Intrinsic Definition of Strong Shape for Compact Metric Spaces

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1 Volume 39, 0 Pages Intrinsic Definition of Strong Shape for Compact Metric Spaces by Nikita Shekutkovski Electronically published on April 4, 0 Topology Proceedings Web: Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA topolog@auburn.edu ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume 39 (0) Pages 7-39 E-Published on April 4, 0 INTRINSIC DEFINITION OF STRONG SHAPE FOR COMPACT METRIC SPACES NIKITA SHEKUTKOVSKI Abstract. The best known approach to the intrinsic definition of shape of compact metric spaces is by use of near continuous functions f : X Y the corresponding notion of homotopy. A new definition of strong shape will be presented based on higher homotopies of this type.. Introduction The best known approach to the intrinsic definition of shape of (metric) spaces is by use of near continuous functions f : X Y. The idea of ε- continuity (continuity up to ε > 0) leads to continuity up to some covering V of Y, i.e., V-continuity the corresponding V-homotopy. The first approach of this type for compact metric spaces was given in []. We also refer the reader to [7]. In [3], the approach is generalized to paracompact spaces. In this paper, we present, for the first time, an intrinsic approach to strong shape theory construct the strong shape category of compact metric spaces. The approach combines continuity up to a covering the corresponding homotopies of second order, used in [4], in constructing the strong shape category of metric compacta. The homotopies of second order are used in actually the same way, for construction of a category of inverse systems in the paper [8]. In [4], it is shown that the approach using continuous homotopies of second order leads to stard strong shape theory in the case of compact metric spaces. It is shown that this approach is equivalent with the first definition of the strong shape category SSh given by J. Brendan Quigley [6] in 973. The author believes that the strong shape category constructed in this paper the category SSh are isomorphic. 00 Mathematics Subject Classification. Primary 54C56, 55P55; Secondary 54C08. Key words phrases. Continuity up to a covering, intrinsic definition of shape, proximate sequence, strong shape. c 0 Topology Proceedings. 7

3 8 N. SHEKUTKOVSKI. Continuity Up to a Covering Let X Y be compact metric spaces. By a covering, we underst an open covering. Definition.. Suppose V is a finite covering of Y. A function f : X Y is V-continuous at point x X if there exists a neighborhood U x of x, V V such that f(u x ) V. A function f : X Y is V-continuous if it is V-continuous at every point x X. In this case, the family of all U x form a covering of X, since X is compact, there exists a finite subcovering. By this, f : X Y is V-continuous if there exists a finite covering U of X, such that for any x X, there exists a neighborhood U U of x, V V such that f(u) V. We denote: there exists U, such that f (U) V. If f : X Y is V-continuous, then f : X Y is W-continuous for any W such that V W. If V is a finite covering of Y V V, the open set st(v ) (star of V ) is the union of all W V such that W V. We form a new covering of Y, st(v) = st(v ) V V}. Theorem.. Suppose V is a finite covering of Y, X = X X, X i closed, i =,, f i : X i Y, V-continuous functions, i =,, such that f (x) = f (x) for all x X X. We define a function f : X Y by f(x) = f i (x), for x X i, i=,. Then () if x IntX or x IntX, then f : X Y is V-continuous at x; () if x FrX x FrX, then f : X Y is st(v)-continuous at x. Proof. () If x Int X or x Int X, then there exists an open subset U of X, x U, such that f(u) V for some V V; i.e., f : X Y is V-continuous at x. () If x FrX x FrX, since X i are closed, i =,, then x X x X. There exist open subsets U i of X i, x U i, V i V, such that f(u i ) V i, for i =,. Then V V, since f (U ) f (U ). There are open subsets U i of X, U i = U i Xi, i=,. We put U = U U. Then U is a neighborhood of x in X f(u) = f (U X ) f (U X ) f (U X ) f (U X ) = f (U ) f (U ) V V. It follows that f : X Y is st(v)-continuous at x.

