open subsets of X with A i ⊂ U i (see [8, Theorems 5.1.17, 5.1.18.]).Fix o ∈ I . Replace U o with X \ S i = o A i . Since { A i } i ∈ I is discrete, theset U o is open. By the definitions, { U i } i ∈ I is an open covering of X .Thus by Proposition 2.6, there exists a locally finite partition of unity { u j } j ∈ J of X subordinated to { U i } i ∈ I . Put K ( o ) = J \ S i ∈ I \{ o } K ( i ).For each i ∈ I \ { o } , put K ( i ) = { j ∈ J | u − j ((0 , ∩ A i = ∅ } . For each i ∈ I , define a map v i : X → [0 ,
1] by v i ( x ) = X j ∈ K ( i ) u j ( x ) . For all i, j ∈ I , if A i ∩ U j = ∅ , then i = j . Thus { K ( i ) } i ∈ I is mutuallydisjoint, and for every i ∈ I and for every x ∈ A i we have u i ( x ) = 1.Since { u j } j ∈ J is a locally finite partition of unity of X , so is { v i } i ∈ I . (cid:3) The Kuratowski embedding theorem.
For a metric space(
X, d ), we denote by C b ( X ) the Banach space of all bounded continu-ous functions on X equipped with the supremum norm. For x ∈ X , wedenote by d x the function from X to R defined by d x ( p ) = d ( x, p ). Thefollowing theorem, which states that every metric space is isometricallyembeddable into Banach spaces, is known as the Kuratowski embed-ding theorem. The proof can be seen in, for example, [8, Theorem4.3.14.]. Theorem 2.8.
Let ( X, d ) be a metric space. Take o ∈ X . Then themap K : X → C b ( X ) defined by K ( x ) = d x − d o is an isometricembedding. Moreover, if ( X, d ) is bounded, the map L : X → C b ( X ) defined by L ( x ) = d x is an isometric embedding. Baire spaces.
A topological space X is said to be Baire if theintersections of countable dense open subsets of X are dense in X .The following is known as the Baire category theorem. Theorem 2.9.
Every completely metrizable space is a Baire space.
N INTERPOLATION OF METRICS AND SPACES OF METRICS 7
Since G δ subset of completely metrizable space is completely metriz-able (see, e.g. [26, Theorem 24.12]), we obtain the following: Lemma 2.10.
Every G δ subset of a completely metrizable space is aBaire space. An interpolation Theorem of metrics
In this section, we prove Theorem 1.1.3.1.
Amalgamation lemmas.
Let X and Y be sets, and let τ : X → Y be a bijection. For a metric d on Y , we denote by τ ∗ d the metric on X defined by ( τ ∗ d )( x, y ) = d ( τ ( x ) , τ ( y )). Note that the map τ is anisometry between ( X, τ ∗ d ) and ( Y, d ).The following proposition can be considered as a specific case of therealization of a Gromov–Hausdorff distance of two metric spaces (see[5, Chapter 7]).
Proposition 3.1.
Let X be a metrizable space. For r ∈ (0 , ∞ ) , let d, e ∈ M( X ) with D X ( d, e ) ≤ r . Put X = X , and let X be a set with card( X ) = card( X ) and X ∩ X = ∅ . Let τ : X → X be a bijection.Then there exists a metric h ∈ M( X ⊔ X ) such that (1) h | X = d ; (2) h | X = ( τ − ) ∗ e ; (3) for every x ∈ X we have h ( x, τ ( x )) = r/ .Proof. We define a symmetric function h : ( X ⊔ X ) → [0 , ∞ ) by h ( x, y ) = d ( x, y ) if x, y ∈ X ; e ( x, y ) if x, y ∈ X ;inf a ∈ X ( d ( x, a ) + r/ e ( τ ( a ) , y )) if ( x, y ) ∈ X × X .By the definition, for every x ∈ X , we have h ( x, τ ( x )) ≥ r/
2, and h ( x, τ ( x )) ≤ d ( x, x ) + r/ e ( τ ( x ) , τ ( x )) = r/ . Therefore for every x ∈ X we have h ( x, τ ( x )) = r/ h satisfies the triangle inequality. In the casewhere x, y ∈ X and z ∈ X , for all a, b ∈ X we have h ( x, y ) = d ( x, y ) ≤ d ( x, a ) + d ( a, b ) + d ( b, y ) ≤ d ( x, a ) + r + e ( τ ( a ) , τ ( b )) + d ( b, y ) ≤ d ( x, a ) + r + e ( τ ( a ) , z ) + e (( τ ( b ) , z ) + d ( b, y ) ≤ ( d ( x, a ) + r/ e ( τ ( a ) , z )) + ( d ( y, b ) + r/ e ( τ ( b ) , z )) , and hence we obtain h ( x, y ) ≤ h ( x, z ) + h ( z, y ). In the case where x, z ∈ X and y ∈ X , for all a ∈ X we have h ( x, y ) ≤ d ( x, a ) + r/ e ( τ ( a ) , y ) ≤ d ( x, z ) + ( d ( z, a ) + r/ e ( τ ( a ) , y )) , YOSHITO ISHIKI and hence h ( x, y ) ≤ h ( x, z ) + h ( z, y ). By replacing the role of X withthat of X , we conclude that h satisfies the triangle inequality. (cid:3) The following two propositions are known as amalgamations of met-rics. The proof can be seen in, for example, [4] (cf [25], [9]). For thesake of self-containedness, we give a proof.
