shape as a cantor completion process

Math. Z. 225, 67–86 (1997)

Shape as a Cantor completion process?

M.A. Mor�on1, F.R. Ruiz del Portal1;??

1 U.D. Matem�aticas. E.T.S.I. de Montes, Universidad Polit�ecnica 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 morphismsin 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 andresults, and as an adequate framework to simplify old de�nitions, results andproofs. Moreover, sometimes, it allows us to obtain some connections betweenapparently 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 com-pletion process, to obtain the real numbers from the rationals (or the com-pletion of a metric space). But our starting-point is not a metric, even nota pseudometric one. The �rst advantage of this new point of view is thatwe construct complete metrics in spaces of shape morphisms, which, in ad-dition, are non-Archimedean (or ultrametrics). Following with the analogyof the construction of the irrationals from the rationals, and as a survey ofthe topological properties of the spaces herein constructed, we obtain thatthese spaces are homeomorphic to closed subspaces of the irrationals. Later,we associate, to shape morphisms, in a functorial way, uniformly continu-ous maps, obtaining that the uniform homeomorphism types of our spaces areshape invariants. Consequently, we describe many new shape invariants. Wepoint out later that some of them are new reformulations of other alreadyknown.

? The authors have been supported by DGICYT, PB93-0454-C02-02.?? Current address: Departamento de Geometr��a y Topolog��a, Facultad de CC. Matem�aticas,Universidad Complutense de Madrid, Madrid, 28040, Spain(e-mail: rrportal @sungt1.mat.ucm.es)

68 M.A. Mor�on, F.R. Ruiz del Portal

We also show that two close maps (with the supremum metric) generateclose shape morphisms. This closeness is strongly uniform in the sense thatit depends only on the image space. At this point we show another di�er-ence on the behaviour of the homotopy and shape functors, because there isno possibility to de�ne a Hausdor� topology, in the sets of homotopy classesof maps between metric compacta, such that the natural projection, from thespaces of maps with the supremum metric is continuous (for every pair ofcompacta).The remaining part of the paper is dedicated to point out how to use the

spaces of shape morphisms, here constructed, to obtain new characterizationsof some known notions (movability, internal movability, FANR: : :). Anotheradvantage we obtain is the possibility to construct, in a natural way, newmetrics in hyperspaces following Borsuk’s ideas in the de�nition of the metricof continuity. We also introduce the concept of equimovable family of setswhich seems to be useful to obtain �niteness theorems in shape theory andallows 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 con-struction which, in particular, allows us to obtain, up a gap of set the-ory, a new characterization of N-compactness and a new construction ofN-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 clopennormal subgroups.

We think that quite a number of new problems can be implicitly derivedfrom this paper. We also note that the construction here made could be trans-ferred 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 beenused 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 Hilbertcube 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 homo-topy relation).We have

(a) F(f; g)=0

Cantor completion process 69

(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. theyare 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 sequenceif for every �¿0 there is a k0 ∈N such that F(fk; fk′)¡� for every k; k ′= k0.

De�nition 1.2 Two Cauchy sequences {fk}k∈N; {gk}k∈N⊂C(X;Q) are saidto be F-related; written {fk}k∈NF{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; {f′k}k∈N; {gk}k∈N and {g′k}k∈N be Cauchy sequences

such that {fk}k∈NF{f′k}k∈N and {gk}k∈NF{g′k}k∈N then limk→∞ F(fk; gk)=limk→∞ F(f′k ; g

′k):

Proof. (a) is clear, (b) is a consequence of the fact that |F(fk; gk)−F(fk′ ; gk′)|5F(fk; fk′) + F(gk ; gk′) and (c) is obtained by applying (b) to the pairf1; f′1 ; f2; f

′2 ; : : : and g1; g

′1; g2; g

′2; : : : 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 itis 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 theshape morphisms (see [MS1]), then the �nal step in this construction is givenin 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∈NCauchy sequences in the classes �; � respectively. Then (Sh(X; Y ); d) isa complete non-Archimedean metric space.