4 INTRINSIC DEFINITION OF STRONG SHAPE 9 Definition.3. The functions f, g : X Y are V-homotopic if there exists a function F : I X Y such that () F : I X Y is st(v)-continuous ; () there exists a neighborhood N = [0, ε) ( ε, ], ϵ > 0, of I = 0, } such that F N X is V-continuous ; (3) F (0, x) = f(x) F (, x) = g(x). We denote the relation of homotopy by f V g. Proposition.4. The relation of homotopy f V g is an equivalence relation. Proof. If f V g by a V-homotopy F : I X Y g V h by a V- homotopy G : I X Y, we define a homotopy H : I X Y by F (s, x), 0 s H(s, x) = G(s, x), s. Since F is V-continuous on ( δ, ] X for some δ, > δ > 0, G is V-continuous on [, + ε) X for some ε, > ε > 0, by Theorem., it follows that H is st(v)-continuous on ( δ, + ε) X; i.e., H : I X Y is st(v)-continuous. We have checked only one of several conditions. Remark.5. In Definition.3, we cannot replace conditions () () by the expected condition: F : I X Y is V-continuous. A simple example in [] shows that in this case the usual concatenation of two homotopies which are V-continuous is not always V-continuous. At the end of this section, an intrinsic definition of shape is presented which somewhat differs from the definition in []. However, it is shown that the definition of a shape morphism in this paper coincides with the definition in [], consequently, coincides with the stard definition. The main notion for the intrinsic definition of shape for compact metric spaces is the notion of proximate sequence from X to Y. A sequence of finite coverings, V V... of a compact metric space such that for any covering V, there exists n, such that V V n we call a cofinal sequence of finite coverings. Definition.6. A sequence (f n ) of functions f n : X Y is a proximate sequence from X to Y if there exists a cofinal sequence of finite coverings of Y, V V..., for all indices m n, f n f m are V n - homotopic. In this case, we say that (f n ) is a proximate sequence over (V n ).

5 30 N. SHEKUTKOVSKI If (f n ) (f n) are proximate sequences from X to Y, then there exists a cofinal sequence of finite coverings V V... such that (f n ) (f n) are proximate sequences over (V n ). Two proximate sequences (f n ) (f n) : X Y are homotopic if for some cofinal sequence of finite coverings V V..., (f n ) (f n) are proximate sequences over (V n ), for all integers, f n f n are (V n )-homotopic. We say that (f n ) (f n) are homotopic over (V n ). By Remark.5 Proposition.4, it follows that the relation of homotopy of two proximate sequences (f n ) (f n) is an equivalence relation. We denote the equivalence class by [(f n )]. In [], the set of V-homotopy classes of V-continuous functions f : X Y is denoted by [(X, Y )] V. The homotopy class of f is denoted by [f] V. If V V f V g, then f V g. So the map p VV : [X, Y ] V [X, Y ] V, defined by p VV ([f] V ) = [f] V, is well defined, ([X, Y ] V, p VV, V finite covering) is an inverse system in the category of sets functions. The inverse limit of this inverse system is denoted by V lim [X, Y ] V. In [], a bijection is established between V lim [X, Y ] V the set of all shape morphisms from X to Y. Theorem.7. There is a bijection between the set lim V [X, Y ] V the set of homotopy classes [(f n )] of proximate sequences (f n ) : X Y. Proof. Suppose (f n ) : X Y is a proximate sequence over V V.... Then the inverse limit V lim [X, Y ] V is isomorphic to the inverse limit lim n [X, Y ] Vn of the inverse sequence ([X, Y ] Vn, p Vn V m, n N), where for n < m, p Vn V m ([f] Vm ) = [f] Vn. With the homotopy class [(f n )] of the proximate sequence (f n ) : X Y, we associate the element ([f n ] Vn ) = ([f ] V, [f ] V,...) of lim n [X, Y ] Vn. First, this is an element of n lim [X, Y ] Vn since for n < m, from f n Vn f m, it follows p Vn V m ([f m ] Vm ) = [f n ] Vn. Second, if (f n ) (f n) are homotopic over (Vn), then for all integers f n Vn f n, it follows ([f n ] Vn ) = ([f n] Vn ).

