Shape as a Cantor completion process

Math. Z. 225, 67– 86 (1997)

Shape as a Cantor completion process? M.A. Moron1 , F.R. Ruiz del Portal1; ?? 1 U.D.

Matematicas. E.T.S.I. de Montes, Universidad Politecnica de Madrid, Madrid, 28040, Spain (e-mail: [email protected]) Received 14 October 1994; in nal form 31 October 1995

Introduction It has been proved that giving certain structures on the sets of morphisms in categories is a useful tool for classi cation problems in such categories. On the other hand, doing this appears as a source of new de nitions and results, and as an adequate framework to simplify old de nitions, results and proofs. Moreover, sometimes, it allows us to obtain some connections between apparently unrelated facts. The above paragraph is just an outline of our intentions in this paper on shape theory. The main idea in this paper is to consider shape morphisms as certain classes of Cauchy sequences obtained following, step by step, Cantor’s completion process, to obtain the real numbers from the rationals (or the completion of a metric space). But our starting-point is not a metric, even not a pseudometric one. The rst advantage of this new point of view is that we construct complete metrics in spaces of shape morphisms, which, in addition, are non-Archimedean (or ultrametrics). Following with the analogy of the construction of the irrationals from the rationals, and as a survey of the topological properties of the spaces herein constructed, we obtain that these spaces are homeomorphic to closed subspaces of the irrationals. Later, we associate, to shape morphisms, in a functorial way, uniformly continuous maps, obtaining that the uniform homeomorphism types of our spaces are shape invariants. Consequently, we describe many new shape invariants. We point out later that some of them are new reformulations of other already known. ?

The authors have been supported by DGICYT, PB93-0454-C02-02. address: Departamento de Geometra y Topologa, Facultad de CC. Matematicas, Universidad Complutense de Madrid, Madrid, 28040, Spain (e-mail: rrportal @sungt1.mat.ucm.es)

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We also show that two close maps (with the supremum metric) generate close shape morphisms. This closeness is strongly uniform in the sense that it depends only on the image space. At this point we show another dierence on the behaviour of the homotopy and shape functors, because there is no possibility to de ne a Hausdor topology, in the sets of homotopy classes of maps between metric compacta, such that the natural projection, from the spaces of maps with the supremum metric is continuous (for every pair of compacta). The remaining part of the paper is dedicated to point out how to use the spaces of shape morphisms, here constructed, to obtain new characterizations of some known notions (movability, internal movability, FANR: : :). Another advantage we obtain is the possibility to construct, in a natural way, new metrics in hyperspaces following Borsuk’s ideas in the de nition of the metric of continuity. We also introduce the concept of equimovable family of sets which seems to be useful to obtain niteness theorems in shape theory and allows us to connect with recent results in controlled Topology (Dranishnikov, Ferry, Grove, Petersen, Wu, Yamaguchi,: : : see references herein). In forthcoming papers the authors will follow the line, here initiated, to give: (a) A topological extension, for arbitrary topological spaces, of our construction which, in particular, allows us to obtain, up a gap of set theory, a new characterization of N-compactness and a new construction of N-compacti cations. (b) An invariant ultrametric on the shape groups obtaining that such metric groups are shape invariants. This metric allows us to recognize many clopen normal subgroups. We think that quite a number of new problems can be implicitly derived from this paper. We also note that the construction here made could be transferred to other categories, not only in topological shape. Information about shape theory can be found in [B1, DS, and MS1]. The introductory chapters of [Sc, V] and their corresponding references, have been used by the authors for getting acquainted about ultrametrics. Along this article we shall assume the Hilbert Q to be endowed with a xed metric. 1 The construction and basic properties Let X; Y be two compacta and consider Y to be embedded in the Hilbert cube Q. Denote by C(X; Q) the set of maps from X to Q and de ne F : C(X; Q) × C(X; Q) →R where F(f; g)=inf { ¿ 0 : f ' g in B(Y; )} (' denotes the homotopy relation). We have (a) F(f; g)=0

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(b) F(f; g) = F(g; f) (c) F(f; g)5max{F(f; h); F(h; g)} for every h ∈ C(X; Q). Note that F is not a pseudometric on C(X; Q) but if we restrict ourselves to maps whose images are contained in Y; then it is a pseudometric on C(X; Y ) such that F(f; g) = 0 if and only if f and g are weakly homotopic, i.e. they are homotopic in every neighbourhood of Y in Q. Let us initiate, step by step, the Cantor process to obtain the real numbers from the rationals, and then we need the following De nition 1.1 A sequence {fk }k∈N ⊂ C(X; Q) is said to be a Cauchy sequence if for every  ¿0 there is a k0 ∈ N such that F(fk ; fk 0 )¡ for every k; k 0 = k0 . De nition 1.2 Two Cauchy sequences {fk }k∈N ; {gk }k∈N ⊂ C(X; Q) are said to be F-related; written {fk }k∈N F{gk }k∈N ; if the sequence f1 ; g1 ; f2 ; g2 ; f3 ; g3 ; : : : is again a Cauchy sequence. Let us prove now the following Proposition 1.3 (a) The F-relation is an equivalence relation. (b) For every pair of Cauchy sequences {fk }k∈N ; {gk }k∈N there exists limk → ∞ F(fk ; gk ). (c) Let {fk }k∈N ; {fk0 }k∈N ; {gk }k∈N and {gk0 }k∈N be Cauchy sequences such that {fk }k∈N F{fk0 }k∈N and {gk }k∈N F{gk0 }k∈N then limk→∞ F(fk ; gk ) = limk→∞ F(fk0 ; gk0 ): Proof. (a) is clear, (b) is a consequence of the fact that |F(fk ; gk )−F(fk 0 ; gk 0 )| 5 F(fk ; fk 0 ) + F(gk ; gk 0 ) and (c) is obtained by applying (b) to the pair f1 ; f10 ; f2 ; f20 ; : : : and g1 ; g10 ; g2 ; g20 ; : : : of Cauchy sequences. In order to relate the last construction with shape theory we have Proposition 1.4 With the above notations; we have (a) A sequence {fk }k∈N ⊂ C(X; Q) is a Cauchy sequence if and only if it is an approximative map (in the sense of Borsuk [B1]). (b) Two Cauchy sequences {fk }k∈N ; {gk }k∈N ⊂ C(X; Q) are F-related if and only if they are homotopic approximative maps (in the sense of Borsuk [B1]). As we know, the homotopy classes of approximative maps represent the shape morphisms (see [MS1]), then the nal step in this construction is given in the next Theorem 1.5 Let Sh(X; Y ) be the set of shape morphisms from X to Y . Take ; ∈ Sh(X; Y ) and de ne d(; ) = limk→∞ F(fk ; gk ) being {fk }k∈N ; {gk }k∈N Cauchy sequences in the classes ; respectively. Then (Sh(X; Y ); d) is a complete non-Archimedean metric space.