Proof. It is clear that d(�; �)= 0 if and only if �= � and that d(�; �)=d(�; �).From property (c) of F it follows that d(�; �)5max{d(�; ); d( ; �)} for ev-ery �; �; ∈ 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 metricspace.

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 cov-

ering dimension).

Corollary 1.7 A sequence {�k}k∈N⊂ Sh(X; Y ) converges if and only ifd(�k ; �k+1)→0 as k→∞.

Corollaries 1.6 and 1.7 follow from properties of ultrametrics. We usedmainly [Sc and V]. We are aware that deeper results, about the metric d; canbe obtained if we read carefully the last references, but it is not our intentionnow.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 viewof 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 em-bedded 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 shapemorphism induced by the map h; and B(Y; �) is the generalized open ball ofcenter 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′ ∈R such that r¡r′¡� there exists k0 ∈N with fk ' gk in B(Y; r′)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)5max{F(fk0 ; fk0+j); F(fk0 ; gk0 ); F(gk0 ; gk0+j)} therefore

F(fk; gk)5max{�; �=2} for every k=k0 and consequently d(�; �)¡�.

The next result is a representation of the topological space Sh(X; Y ) whichstrengthens the relationships between the construction, given in this paper, with


the usual, to obtain the reals from the rationals. Moreover, it points out themain topological properties of the space Sh(X; Y ).

Theorem 1.9 For every pair X; Y of compacta; the space Sh(X; Y ) is homeo-morphic 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 mapsf; g :X→Y such that S(f)= � and S(g)= �. Let �0¿0 be such that thereis a retraction r :B(Y; �0)→Y . Suppose that d(�; �)¡�0; then there existsa homotopy H :X × I→B(Y; �0) connecting f and g. Consequently the homo-topy 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. Take a neighbourhood base {Vn}n∈N of compact ANR’sof 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 theclaim it su�ces to prove that � maps Sh(X; Y ) homeomorphically ontoa 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. In-

deed: 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∈Nthere exists kn ∈N such that pn(�(�k))=pn(�(�)) for k=kn. From the factthat {Vn}n∈N is a neighbourhood base of Y it follows that base {�k}k∈N con-verges 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 ontoImrn for every n∈N. Now, it is straightforward to see that �(Sh(X; Y ))=⋂n∈N Imrn; 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, whosespaces are Sh(K; Xn). Since a more general result will be given in relatedforthcoming papers, and in order to avoid repetitions we have not used suchlanguage.


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. Sup-pose 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) (IdX )∗= IdSh(X;Z) and (IdX )∗= IdSh(Z;X ).

Proof. Only the proof of (a) is needed. Let us denote by d the metrics in allspaces 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))5d(�1; �2).Now let �¿0 and f′ :X →B(Y; �) be a map such that [f′] = S(iY;B(Y; �)) ◦f.

Take �¿0 such that f′ 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 f′ and on the chosen extension F . In orderto avoid it we can take � as the supremum of positive real numbers r such thatthe homotopy class generating S(iY;B(Y; �)) ◦f can be extended to a homotopyclass 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 anyarbitrary compactum; then Sh(Y; Z) is uniformly homeomorphic to a uniformretract of Sh(X; Z). Moreover; Sh(Z; Y ) is uniformly homeomorphic to a uni-form retract of Sh(Z; X ).(b) Suppose now that Sh(X )=Sh(Y ); then Sh(X; Z) and Sh(Y; Z) are uni-

formly homeomorphic. Moreover; Sh(Z; X ) and Sh(Z; Y ) are also uniformlyhomeomorphic (uniformly homeomorphic means that there is a uniformly con-tinuous 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 . Onthe other hand, as an easy consequence of Corollary 2.2, we have that for everycompactum 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 Xto an FANR Y there are at most countable many di�erent shape morphisms.Since this result will be improved later, we do not state it as a proposition.