Proposition 3.2.
Let ( X, d X ) and ( Y, d Y ) be metric spaces, and let Z = X ∩ Y . If Z = ∅ and d X | Z = d Y | Z , then there exists a metric h on X ∪ Y such that (1) h | X = d X ; (2) h | Y = d Y .Proof. We define a symmetric function h : ( X ∪ Y ) → [0 , ∞ ) by h ( x, y ) = d X ( x, y ) if x, y ∈ X ; d Y ( x, y ) if x, y ∈ Y ;inf z ∈ Z ( d X ( x, z ) + d Y ( z, y )) if ( x, y ) ∈ X × Y .Since d X | Z = d Y | Z , the function h is well-defined. By the definition, h satisfies the conditions (1) and (2).We next prove that h satisfies the triangle inequality. In the casewhere x, y ∈ X and z ∈ Y , for all a, b ∈ Z we have h ( x, y ) = d X ( x, y ) ≤ d X ( x, a ) + d X ( a, b ) + d X ( b, y )= d X ( x, a ) + d Y ( a, b ) + d X ( b, y ) ≤ ( d X ( x, a ) + d Y ( a, z )) + ( d Y ( z, b ) + d X ( b, y )) , and hence we obtain h ( x, y ) ≤ h ( x, z ) + h ( z, y ). In the case where x, z ∈ X and y ∈ Y , for all a ∈ Z we have h ( x, y ) ≤ d X ( x, a ) + d Y ( a, y ) ≤ d X ( x, z ) + d X ( z, a ) + d Y ( a, y ) , and hence we have h ( x, y ) ≤ h ( x, z )+ h ( z, y ). By replacing the role of X with that of Y , we conclude that h satisfies the triangle inequality. (cid:3) For a mutually disjoint family { T i } i ∈ I of topological spaces, we con-sider that the space ` i ∈ I T i is always equipped with the direct sumtopology. Proposition 3.3.
Let ( X, d X ) and ( Y, d Y ) be metric spaces. If X ∩ Y = ∅ , then for every r ∈ (0 , ∞ ) there exists a metric h ∈ M( X ⊔ Y ) suchthat (1) h | X = d X ; (2) h | Y = d Y ; (3) for all x ∈ X and y ∈ Y we have r ≤ h ( x, y ) . N INTERPOLATION OF METRICS AND SPACES OF METRICS 9
Proof.
Take two fixed points a ∈ X and b ∈ Y . Take r ∈ (0 , ∞ ). Wedefine a symmetric function h : ( X ∪ Y ) → [0 , ∞ ) by h ( x, y ) = d X ( x, y ) if x, y ∈ X ; d Y ( x, y ) if x, y ∈ Y ; d X ( x, a ) + r + d Y ( b, y ) . if ( x, y ) ∈ X × Y .Then the conditions (1), (2) and (3) are satisfied.We prove that the function h satisfies the triangle inequality. In thecase where x, y ∈ X and z ∈ Y , we have h ( x, y ) = d X ( x, y ) ≤ d X ( x, a ) + d X ( a, y ) ≤ ( d X ( x, a ) + r + d Y ( b, z )) + ( d X ( y, a ) + r + d Y ( b, z ))= h ( x, z ) + h ( z, y ) . In the case where x, z ∈ X and y ∈ Y , we have h ( x, y ) = d X ( x, a ) + r + d Y ( b, y ) ≤ d X ( x, z ) + d X ( z, a ) + r + d Y ( b, y ) = h ( x, z ) + h ( z, y ) . By replacing the role of X with that of Y , we conclude that h satisfiesthe triangle inequality. (cid:3) Lemma 3.4.