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Proof. It is clear that d(; ) = 0 if and only if  = and that d(; ) = d( ; ). From property (c) of F it follows that d(; )5 max{d(; ); d( ; )} for every ; ; ∈ Sh(X; Y ); then d is an ultrametric (or a non-Archimedean metric). Completeness can be obtained following the proof of the completeness of R; constructed via Cauchy sequences of rationals, or a completion of a metric space. Corollary 1.6 (a) For every  ¿ 0 and  ∈ Sh(X; Y ); B(; ) is a clopen subset of Sh(X; Y ). (b) dim(Sh(X; Y ); d) = 0 for every pair of compacta (dim denotes the covering dimension). Corollary 1.7 A sequence {k }k∈N ⊂ Sh(X; Y ) converges if and only if d(k ; k+1 ) →0 as k → ∞. Corollaries 1.6 and 1.7 follow from properties of ultrametrics. We used mainly [Sc and V]. We are aware that deeper results, about the metric d; can be obtained if we read carefully the last references, but it is not our intention now. The next result is useful because it points out a geometrical characterization of the relation d(; )¡ . It attempts to be an analogue to the geometrical view of the distance of reals numbers, if you consider R as a line. Proposition 1.8 Let X; Y be two compacta and consider; as above; Y to be embedded in the Hilbert cube Q. Take ; ∈ Sh(X; Y ) and  ¿ 0; then; d(; )¡ if and only if S(iY; B(Y; ) ) ◦  = S(iY; B(Y; ) ) ◦ ; where S(h) denotes the shape morphism induced by the map h; and B(Y; ) is the generalized open ball of center Y; in the Hilbert cube Q. Proof. Let {fk }k∈N ∈ ; {gk }k∈N ∈ and r ∈ R such that limk→∞ F(fk ; gk ) = r ¡ : For r 0 ∈ R such that r ¡r 0 ¡  there exists k0 ∈ N with fk ' gk in B(Y; r 0 ) for k = k0 ; then S(iY; B(Y; ) ) ◦  = S(iY; B(Y; ) ) ◦ . On the other hand, take k0 ∈ N such that F(fk ; fk+1 )¡=2 and F(gk ; gk+1 )¡ =2 for every k = k0 : Thus S(iY; B(Y; ) ) ◦  = [fk0 ] = S(iY; B(Y; ) ) ◦ = [gk0 ]. Let H : X × I → B(Y; ) be a homotopy with H0 = fk0 and H1 = gk0 from the compactness of H (X × I ) there exists ¡ such that H (X × I ) ⊂ B(Y; ) ⊂ B(Y; ). F(fk0 +j ; gk0 +j )5 max{F(fk0 ; fk0 +j ); F(fk0 ; gk0 ); F(gk0 ; gk0 +j )} therefore F(fk ; gk ) 5 max{; =2} for every k =k0 and consequently d(; )¡. The next result is a representation of the topological space Sh(X; Y ) which strengthens the relationships between the construction, given in this paper, with

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the usual, to obtain the reals from the rationals. Moreover, it points out the main topological properties of the space Sh(X; Y ). Theorem 1.9 For every pair X; Y of compacta; the space Sh(X; Y ) is homeomorphic to a closed subset of the irrationals. First we need the following Claim. If Y is an ANR then there exists 0 ¿0 such that if ; ∈ Sh(X; Y ) and d(; ) ¡ 0 then  = . Moreover Card(Sh(X; Y )) 5ℵ0 : Proof of the claim. Let ; ∈ Sh(X; Y ). Since Y ∈ ANR, there exist maps f; g : X → Y such that S(f) =  and S(g) = . Let 0 ¿0 be such that there is a retraction r : B(Y; 0 )→Y . Suppose that d(; ) ¡0 ; then there exists a homotopy H : X × I →B(Y; 0 ) connecting f and g. Consequently the homotopy r ◦ H joins f and g in Y . Thus  = . On the other hand, the shape classes of the maps form a discrete covering of C(X; Y ) and from separability we deduce that Card(Sh(X; Y )) 5ℵ0 : Proof of Theorem 1.9. Q Take a neighbourhood base {Vn }n∈N of compact ANR’s of Y in Q. Let Z = n∈N Sh(X; Vn ). De ne  : Sh(X; Y )→ Z by () = (n )n∈N where n = S(iY;Vn ) ◦ . Using the claim it suces to prove that  maps Sh(X; Y ) homeomorphically onto a closed subset of Z. Obviously  is injective. Take {k }k∈N →  as k → ∞ in Sh(X; Y ) and n ∈ N. Let  ¿ 0 such that B(Y; ) ⊂ Vn . Take k0 such that d(; k )¡  for every k =k0 . Then pn (k ) = pn () for every k = k0 where pn : Z →Sh(X; Vn ) is the corresponding projection. This shows that  is continuous. On the other hand, −1 : (Sh(X; Y ) →Sh(X; Y )) is also continuous. Indeed: take {(k )}k∈N → () as k → ∞ in (Sh(X; Y )), then, for every n ∈ N, {pn ((k ))}k∈N converges to pn (()). Using Lemma 1.10, for every n ∈ N there exists kn ∈ N such that pn ((k )) = pn (()) for k =kn . From the fact that {Vn }n∈N is a neighbourhood base of Y it follows that base {k }k∈N converges to . It only remains to check that (Sh(X; Y )) is closed in Z. For every n ∈ N de ne rn : Z →Z by rn ((n )n∈N ) = ( n )n∈N where k = k for k =n; k = S(iVn ; Vk ) ◦n for k ¡ n. It is clear that rn is a continuous retraction onto Imr T n for every n ∈ N. Now, it is straightforward to see that (Sh(X; Y )) = n∈N Imr n ; this concludes the proof. Remark. The last proof points out that if X is compact and X = lim{Xn ; ← Pn; n+1 }, where Xn is a compact ANR for each n ∈ N; then, for every compact K; Sh(K; X ) can be obtained as the inverse limit of an inverse sequence, whose spaces are Sh(K; Xn ). Since a more general result will be given in related forthcoming papers, and in order to avoid repetitions we have not used such language.