Another consequence of Theorem 2.1 is the following characterization ofcompactness in the spaces of shape morphisms.

Corollary 2.3 A⊂ Sh(X; Y ) is relatively compact if and only if for every �¿0the set {S(iY;B(Y; �)) ◦ �: �∈A} is �nite.Proof. Let us suppose that A is relatively compact and take �¿0. ConsiderV to be a compact neighbourhood of Y in Q such that V ⊂B(Y; �). It followsthat S(iY;V )∗( �A) is compact in Sh(X; V ) and then �nite (see claim of 1.9).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 iscompact.

Corollary 2.4 Let X; Y; Z be compacta.Consider the functions D∗Z : Sh(X; Y )→C(Sh(Y; Z); Sh(X; Z)) and DZ∗ :

Sh(X; Y )→C(Sh(Z; X ); Sh(Z; Y )) de�ned by D∗Z(�)= �∗ and DZ∗ (�)= �∗. ThenD∗Z and DZ∗ are continuous (the functional spaces are endowed with thecompact-open topology).

Proof. First we prove that D∗Z is continuous. Let A⊂ Sh(Y; Z) be a com-pact and let �¿0. Then {S(iZ;B(Z; �)) ◦ �: �∈A} is �nite. There exist mapsf1; 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 extendedto maps F1; F2; : : : ; Fm :B(Y; �)→B(Z; �). Suppose now �1; �2 ∈ Sh(X; Y ) withd(�1; �2)¡�, then, if �∈A we have that d(�∗1 (�); �∗2 (�))¡�. This implies thecontinuity of D∗Z .

Now let �1; �2 ∈ Sh(X; Y ) and d((�1)∗(�); (�2)∗(�))=d(�1 ◦ �; �2 ◦ �)5d(�1; �2). Consequently DZ∗ is also continuous.

In order to end this section we will say a little about the new shape in-variants obtained in Corollary 2.2. It would be interesting to give geometricaldescriptions 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 enoughto see what Sh( · ; Y ) is, where · is a one point space. The answer is givenin 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 thecomponents 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 com-ponents of an open set of Q are arcwise connected, it follows that [f] = [g] ifand 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


such a component by �(�). It is not di�cult to see (using the compactness ofY ) that

⋂�¿0 �(�) is a component of Y .

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 isclear that �(S(h))=Y0, then � is a bijective function.

In order to check the continuity of �, consider a sequence {�n}n∈N con-verging 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 par-tition 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 thequotient topology induced by the natural projection p : Y → Y . In fact, if weassume Y to be embedded in the Hilbert cube Q, and � is a �xed metric in Q.For a pair of components Y0; Y ′0 of Y we de�ne d∗(Y0; Y ′0)= inf{�¿0 suchthat Y0 and Y ′0 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 poly-hedra 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→ �(�); iscontinuous.

3 A uniform continuity property of the shape functor

As above, we consider the Hilbert cube Q with a �xed metric � and supposeY ⊂Q to be a subcompactum.

Theorem 3.1 Let Y ⊂Q be a subcompactum. Then; for every �¿0 there ex-ists �¿0 (depending only on Y ) such that for every pair of maps f; g :X → Ywith max{�(f(x); g(x)): x∈X }¡�; we have that d(S(f); S(g))¡� inSh(X; Y ). In particular; for any compactum X; the natural projectionS :C(X; Y )→ Sh(X; Y ); is uniformly continuous.Proof. Let �¿0. There is a compact neighbourhood of Y in Q; V , which isan 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.


Theorem 3.1 seems to point out one of the main di�erences between homo-topy and shape, in the realm of compacta. The following example explains whatwe want to mean.

Example Let X = {(x; y)∈R2: 0¡x5 1; y=sen1=x}∪ {(0; y)∈R2: −15y5 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 havethat is not possible to give a Hausdor� topology in H (X; X ) such that thenatural 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 ) whichare 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 openand 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 topologicalstructure of the functional space C(X; Y ).

Proposition 3.3 Let X; Y be two compacta. Suppose that f; g ∈ C(X; Y ) be-long 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 ofC(X; Y ) and suppose that the shape morphism S(f) and S(g) are di�erent.Since Sh(X; Y ) is 0-dimensional, there exists a clopen set A⊂ Sh(X; Y ) suchthat S(f)∈A and S(g)⊂ Sh(X; Y )\A. Now the continuity of S leads us to acontradiction. Thus S(f)= S(g).

Another important di�erence between the homotopy and shape categories,is that, in homotopy every morphism is represented by a map while in shapeit is not. That is, the projection S :C(X; Y )→ Sh(X; Y ) is not, in general, sur-jective. As we know, see [MS], shape morphisms can be represented by ap-proximative maps (or equivalently by F-Cauchy sequences as in Sect. 1 ofthis paper). It is now natural to look for topologies on the sets of approxi-mative maps between compacta such that the natural projection is continuous.In fact, in Sect. 1, we have found one of them. In the following propositionwe will write it explicitly.

Proposition 3.4 The relation �({fk}; {gk})= limk→∞ F(fk; gk) de�nes a pseu-dometric 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 (thatis; the shape morphisms):


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 ofapproximative 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 denotedby A(X; Y ) and A∗(X; Y ) the corresponding spaces so obtained. Among otherthings, they proved that approximative maps in the same path-component ofA(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 thatthe shape morphisms were represented just as the path-components on suchspace (giving so an analogue of the classical fact that path-components of thespaces of maps are the homotopy classes). In order to connect our constructionwith [LS1, 2, 3] we have:

Proposition 3.5 The identity function A∗(X; Y )→A∗∗(X; Y ) is uniformly con-tinuous (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 samequasicomponent of A(X; Y ) or A∗(X; Y ); then; they are homotopic.

Finally, in order to obtain relationships between our results and thoseof [LS3], we have that the space A∗∗(X; Y ) has the property that the path-components are representations of shape morphisms as in the space A(X; Y )there constructed. However, topological properties of A∗∗(X; Y ) are not so goodas 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 spacesof shape morphisms and some of the main concepts in shape theory.The �rst result allows us, in particular, to count the amount of shape mor-

phisms 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 subsetsin C(X; Y ) when Y is calm.


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 iffor every �¿0; there is a compact neighbourhood V of Y in Q and a shapemorphism � :V → Y such that d(�|Y ; S(IdY ))¡� in Sh(Y; Y ).

Proof. Suppose Y to be movable. Let �¿0 then, there exist a compact neigh-bourhood 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(IdY ))¡� inSh(Y; Y ).Conversely, let �¿0, V and � be with d(�|Y ; S(IdY ))¡� in Sh(Y; Y ). This

implies that S(iY;B(Y; �))= S(iY;B(Y; �)) ◦ �|Y . Take a map f :V →B(Y; �) such thatS(f)= S(iY;B(Y; �))◦�, since f|Y ≈ IdY in B(Y; �), there is a compact neighbour-hood V ′⊂V of Y such that S(f|V ′)=S(iV ′; B(Y; �)), then S(iV ′; B(Y; �))=S(iV;B(Y; �))◦ �|V ′ and Y is movable.

Remark. 1. Note that the last theorem says that movability is an approxi-mate ANR concept in shape theory and that its role, related with shape mor-phisms, is the same as the Approximative Absolute Neighbourhood Retract(shortly AANRC) in the sense of Clapp ([C1]) related with maps. Finally, asS :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 thelast question is:

A compactum Y ⊂Q is an FANR if and only if there is a compact neigh-bourhood V of Y in Q such that for every �¿0 there exists a shape morphism�� :V → Y with d(��|Y ; S(IdY ))¡� 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 di�cult to show that if X is any compactum and�; �∈ Sh(X; Y ) with d(�; �)¡�0 then �= �. Take ��0 :V → Y, then ��0 |Y = IdYand 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 notthe same if we use maps instead shape morphisms.