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2 Uniformly continuous functions induced by shape morphisms The key result in this section is the following Theorem 2.1 Let X; Y be compacta and f : X →Y be a shape morphism. Suppose Z to be a compactum and de ne f∗ : Sh(Y; Z)→ Sh(X; Z) by f∗ () =  ◦ f and f∗ : Sh(Z; X ) →Sh(Z; Y ) by f∗ ( ) = f ◦ . Then; (a) f∗ and f∗ are uniformly continuous functions. (b) (g ◦ f)∗ = f∗ ◦ g∗ and (g ◦ f)∗ = g∗ ◦ f∗ . (c) (Id X )∗ = Id Sh(X; Z) and (Id X )∗ = Id Sh(Z; X ) . Proof. Only the proof of (a) is needed. Let us denote by d the metrics in all spaces of shape morphisms considered. The rst assertion is a consequence of the fact that for every 1 ; 2 ∈ Sh(Y; Z) d(f∗ (1 ); f∗ (2 )) 5 d(1 ; 2 ). Now let ¿0 and f0 : X → B(Y; ) be a map such that [ f0 ] = S(iY; B(Y; ) ) ◦ f. Take  ¿0 such that f0 admits an extension F : B(X; ) → B(Y; ). Suppose 1 ; 2 ∈ Sh(Z; X ) with d( 1 ; 2 )¡ , then it follows that d(f∗ ( 1 ); f∗ ( 2 ))¡ and f∗ is also uniformly continuous. Remark. Note that  depends on f0 and on the chosen extension F. In order to avoid it we can take  as the supremum of positive real numbers r such that the homotopy class generating S(iY; B(Y; ) ) ◦ f can be extended to a homotopy class with domain B(X; r). Among the possible consequences of Theorem 2.1 we see the following. Corollary 2.2 (Shape invariance of spaces of shape morphisms). (a) Suppose X; Y to be compacta with Sh(X ) = Sh(Y ) and let Z be any arbitrary compactum; then Sh(Y; Z) is uniformly homeomorphic to a uniform retract of Sh(X; Z). Moreover; Sh(Z; Y ) is uniformly homeomorphic to a uniform retract of Sh(Z; X ). (b) Suppose now that Sh(X ) = Sh(Y ); then Sh(X; Z) and Sh(Y; Z) are uniformly homeomorphic. Moreover; Sh(Z; X ) and Sh(Z; Y ) are also uniformly homeomorphic (uniformly homeomorphic means that there is a uniformly continuous homeomorphism such that its inverse is also uniformly continuous; uniform retract means that there is a uniformly continuous retraction): Remark. Note that we have found many new shape invariants, because the (uniform) metric type of Sh(X; Y ) only depends on the shape of X and Y . On the other hand, as an easy consequence of Corollary 2.2, we have that for every compactum Y shape dominated by an ANR (i.e. Y is an FANR), there exists ¿0 such that for every compactum X and every pair of shape morphisms ; ∈ Sh(X; Y ), with d(; )¡ , then  = . Moreover, from a compactum X to an FANR Y there are at most countable many dierent shape morphisms. Since this result will be improved later, we do not state it as a proposition.

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Another consequence of Theorem 2.1 is the following characterization of compactness in the spaces of shape morphisms. Corollary 2.3 A ⊂ Sh(X; Y ) is relatively compact if and only if for every  ¿ 0 the set {S(iY; B(Y; ) ) ◦ :  ∈ A} is nite. Proof. Let us suppose that A is relatively compact and take  ¿0. Consider V to be a compact neighbourhood of Y in Q such that V ⊂ B(Y; ). It follows  is compact in Sh(X; V ) and then nite (see claim of 1.9). that S(iY; V )∗ (A) On the other hand, if {S(iY; B(Y; ) ) ◦ :  ∈ A} is nite for every  ¿0, then A is totally bounded. Since Sh(X; Y ) is complete it follows that A is compact. Corollary 2.4 Let X; Y; Z be compacta. Consider the functions DZ∗ : Sh(X; Y ) → C(Sh(Y; Z); Sh(X; Z)) and D∗Z : Sh(X; Y ) → C(Sh(Z; X ); Sh(Z; Y )) de ned by DZ∗ () = ∗ and D∗Z ( ) = ∗ . Then DZ∗ and D∗Z are continuous (the functional spaces are endowed with the compact-open topology). Proof. First we prove that DZ∗ is continuous. Let A ⊂ Sh(Y; Z) be a compact and let  ¿0. Then {S(iZ; B(Z; ) ) ◦ :  ∈ A} is nite. There exist maps f1 ; f2 ; : : : ; fm : Y → B(Z; ) such that for every  ∈ A there exists j ∈ {1; 2; : : : ; m} with S(iY; B(Y; ) ) ◦  = S(fj ). Take ¿ 0 such that f1 ; f2 ; : : : ; fm can be extended to maps F1 ; F2 ; : : : ; Fm : B(Y; ) → B(Z; ). Suppose now 1 ; 2 ∈ Sh(X; Y ) with d( 1 ; 2 ) ¡, then, if  ∈ A we have that d( 1∗ (); 2∗ ()) ¡. This implies the continuity of DZ∗ . Now let 1 ; 2 ∈ Sh(X; Y ) and d(( 1 )∗ (); ( 2 )∗ ()) = d( 1 ◦ ; 2 ◦ ) 5 d( 1 ; 2 ). Consequently D∗Z is also continuous. In order to end this section we will say a little about the new shape invariants obtained in Corollary 2.2. It would be interesting to give geometrical descriptions of the spaces Sh(X; Y ) for certain classes of spaces X (or Y ). In particular one can ask what Sh(X; Y ) is if X has trivial shape. It is enough to see what Sh( · ; Y ) is, where · is a one point space. The answer is given in the next proposition. Proposition 2.5 Let Y be a compactum. Then; Sh( · ; Y ) is homeomorphic (and from compactness; uniformly homeomorphic) to the space Y; of the components of Y . Proof. Let  ¿0 and consider B(Y; ). The components of B(Y; ) induce a nite partition of Y in clopen sets. Take ; ∈ Sh(X; Y ) and maps f; g : · → B(Y; ) such that [f] = S(iY; B(Y; ) ) ◦  and [g] = S(iY; B(Y; ) ) ◦ . Since the components of an open set of Q are arcwise connected, it follows that [ f] = [g] if and only if f( · ) and g( · ) are in the same component of B(Y; ). Therefore, using Corollary 2.3, Sh( · ; Y ) is compact. Now let  ∈ Sh( · ; Y ) and  ¿0. As we have just seen, the homotopy class of S(iY; B(Y; ) ) ◦  can be identi ed with a component of B(Y; ). We denote

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such a component by (). It is not dicult to see (using the compactness of T Y ) that ¿0 () is a component of Y . T De ne : Sh( · ; Y ) → Y by () = ¿0 (). For ; ∈ Sh( · ; Y ) with d(; ) = r ¿ 0, since d(; )¿r=2 it follows that (r=2) ∩ (r=2) =∅. Then ()- ( ) and is injective. On the other hand, for every component Y0 of Y we take a map h : · → Y with h( · ) ∈ Y0 . It is clear that (S(h)) = Y0 , then is a bijective function. In order to check the continuity of , consider a sequence {n }n∈N converging to ; in Sh( · ; Y ). Thus, for every  ¿ 0 there exists n0 ∈ N such that n () = () for every n = n0 . Using now the upper-semicontinuity of the partition of Y into quasicomponents, we have that { (n )}n∈N converges to () in Y . Remark. 1. Note that we have constructed a metric on Y , generating the quotient topology induced by the natural projection p : Y → Y . In fact, if we assume Y to be embedded in the Hilbert cube Q, and  is a xed metric in Q. For a pair of components Y0 ; Y00 of Y we de ne d∗ (Y0 ; Y00 ) = inf { ¿0 such that Y0 and Y00 are contained in the same component of B(Y; ) in Q}. Then, d∗ is the mentioned metric. 2. Observe that we also construct ultrametrics on compact 0-dimensional spaces, inducing the original topologies. 3. It seems to us that it would be interesting to describe, as in the last proposition, the spaces Sh(X; Y ), for bigger classes of spaces X such as polyhedra or manifolds. From Corollary 2.4 and Proposition 2.5 we have Corollary 2.6 Let X; Y be two compacta. Then; every shape morphism  : X → Y induces a map () : X → Y such that the assignment  : Sh(X; Y ) →C( X; Y ) (with the uniform convergence topology); given by  7→ (); is continuous.