Corollary 4.4 A compactum X is movable if and only if for every �¿0; thereis an ANR P (depending on �) and shape morphisms � :X →P; � :P→X suchthat d(� ◦ �; S(IdX ))¡�.


Corollary 4.5 A compactum X is movable if and only if for every compactumY; the set of neighbourhood extendable shape morphisms (i.e. shape morphismshaving 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 andSh(X; X ) is discrete or; equivalently; X is movable and S(IdX ) is an isolatedpoint in Sh(X; X ).

Corollary 4.7 A compactum X is movable if and only if for every compactumY the set of all neighbourhood extendable shape morphism from Y to X isdense in Sh(Y; X ).

Another concept, although it is not a shape invariant, which can be rede�nedusing 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 ifand only if for every �¿0 there is a compact neighbourhood V of X in Qand a map f :V →X such that d(S(f)|X ; S(IdX ))¡�.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 beinternal if it belongs to the closure; in Sh(X; Y ); of the set of shape morphismsinduced 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 spacesSh(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 aboutthe metric of continuity. We have, only, one additional trouble. In the metricof continuity, we use maps (then one has a point to point assignment) then, theproperty dC(A; B) = 0 if and only if A=B, holds obviously, but if one wantsto give the new de�nition changing, in the original Borsuk’s notion, maps byshape morphisms, the result is not a metric because the shape morphisms donot detect near subspaces of Q.


An inadequate (for redundance) but correct rede�nition of the metric ofcontinuity, 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 supremunmetric in the corresponding spaces of maps. Now, if in this rede�nition wechange maps by shape morphisms we obtain a metric in 2Q.

Proposition 4.12 Let X; Y be two subcompacta of Q. The assignmentD(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 sub-compacta of Q. Besides; dH5D5dc.

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, thereare shape morphisms. � :X → Y; � : Y →X and �′ : Y → Z , �′ : Z→ Y , withd(S(iX;X∪Y ) ◦ �; S(iY;X∪Y )¡a, d(S(iY;X∪Y ) ◦ �; S(iX;X∪Y )¡a; d(S(iY; Y∪Z) ◦�′; S(iZ; Y∪Z)¡b and d(S(iZ; Y∪Z) ◦ �′; S(iY; Y∪Z)¡b.

Therefore,

S(iZ;B(X∪Z; a+b) ◦ �′ ◦ � = S(iB(Y∪Z; b); B(X∪Z; a+b)) ◦ S(iZ;B(Y∪Z;b)) ◦ �′ ◦ �= 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)) ◦ � ◦ �′= S(iZ;B(X∪Z; a+b)).It follows that d(S(iZ;X∪Z) ◦ �′ ◦ �; S(iX;X∪Z))¡a+ b and d(S(iX;X∪Z) ◦ � ◦ �′;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 met-ric, 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 withd(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; dH5D5DI5dc.

Before this paper was conceived, in [LMNS] appeared a metric in 2Q, calledshape metric by the authors, which seems to have good properties, some ofthem established there, and it saves some inconveniences of the fundamentalmetric 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 identity Id2Q : (2Q; D)→ (2Q; dS) is a bilipschitzhomeomorphism.


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 thatthe 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 andseparable subspace of 2Q. We want to point out that now, it is very easy forus 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 neighbourhoodand a map (in the case of dc); a shape morphism (in the case of D) anda shape morphism generated by a map (in the case of DI) f :V →A suchthat d(f|A; iA)¡� (d is the supremun metric in C(X; Y ); d in Sh(X; Y ) or therestriction 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 limitof 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 Theorem3.1 for the �rst implicationthat 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. Usingsome Nowak’s ([No]) results and standard facts on the convergence, in theHausdor� metric, of polyhedra one can prove

Proposition 4.15 Given a subcompactum X ⊂Q the following conditions areequivalent:

(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 �nitesubsets of Q, with the metric D.

Next result is a key to connect, up to shape, inverse limits and limits ofpolyhedra 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 com-pactum such that {Pn}n∈N D→ X . Then; there exist a subsequence {Pnk}k∈Nand a compactum Y such that Sh(X )=Sh(Y ); Y = lim←− Pnk ; and the bonding

maps Pnk+1→Pnk factorize; in shape; through X .


Conversely; if a movable space X is inverse limit of a sequence of poly-hedra {Pn}n∈N one can embed; up to homeomorphism the polyhedra Pn n∈Nand X; in the Hilbert cube in such a way that {Pn}n∈N D→ X .

Proof. Take �1 = 1 and Pn1 such that dS(Pn1 ; X )¡�1=2. Then, there are shapemorphisms �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 →Pn1inducing the shape morphism �1 and �2¡�1=2 such that g1 extends to acontinuous map g̃ :B(X; �2)→Pn1 homotopic, inside of B(X; 1=2), to the inclu-sion 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 g̃n :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 di�cult 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 resultin [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 suchthat Sh(A )-Sh(A ′) if and only if - ′. Assume all of them to be embeddedin 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. Fromthe separability we have that for every � there is a pair (�; �′) such thatD(A�; A�′)¡�. Taking �¡� we produce a contradiction.

Remark. From Proposition 4.1 and the fact that solenoids are calm [Ce], anal-ogous 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 themetric D (see Theorem 3.7 in [LMNS]).

In [B2], Borsuk used the concept of equally locally contractible family ofANR-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 otherauthors ([DF, Fe, GP, GPW, Pe1, Pe2, Y] : : :) exploited these ideas to ob-tain, in particular, �niteness theorems for homotopical, topological and smoothcategories 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 metricspace X is called locally geometrically contractible of size � : [0; r]→ [0;∞),


written X ∈LGC(�), if for every x∈X the ball B(x; �) is contractible insideB(x; �(�)) for �∈ (0; r].

Note that a compact metric space X is locally contractible if and onlyif 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 �nitenesstheorems of n-dimensional closed manifolds and, special results are obtained, inriemannian 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. Wesay that a function � : [0; r]→ [0;∞) is a movability function if � has exactlythe 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 isa movability function � such that X ∈M (�).Proof. It is obvious that X ∈M (�) for some movability function implies thatX is movable. For the converse, take �(�)= �+inf{�= �: there exists a shapemorphism �� :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 � withA∈M (�) for every A∈A (we call such a family A to be an equimovablefamily of size �). Then Id: (A; D)→ (A; dH ) is a uniformly continuoushomeomorphism. Consequently (A; D) is totally bounded.

Proof. We only need to show that Id: (A; dH )→ (A; D) is uniformly contin-uous. Take �¿0 and �¿0 such that 2�(�)¡�. For every A∈A there existsa 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�(�))) :


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 toconsider 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 ofA in (2Q; D) is compact.

Proof. It is enough to prove that if {Xn}n∈N⊂M (�) and {Xn}n∈N→dHX then

{Xn}n∈N→DX .

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

let �¿0 and �¿0 such that 2�(�)¡�. Consider �¿�¿0. There is n0 ∈Nsuch that dH (X; Xn)¡�=2 for n05 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 mov-

able.Consequently, it is obvious that {X; {Xn}n∈N} is equimovable, (for a

movability function �′= �). Now, Proposition 4.20 implies that {Xn}n∈N→DX .

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 everypair �; �∈ 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 di�erent shapes.

Proof. It follows from the fact that D(A; B)¡�=4 implies Sh(A)=Sh(B) forA; B∈ FANR(�).Corollary 4.23 Every equimovable subfamily of FANR(�) contains only �ni-tely many di�erent shapes.