3 A uniform continuity property of the shape functor As above, we consider the Hilbert cube Q with a xed metric  and suppose Y ⊂ Q to be a subcompactum. Theorem 3.1 Let Y ⊂ Q be a subcompactum. Then; for every  ¿ 0 there exists  ¿ 0 (depending only on Y ) such that for every pair of maps f; g : X → Y with max{(f(x); g(x)): x ∈ X }¡ ; we have that d(S(f); S(g))¡ in Sh(X; Y ). In particular; for any compactum X; the natural projection S : C(X; Y ) → Sh(X; Y ); is uniformly continuous. Proof. Let  ¿0. There is a compact neighbourhood of Y in Q; V , which is an ANR, such that Y ⊂ V ⊂ B(Y; ). Take ¿0 such that -near maps into V , are homotopic in V . Such a  satis es the desired conclusion.

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Theorem 3.1 seems to point out one of the main dierences between homotopy and shape, in the realm of compacta. The following example explains what we want to mean. Example Let X = {(x; y) ∈ R2 : 0¡ x 5 1; y = sen1=x} ∪ {(0; y) ∈ R2 : −15 y 5 1}. Let H (X; X ) the set of homotopy classes of maps from X into X . Since the IdX can be uniformly approximated by nulhomotopic maps we have that is not possible to give a Hausdor topology in H (X; X ) such that the natural projection H : C(X; X ) → H (X; X ) is continuous. Theorem 3.1 can be used, for example, to nd subsets of C(X; Y ) which are open and closed in the topology of the uniform convergence. Corollary 3.2 Let X; Y be compacta. Then; for every f ∈ C(X; Y ) and every  ¿ 0; the set {g ∈ C(X; Y ) such that d(S(f); S(g))¡ in Sh(X; Y )} is open and closed in C(X; Y ). Proof. It is consequence of Theorem 3.1 and Corollary 1.6(a). Let us prove now an useful proposition in order to study the topological structure of the functional space C(X; Y ). Proposition 3.3 Let X; Y be two compacta. Suppose that f; g ∈ C(X; Y ) belong to the same quasicomponent of C(X; Y ); then S(f) = S(g). Proof. Consider f; g ∈ C(X; Y ) belonging to the same quasicomponent of C(X; Y ) and suppose that the shape morphism S(f) and S(g) are dierent. Since Sh(X; Y ) is 0-dimensional, there exists a clopen set A ⊂ Sh(X; Y ) such that S(f) ∈ A and S(g) ⊂ Sh(X; Y )\A. Now the continuity of S leads us to a contradiction. Thus S(f) = S(g). Another important dierence between the homotopy and shape categories, is that, in homotopy every morphism is represented by a map while in shape it is not. That is, the projection S : C(X; Y ) → Sh(X; Y ) is not, in general, surjective. As we know, see [MS], shape morphisms can be represented by approximative maps (or equivalently by F-Cauchy sequences as in Sect. 1 of this paper). It is now natural to look for topologies on the sets of approximative maps between compacta such that the natural projection is continuous. In fact, in Sect. 1, we have found one of them. In the following proposition we will write it explicitly. Proposition 3.4 The relation ({fk }; {gk }) = limk→∞ F(fk ; gk ) de nes a pseudometric on the set of approximative maps between two compacta X and Y . Let us denote by A∗∗ (X; Y ) the pseudometric space obtained. Then; we have: (a) The projection S : A∗∗ (X; Y ) → Sh(X; Y ) is uniformly continuous. (b) The quasicomponents; components and path-components of A∗∗ (X; Y ) coincide and they are just the homotopy classes of approximative maps (that is; the shape morphisms):

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Proof. (a) Indeed, from Sect. 1, ({fk }; {gk }) = d([{fk }]; [{gk }]) where [ ◦ ] is the shape morphism induced by the approximative map ◦ . (b) It follows from (a) and the fact that the closure of a point of A∗∗ (X; Y ) is its homotopy class. We must say that the rst authors, who considered topologies in the set of approximative maps, were Laguna and Sanjurjo, see [LS1, LS2 and LS3]. In [LS1 and LS2], they gave a metric d and a pseudometric d∗ and they denoted by A(X; Y ) and A∗ (X; Y ) the corresponding spaces so obtained. Among other things, they proved that approximative maps in the same path-component of A(X; Y ) or A∗ (X; Y ), are homotopic. But converses are not true. Finally, in [LS3], the authors got a metric on the set of approximative maps, such that the shape morphisms were represented just as the path-components on such space (giving so an analogue of the classical fact that path-components of the spaces of maps are the homotopy classes). In order to connect our construction with [LS1, 2, 3] we have: Proposition 3.5 The identity function A∗ (X; Y ) → A∗∗ (X; Y ) is uniformly continuous (respect to the corresponding pseudometric). The proof of the last proposition is analogous to that of Theorem 3.1. As a consequence of Proposition 3.5 we can improve some results of [LS1, 2], for example: Corollary 3.6 If two approximative maps {fk }; {gk } : X → Y are in the same quasicomponent of A(X; Y ) or A∗ (X; Y ); then; they are homotopic. Finally, in order to obtain relationships between our results and those of [LS3], we have that the space A∗∗ (X; Y ) has the property that the pathcomponents are representations of shape morphisms as in the space A(X; Y ) there constructed. However, topological properties of A∗∗ (X; Y ) are not so good as those of A(X; Y ).

4 Some geometrical consequences of the constructed spaces In this chapter we are going to obtain some relationships between the spaces of shape morphisms and some of the main concepts in shape theory. The rst result allows us, in particular, to count the amount of shape morphisms from a compactum to, for example, an FANR. Next proposition is easy to be proved. Proposition 4.1 Let X be a compactum and Y be a calm ([Ce]) or AWNR ([Bo]) compactum. Then there is an  ¿ 0 (depending only on Y ) such that ; ∈ Sh(X; Y ) with d(; )¡  implies  = . Remark. Note that above proposition allows us to give many clopen subsets in C(X; Y ) when Y is calm.