As we know, see [BO], a compactum X is shape dominated by a �nitepolyhedron P if and only if X is homeomorphic to a shape retract of theQ-manifold P×Q. One can embed P×Q in Q, then there is �¿0 such thatP×Q∈ FANR(�) and it is not di�cult to prove that FANR(�) is closed undershape retractions. Maybe this context is adequate to count di�erent shapesamong 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


Theorem 4.24 Let In be the n-dimensional cube (with the usual euclideanmetric) and A⊂ 2In be an LGC(�) family. Then; A×Q= {A×Q: A∈A} isan 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 di�erent shapes (equivalently ho-motopy types).

Proof. From Lemma in [B2], page 187, there is an increasing continuous func-tion � : (0; p]→R (p¿0) such that 0¡�(�)5 � and for every A∈A there ex-ists a retraction rA :B(A; �(p))→A satisfying the inequality ‖x− rA(x)‖¡� forx∈B(A; �(�)). Denote A′=A×Q⊂ In×Q for each A∈A. We have retrac-tions rA′ :B(A′; �(p))→A′, B(A′; �(p))⊂ In×Q≡Q and ‖x − rA′(x)‖¡� forx∈B(A′; �(�)). The linear homotopy tx + (1 − t)rA′(x) allows us check thatS(iB(A′ ; �(�)); B(A′ ; �))= S(iA′ ; B(A′ ; �)) ◦ S(rA′), then �= �−1 is a movability functionfor A′. On the other hand, A′⊂ FANR(�(p)).

In order to end this section let us recall that Chapman, in [Ch1, Ch3] seealso [Ch2] for �nite dimensional analogues, obtained some of the deepest andmost beautiful geometrical results in shape theory. In particular he constructedan isomorphism between the categories S and P where S denotes the cat-egory whose objects are Z-sets compacta of Q (see [Ch3], for example, forde�nition) and shape morphisms between them and P is the category whoseobjects are open subsets of Q with complement in Q being a Z-set and whosemorphisms are weak proper homotopy classes of maps between them. Usingthe results in this paper and Chapman’s isomorphism one can give a metricin the set of morphism of the category P such that the isomorphism is anisometry in some sense. Another possibility is to construct such metric andobtain later that the isomorphism is an isometry. First of all, we will recallsome 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 com-pact).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 homotopyFB :X × I→ Y connecting f and g such that FB((X \A)× I)∩B= ∅ (equiva-lently F−1B (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 functionG :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 homo-topy F :Q\X × I→Q\Y connecting f and g and F−1(Q\B(Y; �))⊂A× I}. Itis not hard to prove the following

Proposition 4.25 For every pair X; Y of compact Z-subsets of Q we have thatG is a pseudometric on CP(Q\X;Q\Y ). Moreover; G(f; g)5max{G(f; h);


G(h; g)} for every f; g; h∈CP(Q\X;Q\Y ). Finally G(f; g)= 0 if and only iff 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 properlyhomotopic classes of maps.

Let us denote now T :P →S the Chapman’s isomorphism, using argu-ments as in [Ch1], we have

Theorem 4.26 Let X; Y ⊂Q be two compact Z-sets. Suppose �; � to betwo 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 onSh(X; Y ).

From above result we can translate all metric properties obtained for shapemorphisms 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 se-quence of proper maps {fn}n∈N :Q\X →Q\Y such that for every n∈Nthere is a compactum An⊂Q\X and homotopies Fn :Q\X × I→Q\Y withFn(x; 0)=fn(x); Fn(x; 1)=fn+1(x) and F−1n (Q\B(Y; 1=n))⊂An× I . Then; thereis 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 G−1n (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. Kieboomwho helped us to obtain information about non-Archimedean metrics. We also thank professorJ. Dydak who clari�ed for us some questions.

We are also indebted with J.M.R. Sanjurjo and F. Blasco who improved the �nal versionof the paper.

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shape as a cantor completion process

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