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From the separability of Sh(X; Y ) we have Corollary 4.2 If X is a compactum and Y be a calm or AWNR compactum (in particular if Y is an FANR); Card(Sh(X; Y ) 5ℵ0 . Now we are going to characterize one of the main concepts in shape theory, in terms of the spaces of shape morphisms. Let us prove Theorem 4.3 Let Y ⊂ Q be a compactum. Then; Y is movable if and only if for every  ¿0; there is a compact neighbourhood V of Y in Q and a shape morphism  : V → Y such that d(|Y ; S(Id Y ))¡ in Sh(Y; Y ). Proof. Suppose Y to be movable. Let  ¿ 0 then, there exist a compact neighbourhood V of Y in Q and a shape morphism  : V → Y, such that S(iV; B(Y; ) ) = S(iY; B(Y; ) ) ◦ . This implies, see Proposition 1.8, that d(|Y ; S(Id Y ))¡ in Sh(Y; Y ). Conversely, let  ¿ 0, V and  be with d(|Y ; S(Id Y ))¡ in Sh(Y; Y ). This implies that S(iY; B(Y; ) ) = S(iY; B(Y; ) ) ◦ |Y . Take a map f : V → B(Y; ) such that S(f) = S(iY; B(Y; ) )◦, since f|Y ≈ Id Y in B(Y; ), there is a compact neighbourhood V 0 ⊂ V of Y such that S(f|V 0 )=S(iV 0; B(Y; ) ), then S(iV 0; B(Y; ) )= S(iV; B(Y; ) ) ◦ |V 0 and Y is movable. Remark. 1. Note that the last theorem says that movability is an approximate ANR concept in shape theory and that its role, related with shape morphisms, is the same as the Approximative Absolute Neighbourhood Retract (shortly AANRC ) in the sense of Clapp ([C1]) related with maps. Finally, as S : C(X; Y ) → Sh(X; Y ) is continuous, every AANRC is movable. 2. Notice that the neighbourhood V in Theorem 4.3 depends on . What would we obtain if we could choose V independently of ? The answer to the last question is: A compactum Y ⊂ Q is an FANR if and only if there is a compact neighbourhood V of Y in Q such that for every  ¿0 there exists a shape morphism  : V → Y with d( |Y ; S(Id Y ))¡  in Sh(Y; Y ). Indeed, suppose the existence of V with the above properties. Take 0 ¿0 such that B(Y; 0 )⊂ V . It is not dicult to show that if X is any compactum and ; ∈ Sh(X; Y ) with d(; )¡ 0 then  = . Take 0 : V → Y, then 0 |Y = Id Y and Y is an FANR. The converse is obvious. Then we have that the exact concept and the approximative one in the sense of Noguchi, [N], are the same in this context. Note that the behaviour is not the same if we use maps instead shape morphisms. Corollary 4.4 A compactum X is movable if and only if for every  ¿ 0; there is an ANR P (depending on ) and shape morphisms  : X → P; : P → X such that d( ◦ ; S(IdX ))¡ .

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Corollary 4.5 A compactum X is movable if and only if for every compactum Y; the set of neighbourhood extendable shape morphisms (i.e. shape morphisms having an extension to some neighbourhood V of X ) is dense in Sh(X; Y ). Corollary 4.6 A compactum X is an FANR if and only if X is movable and Sh(X; X ) is discrete or; equivalently; X is movable and S(Id X ) is an isolated point in Sh(X; X ). Corollary 4.7 A compactum X is movable if and only if for every compactum Y the set of all neighbourhood extendable shape morphism from Y to X is dense in Sh(Y; X ). Another concept, although it is not a shape invariant, which can be rede ned using the spaces of shape morphisms is that of internally movable space (see [Bo], for de nition). Proposition 4.8 Let X ⊂ Q be a compactum. Then; X is internally movable if and only if for every  ¿ 0 there is a compact neighbourhood V of X in Q and a map f : V → X such that d(S(f)|X ; S(Id X ))¡. Some consequences, as from Theorem 4.3 for movability, can be obtained for internal movability. In order to give the relation between movability and internal movability we need: De nition 4.9 A shape morphism between compacta  : X → Y is said to be internal if it belongs to the closure; in Sh(X; Y ); of the set of shape morphisms induced by maps. Proposition 4.10 For a compactum X the following conditions are equivalent (a) X is internally movable. (b) X is movable and; for every compactum Y; all elements of Sh(Y; X ) are internal. If Y is calm or AWNR compactum, from the discreteness of the spaces Sh(X; Y ) we have Proposition 4.11 If an internally movable compactum Y is calm or AWNR; then; for every compactum X; all elements of Sh(X; Y ) are generated by maps. Another option we have, from the construction of the metric in Sh(X; Y ), is to obtain a new metric in hyperspace 2Q , if we follow Borsuk’s ideas about the metric of continuity. We have, only, one additional trouble. In the metric of continuity, we use maps (then one has a point to point assignment) then, the property dC (A; B) = 0 if and only if A = B, holds obviously, but if one wants to give the new de nition changing, in the original Borsuk’s notion, maps by shape morphisms, the result is not a metric because the shape morphisms do not detect near subspaces of Q.

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An inadequate (for redundance) but correct rede nition of the metric of continuity, once we have a xed metric in Q, is: Given A; B ⊂ Q subcompacta, dc (A; B) = inf { ¿ 0 : dH (A; B) ¡ and there are maps f : A → B y g : B → A such that d(iB; A∪B ◦ f; iA; A∪B )¡ and d(iB; A∪B ; iA; A∪B ◦ g) ¡}, where the distance d between functions is the usual supremun metric in the corresponding spaces of maps. Now, if in this rede nition we change maps by shape morphisms we obtain a metric in 2Q . Proposition 4.12 Let X; Y be two subcompacta of Q. The assignment D(X; Y ) = inf { ¿ 0 : dH (X; Y )¡  and such that there exist shape morphisms  : X → Y : Y → X with d(S(iX; X ∪Y ) ◦ ; S(iY; X ∪Y )¡ and d(S(iY; X ∪Y ) ◦ ; S(iX; X ∪Y ) ¡} de nes a metric in the hyperspace 2Q of all nonempty subcompacta of Q. Besides; dH 5 D 5 dc . Proof. Only a proof of the triangle inequality is needed. Take X; Y; Z subcompacta of Q and a; b ¿ 0 such that D(X; Y ) ¡a and D(Y; Z)¡b. It is clear that dH (X; Z)¡ a + b. On the other hand, there are shape morphisms.  : X → Y; : Y → X and 0 : Y → Z, 0 : Z → Y , with d(S(iX; X ∪Y ) ◦ ; S(iY; X ∪Y )¡ a, d(S(iY; X ∪Y ) ◦ ; S(iX; X ∪Y )¡a; d(S(iY; Y ∪Z ) ◦ 0 ; S(iZ; Y ∪Z ) ¡b and d(S(iZ; Y ∪Z ) ◦ 0 ; S(iY; Y ∪Z )¡b. Therefore, S(iZ; B(X ∪Z; a+b) ◦ 0 ◦  = S(iB(Y ∪Z; b); B(X ∪Z; a+b) ) ◦ S(iZ; B(Y ∪Z;b) ) ◦ 0 ◦  = S(iB(Y ∪Z; b); B(X ∪Z; a+b) ) ◦ S(iY; B(Y ∪Z; b) ) ◦  = S(iB(X ∪Y; a); B(X ∪Z; a+b) ) ◦ S(iY; B(X ∪Y; a) ) ◦  = S(iX; B(X ∪Z; a+b) ): A similar argument shows S(iX; B(X ∪Z; a+b) ) ◦ ◦ 0 = S(iZ; B(X ∪Z; a+b) ). It follows that d(S(iZ; X ∪Z ) ◦ 0 ◦ ; S(iX; X ∪Z )) ¡a + b and d(S(iX; X ∪Z ) ◦ ◦ 0 ; S(iZ; X ∪Z )) ¡a + b, then D(X; Z)¡ a + b. If we consider only appropriate subsets of Sh(X; Y ), with the induced metric, we obtain another metrics in 2Q . As an example, we have: Proposition 4.13 Let X; Y two subcompacta of Q. The assignment DI (X; Y ) = max{dH (X; Y ); inf { ¿ 0 such that there exist maps f : X → Y; g : Y → X with d(S(iX; X ∪Y ) ◦ S(g); S(iY; X ∪Y )¡  and d(S(iY; X ∪Y ) ◦ S(f); S(iX; X ∪Y )¡}} de nes a metric on the hyperspace 2Q of all nonempty subcompacta of Q. Besides; dH 5D 5 DI 5 dc . Before this paper was conceived, in [LMNS] appeared a metric in 2Q , called shape metric by the authors, which seems to have good properties, some of them established there, and it saves some inconveniences of the fundamental metric of Borsuk ([B3]). In order to state the relation between the shape metric (denoted by dS ) and D just de ned we have Proposition 4.14 The homeomorphism.

identity

Id 2Q : (2Q ; D) → (2Q ; dS )

is

a

bilipschitz

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Proof. It is not hard to see that D(X; Y )¡  if dS (X; Y )¡ and that dS (X; Y )¡ if D(X; Y )¡=2. Remark. From the last proposition we assume as known the results proved in [LMSN]. It is of special interest to recall, that there, the main result states that the movable subspaces of Q are just the limits of sequences of polyhedra and, consequently, the subfamily of all movable subspaces of Q is a closed and separable subspace of 2Q . We want to point out that now, it is very easy for us to prove, in a uni ed way the following: Let (2Q ; h) where h = dc ; D or DI; then A ∈ 2Q is a limit of polyhedra in (2Q ; h) if and only if for every  ¿0 there is a compact neighbourhood and a map (in the case of dc ); a shape morphism (in the case of D) and a shape morphism generated by a map (in the case of DI ) f : V → A such that d(f|A ; iA )¡  (d is the supremun metric in C(X; Y ); d in Sh(X; Y ) or the restriction of d in the third case). As a consequence; we have (a) A is AANRc in the sense of Clapp ([C1]) if and only if A is a limit of polyhedra in (2Q ; dc ). (b) A is internally movable if and only if A is limit of polyhedra in (2Q ; DI ). (c) A is movable if and only if A is limit of polyhedra in (2Q ; D). On the other hand, it is obvious, using Theorem 3.1 for the rst implication that AANRc ⇒ internally movable ⇒ movable (as was proved by Bogatyi in [Bo]). It is also possible to control the shape dimension of a movable space, if we bound the dimension of the polyhedra, in the considered sequence. Using some Nowak’s ([No]) results and standard facts on the convergence, in the Hausdor metric, of polyhedra one can prove Proposition 4.15 Given a subcompactum X ⊂ Q the following conditions are equivalent: (a) X is movable and Sd(X ) 5n (Sd denotes the shape dimension [MS1, DS]). (b) X is limit of a sequence of polyhedra; of dimension less or equal than n; in (2Q ; D). Remark. Note that, in particular, the spaces of trivial shape are limit of nite subsets of Q, with the metric D. Next result is a key to connect, up to shape, inverse limits and limits of polyhedra in the space (2Q ; D) for movable spaces. Theorem 4.16 Let {Pn }n∈N be a sequence of polyhedra and let X be a comD pactum such that {Pn }n∈N → X . Then; there exist a subsequence {Pnk }k∈N and a compactum Y such that Sh(X ) = Sh(Y ); Y = lim Pnk ; and the bonding maps Pnk+1 → Pnk factorize; in shape; through X .

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Conversely; if a movable space X is inverse limit of a sequence of polyhedra {Pn }n∈N one can embed; up to homeomorphism the polyhedra Pn n ∈ N D and X; in the Hilbert cube in such a way that {Pn }n∈N → X . Proof. Take 1 = 1 and Pn1 such that dS (Pn1 ; X )¡1 =2. Then, there are shape morphisms 1 : Pn1 → X and 1 : X → Pn1 such that S(iX; B(Pn1; 1=2) ) ◦ 1 = S(iPn1; B(Pn1 ; 1=2) ) and S(iPn1; B(X; 1=2) ) ◦ 1 = S(iX; B(X;1=2) ). Take a map g1 : X → Pn1 inducing the shape morphism 1 and 2 ¡ 1 =2 such that g1 extends to a continuous map g˜ : B(X; 2 ) → Pn1 homotopic, inside of B(X; 1=2), to the inclusion B(X; 2 ) → B(X; 1=2). We construct inductively a subsequence, of {Pn }n∈N ; denoted again by {Pn }n∈N , with dS (X; Pn )¡n =2 and the map gn : X → Pn , generating the corresponding shape morphism n : X →Pn , extends to g˜n : B(X; n+1 ) →Pn being homotopic to the inclusion B(X; n+1 ) → B(X; n =2). Let pn : Pn+1 → Pn a map generating the shape morphism n ◦ n+1 : Pn+1 → X → Pn . De ne Y = lim(Pn ; pn ) it is not dicult to see that Sh(X ) = Sh(Y ). ←−

On the other hand, using the space X ∗ de ned in [MS2], the converse is clear. Using the hyperspace (2Q ; D) we can give an easy proof of the main result in [MR]. Proposition 4.17 The set of shape types of FANR’s is countable. Proof. Suppose that we have an uncountable family {A } ∈ of FANR’s such that Sh(A ) - Sh(A 0 ) if and only if - 0 . Assume all of them to be embedded in Q. For every ∈ there is  ¿ 0 as in Proposition 4.1. Then there are  ¿0 and an uncountable subfamily of B ⊂ such that  5 , ∈ B. From the separability we have that for every  there is a pair ( ; 0 ) such that D(A ; A 0 ) ¡. Taking ¡  we produce a contradiction. Remark. From Proposition 4.1 and the fact that solenoids are calm [Ce], analogous arguments, as used in the last proposition, allow us to check, easily, that the subset of all nonmovable subspaces of Q is not separable, with the metric D (see Theorem 3.7 in [LMNS]). In [B2], Borsuk used the concept of equally locally contractible family of ANR-subsets, in order to describe the convergence, in the metric of homotopy, introduced by him. Gromov ([GLP, Gr]) used some analogous ideas in the study of global properties of riemannian manifolds and Grove, Petersen, Wu, Ferry and other authors ([DF, Fe, GP, GPW, Pe1, Pe2, Y] : : :) exploited these ideas to obtain, in particular, niteness theorems for homotopical, topological and smooth categories in some classes of riemannian manifolds. Let us recall, see [Pe1, Pe2] for example, the following de nition: A, not necessarily continuous, function  : [0; r] → [0; ∞) is a contractibility function if (0) = 0, () → 0 as  → 0 and () =  for  ∈ [0; r]. A metric space X is called locally geometrically contractible of size  : [0; r] → [0; ∞),

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written X ∈ LGC(), if for every x ∈ X the ball B(x; ) is contractible inside B(x; ()) for  ∈ (0; r]. Note that a compact metric space X is locally contractible if and only if X ∈ LGC() for some contractibility function . The concept of LGC() is, of course, a local one in nature, which is useful in order to get niteness theorems of n-dimensional closed manifolds and, special results are obtained, in riemannian geometry, when we impose some restrictions on sectional curvature, volume and diameter. Trying to get a global concept adequate for shape theory, playing an analogous role, in some sense, we met again with movability. We say that a function  : [0; r] → [0; ∞) is a movability function if  has exactly the same properties as a contractibility function as above. De nition 4.18 Let X ⊂ Q be a closed subset and  a movability function. X is said to be movable of size , written X ∈ M (), if for every  ∈ (0; r] there is a shape morphism  : B(X; ) → X such that S(iB(X; ); B(X; ()) ) = S(iX; B(X; ()) ) ◦  . Using the equivalence of movability and uniform movability for compacta ([Sp]) we have: Proposition 4.19 A compactum X ⊂ Q is movable if and only if there is a movability function  such that X ∈ M (). Proof. It is obvious that X ∈ M () for some movability function implies that X is movable. For the converse, take () =  + inf { = : there exists a shape morphism  : B(X; ) → X such that S(iB(X; ); B(X; ) ) = S(iX; B(X; ) ) ◦  }. The rst property we want to point out is Proposition 4.20 Let A ⊂ 2Q such that there is a movability function  with A ∈ M () for every A ∈ A (we call such a family A to be an equimovable family of size ). Then Id: (A; D) → (A; dH ) is a uniformly continuous homeomorphism. Consequently (A; D) is totally bounded. Proof. We only need to show that Id: (A; dH ) → (A; D) is uniformly continuous. Take ¿0 and ¿0 such that 2()¡. For every A ∈ A there exists a shape morphism A : B(A; ) → A such that S(iB(A; ); B(A; ()) ) = S(iA; B(A; ()) ) ◦ A : Suppose that dH (A; B)¡, A; B ∈ A, then A ⊂ B(B; ) and B ⊂ B(A; ). De ne shape morphisms  : A → B, : B → A by  = B ◦ S(iA; B(B; ) ) and = A ◦ S(iB; B(A; ) ). Then, S(iA; B(B; 2()) ) ◦ = S(iA; B(B; 2()) ) ◦ A ◦ S(iB; B(A; ) ) = S(iB(A; ()); B(B; 2()) ) ◦ S(iA; B(A; ()) ) ◦ A ◦ S(iB; B(A; ) ) = S(iB(A; ()); B(B; 2()) ) ◦ S(iB(A; ); B(A; ()) ) ◦ S(iB; B(A; ) ) = S(iB; B(B; 2()) ) :

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The proof of S(iB; B(A; 2()) )◦ = S(iA; B(A; 2()) ) is similar. Therefore D(A; B) ¡2()¡. Remark. The converse of Proposition 4.20 does not hold. It is enough to consider the family of all trivial shape spaces. Next proposition states an interesting property of equimovable families of sets. Theorem 4.21 Let A ⊂ 2Q be an equimovable family. Then; the closure of A in (2Q ; D) is compact. Proof. It is enough to prove that if {Xn }n∈N ⊂ M () and {Xn }n∈N → X then dH

{Xn }n∈N → X . D

Assume that {Xn }n∈N → X . We will check that X is movable. Indeed, dH

let ¿0 and ¿0 such that 2()¡. Consider ¿¿0. There is n0 ∈ N such that dH (X; Xn )¡=2 for n0 5 n. Take Y = Xn0 and a shape morphism  : B(Y; ) → Y such that S(iB(Y; ); B(Y; ()) ) = S(iY; B(Y; ()) ) ◦  : De ne = S(iY; B(X; ) ) ◦  ◦ S(iB(X; =2); B(Y; ) ) : B(X; =2) → B(X; ). It is easy to see that S(iB(X; =2); B(X; ) ) = S(iB(X; ); B(X; ) ) ◦ , thus, X is movable. Consequently, it is obvious that {X; {Xn }n∈N } is equimovable, (for a movability function 0 = ). Now, Proposition 4.20 implies that {Xn }n∈N →X. D

Remark. If we recall Proposition 4.1, we can denote by FANR() (for each ¿0) the family of all FANR subsets X of Q such that d(; ) =  for every pair ; ∈ Sh(Y; X )  - , and every compactum Y . The following is a niteness result in shape theory. Proposition 4.22 Every totally bounded subset of FANR() contains only nitely many dierent shapes. Proof. It follows from the fact that D(A; B)¡=4 implies Sh(A) = Sh(B) for A; B ∈ FANR(). Corollary 4.23 Every equimovable subfamily of FANR() contains only nitely many dierent shapes. As we know, see [BO], a compactum X is shape dominated by a nite polyhedron P if and only if X is homeomorphic to a shape retract of the Q-manifold P × Q. One can embed P × Q in Q, then there is ¿0 such that P × Q ∈ FANR() and it is not dicult to prove that FANR() is closed under shape retractions. Maybe this context is adequate to count dierent shapes among spaces shape dominated by a nite polyhedron (equivalently an ANR), but we have not yet been able to clarify such a thing. In order to connect our results with those in [GP, GPW, Pe1, Pe2, Y] : : : we have

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Theorem 4.24 Let In be the n-dimensional cube (with the usual euclidean metric) and A ⊂ 2In be an LGC() family. Then; A × Q = {A × Q: A ∈ A} is an equimovable family of Q-manifolds in In × Q (with the maximum metric). Moreover; there is ¿0 such that A × Q ⊂ FANR(); consequently A × Q (and then A) contains only nitely many dierent shapes (equivalently homotopy types). Proof. From Lemma in [B2], page 187, there is an increasing continuous function  : (0; p] → R (p¿0) such that 0¡() 5  and for every A ∈ A there exists a retraction rA : B(A; (p)) → A satisfying the inequality kx − rA (x)k¡ for x ∈ B(A; ()). Denote A0 = A × Q ⊂ In × Q for each A ∈ A. We have retractions rA0 : B(A0 ; (p)) → A0 , B(A0 ; (p)) ⊂ In × Q ≡ Q and kx − rA0 (x)k¡ for x ∈ B(A0 ; ()). The linear homotopy tx + (1 − t)rA0 (x) allows us check that S(iB(A0 ; ()); B(A0 ; ) ) = S(iA0 ; B(A0 ; ) ) ◦ S(rA0 ), then  = −1 is a movability function for A0 . On the other hand, A0 ⊂ FANR((p)). In order to end this section let us recall that Chapman, in [Ch1, Ch3] see also [Ch2] for nite dimensional analogues, obtained some of the deepest and most beautiful geometrical results in shape theory. In particular he constructed an isomorphism between the categories S and P where S denotes the category whose objects are Z-sets compacta of Q (see [Ch3], for example, for de nition) and shape morphisms between them and P is the category whose objects are open subsets of Q with complement in Q being a Z-set and whose morphisms are weak proper homotopy classes of maps between them. Using the results in this paper and Chapman’s isomorphism one can give a metric in the set of morphism of the category P such that the isomorphism is an isometry in some sense. Another possibility is to construct such metric and obtain later that the isomorphism is an isometry. First of all, we will recall some de nitions ([Ch1], p. 182). A map f : X → Y is said to be proper if for each compactum B ⊂ Y there is a compactum A ⊂ X such that f(X \A) ∩ B = ∅ (equivalently f−1 (B) is compact). Two maps f; g : X → Y are said to be weakly properly homotopic if for each compactum B ⊂ Y there is a compactum A ⊂ X and a homotopy FB : X × I → Y connecting f and g such that FB ((X \A) × I ) ∩ B = ∅ (equivalently FB−1 (B) ⊂ A × I ). Now consider the Hilbert cube Q with a xed metric d. Let X; Y ⊂ Q be two compacta Z-sets. Let CP (Q\X; Q\Y ) be the set of all proper maps from Q\X to Q\Y . De ne a function G : CP (Q\X; Q\Y ) × CP (Q\X; Q\Y ) → R+ ∪ {0} as G(f; g) = inf {¿0: such that there is a compactum A ⊂ Q\X and a homotopy F : Q\X × I → Q\Y connecting f and g and F −1 (Q\B(Y; )) ⊂ A × I }. It is not hard to prove the following Proposition 4.25 For every pair X; Y of compact Z-subsets of Q we have that G is a pseudometric on CP (Q\X; Q\Y ). Moreover; G(f; g) 5 max{G(f; h);

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G(h; g)} for every f; g; h ∈ CP (Q\X; Q\Y ). Finally G(f; g) = 0 if and only if f and g are weakly properly homotopic. Then the relation G(; ) = G(f; g) where  = [f]Wp ; = [g]Wp de nes a ultrametric on the set of weakly properly homotopic classes of maps. Let us denote now T : P → S the Chapman’s isomorphism, using arguments as in [Ch1], we have Theorem 4.26 Let X; Y ⊂ Q be two compact Z-sets. Suppose ; to be two weakly properly homotopy classes of maps from Q\X to Q\Y . Then; G(; ) = d(T (); T ( )) where d is the metric constructed in Chapter 1 on Sh(X; Y ). From above result we can translate all metric properties obtained for shape morphisms to weakly properly homotopy classes of elements of CP (Q\X; Q\Y ). In particular we point out the next existence result. Corollary 4.27 Let X; Y ⊂ Q be two compact Z-sets and consider a sequence of proper maps {fn }n∈N : Q\X → Q\Y such that for every n ∈ N there is a compactum An ⊂ Q\X and homotopies Fn : Q\X × I → Q\Y with Fn (x; 0) = fn (x); Fn (x; 1) = fn+1 (x) and Fn−1 (Q\B(Y; 1=n)) ⊂ An × I . Then; there is a proper map f : Q\X → Q\Y such that for every ¿0 there are n0 ∈ N; a sequence of compacta {Cn }n=n0 of X and a sequence {Gn }n=n0 : Q\X × I → Q\Y of homotopies such that Gn (x; 0) = f(x); Gn (x; 1) = fn (x) and Gn−1 (Q\ B(Y; )) ⊂ Cn . Proof. It is an easy exercise in completeness. Acknowledgements. The authors would like to thank professors J.M. Bayod and R. Kieboom who helped us to obtain information about non-Archimedean metrics. We also thank professor J. Dydak who clari ed for us some questions. We are also indebted with J.M.R. Sanjurjo and F. Blasco who improved the nal version of the paper.

References [B1] Borsuk, K.: Theory of shape. Monogra e Matematyczne 59, Polish Scienti c Publishers, Warszawa (1975) [B2] Borsuk, K.: On some metrization of the hyperspace of compact set. Fund. Math. 41, 168 – 202 (1954) [B3] Borsuk, K.: On a metrization of the hyperspace of a metric space. Fund. Math. 94, 191–207 (1977) [BO] Borsuk, K., Oledzki, J.: Remark on the shape domination. Bull. Acad. Polon. Sci. 28, 67–70 (1980) [Bo] Bogatyi, S.: Approximate and fundamental retracts. Math. USSR Sbornik 22, 91–103 (1974)  [Ce] Cerin, Z.: Homotopy properties of locally compact spaces at in nity – calmness and smoothness. Paci c Journal of Math. 79, 69 –91 (1978) [Ch1] Chapman, T.A.: On some applications of in nite dimensional manifolds to the theory of shape. Fund. Math. 76, 181–193 (1972) [Ch2] Chapman, T.A.: Shape of nite-dimensional compacta. Fund. Math. 76, 261–276 (1972)

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[Ch3] Chapman, T.A.: Lectures on Hilbert cube manifolds. Regional Conferences Series in Math., Amer. Math. Soc. 28 (1976) [C1] Clapp, M.H.: On a generalization of absolute neighborhood retracts. Fund. Math. 70, 117–130 (1971) [DF] Dranisnikov, A.N., Ferry, S.C.: Cell like images of topological manifolds and limits of manifolds in Gromov–Hausdor space. Preprint [DS] Dydak, J., Segal, J.: Shape Theory: an introduction. Lecture Notes in Math. 688, Springer-Verlag, Berlin (1978) [Fe] Ferry, S.C.: Topological niteness theorems for manifolds in Gromov–Hausdor space. Preprint [GLP] Gromov, M., Lafontaine, J., Pansu, P.: Structures metriques pour les varietes riemanniennes. Cedic=Fernand Nathan (1981) [Gr] Gromov, M.: Groups of polynomials growth and expanding maps. Inst. Hautes Etude Sci. Publ. Math. 53, 53 –78 (1981) [GP] Grove, K., Petersen, P.: Bounding homotopy types by geometry. Annals of Math. 128, 195 –200 (1988) [GPW] Grove, K., Petersen, P., Wu, J.: Geometric niteness theorems via controlled topology. Inventiones Mathematicae 99, 205 –213 (1990). Correction in Invent. Math. 104, 221–222 (1991) [LMSN] Laguna, V.F., Moron, M.A., To Nhu, N., Sanjurjo, J.M.R.: Movability and limits of polyhedra. Fund. Math. 143, 191–201 (1993) [LS1] Laguna, V.F., Sanjurjo, J.M.R.: Spaces of approximative maps. Math. Japonica 31 (4) 623 – 633 (1986) [LS2] Laguna, V.F., Sanjurjo, J.M.R.: Spaces of approximative maps. II, Pub. Mat. U. A. B. 30, 115 –126 (1986) [LS3] Laguna, V.F., Sanjurjo, J.M.R.: Shape morphisms and spaces of approximative maps. Fund. Math. 133, 225 –235 (1989) [MR] Moron, M.A., Ruiz del Portal, F.R.: Counting shape and homotopy types among FANR’s: an elementary approach. Manuscripta Math. 79, 411– 414 (1993) [MS1] Mardesic, S., Segal, J.: Shape theory. North-Holland (1982) [MS2] Mardesic, S., Segal, J.: Shapes of compacta and ANR systems. Fund. Math. 72, 41–59 (1971) [N] Noguchi, H.: A generalization of absolute neighbourhood retracts. Kodai Math. Sem. Rep. 1, 20 –22 (1953) [No] Nowak, S.: Some properties of fundamental dimension. Fund. Math. 85, 211–227 (1974) [Pe1] Petersen, P.: A niteness theorem for metric spaces. Journal of Di. Geometry 31, 387– 395 (1990) [Pe2] Petersen, P.: Gromov Hausdor convergence of metric spaces. Proc. of Symposia in Pure Math. 54, Part 3, 489 – 504 (1993) [Sc] Schikhof, W.H.: Ultrametric calculus. An introduction to p-adic analysis, Cambridge Univ. Press (1984) [Sp] Spiez, S.: Movability and uniform movability. Bull. Acad. Polon. Sci. 22, 43 – 45 (1974) [V] Van Rooij, A.: Non-Archimedean functional analysis. Marcel Dekker Inc. New York (1978) [Ya] Yamaguchi, T.: Homotopy type niteness theorems for certain precompact families of Riemannian manifolds. Proc. Amer. Math. Soc. 102, 660 – 666 (1988)