6 INTRINSIC DEFINITION OF STRONG SHAPE 3 Third, for a given element ([g n ] Vn ) = ([g ] V, [g ] V,...) of lim n [X, Y ] Vn, we can find a proximate sequence (g n ) : X Y such that ([g n ] Vn ) is associated with the homotopy class [(g n )]. This proves the theorem. Let (f n ) : X Y be a proximate sequence over (V n ), let (g k ) : Y Z be a proximate sequence over (W k ). For a covering W k of Z, there exists a covering V nk of Y such that g(v nk ) W k. Then the composition of these two proximate sequences is the proximate sequence ( h k ) = (g k f nk ) : X Z. Compact metric spaces homotopy classes of proximate sequences [(f n )] form the shape category of metric compacta. Actually, the verification that a category is obtained is given in the next section. 3. Intrinsic Definition of Strong Shape The sequence of pairs (f n, f n,n+ ) of functions f n : X Y f n,n+ : I X Y is a strong proximate sequence from X to Y if there exists a cofinal sequence of finite coverings, V V... of Y, such that for each natural number n, f n : X Y is a (V n )-continuous function f n,n+ : I X Y is a homotopy connecting (V n )-continuous functions f n : X Y f n+ : X Y. We say that (f n, f n,n+ ) is a strong proximate sequence over (V n ). If (f n, f n,n+ ) (f n, f n,n+) are strong proximate sequences from X to Y, then there exists a cofinal sequence of finite coverings (V n ) such that (f n, f n,n+ ) (f n, f n,n+) are strong proximate sequences over (V n ). Two strong proximate sequences (f n, f n,n+ ) (f n, f n,n+) : X Y are homotopic if there exists a strong proximate sequence (F n, F n,n+ ) : I X Y over (V n ) such that () F n : I X Y is a homotopy between V n -continuous maps f n f n; () F n,n+ : I I X Y is a st (V n )-continuous function, there exists a neighborhood N of I such that F n,n+ N X is st(v n )- continuous, F n,n+ (t, 0, x) = f n,n+ (t, x), F n,n+ (t,, x) = f n,n+(t, x). We denote the homotopy relation by (f n, f n,n+ ) (f n, f n,n+). Theorem The relation of homotopy of strong proximate sequences is an equivalence relation. Proof. We will prove the transitivity of the relation. If (f n, f n,n+ ) (f n, f n,n+) by a homotopy (F n, F n,n+ ) : I X Y over (V n ) (f n, f n,n+) (f n, f n,n+) by a homotopy (G n, G n,n+ ) : I X Y

7 3 N. SHEKUTKOVSKI over (V n ), we define a homotopy (H n, H n,n+ ) : I X Y, defining H n : I X Y by H n (s, x) = Fn (s, x), 0 s G n (s, x), s H n,n+ : I I X Y by H n,n+ (t, s, x) = Fn,n+ (t, s, x), 0 s G n,n+ (t, s, x), s. Since F n,n+ is st(v)-continuous on I ( δ, ] X, for some δ, > δ > 0, G n,n+ is st(v)-continuous on I [, + ε) X, for some ε, > ε > 0, by Theorem., it follows that H n,n+ is st (V)- continuous on I ( δ, + ε) X; i.e., H n,n+ : I I X Y is st (V)-continuous. We denote, by [(f n, f n,n+ )], the homotopy class of a strong proximate sequence (f n, f n,n+ ). Suppose (f n, f n,n+ ) : X Y is a strong proximate sequence. If n n are two indices, we define f n,n : I X Y by for n < n, k = n n, f n,n (t, x) = f n (x), f n,n (t, x) = f n,n+ f n+,n+ f n,n (t, x) f n,n+ (kt, x), 0 t k f = n+,n+ (kt, x), k t k... f n,n (kt k +, x), k k t. If (f n, f n,n+ ) is a strong proximate sequence over (V n ), then the function f n,n is a V n -homotopy connecting functions f n f n. Definition A strong proximate subsequence (g k, g k,k+ ) of the strong proximate sequence (f n, f n,n+ ) is defined by a strictly increasing sequence of integers n < n <... by maps g k = f nk g k,k+ : I X Y, g k,k+ = f nk,n k + f nk +,n k +... f nk+,n k+. In order to prove the next theorem, we need the following functions defined on -simplexes. In the plane with orthogonal coordinates (t, s), let e 0, e, e be the vertices of a simplex =[e 0, e, e ], let (b 0, b, b ) be barycentric coordinates in the simplex such that e 0 =(, 0, 0), e =(0,, 0), e =(0, 0, ). The map L : I, defined by L(t, s) = b, is continuous L(e 0 ) = 0, L(e ) = 0, L(e ) =. The restriction of the function L : I to the -simplex [e 0, e ] is a linear map.

8 INTRINSIC DEFINITION OF STRONG SHAPE 33 The same is true for the -simplex [e, e ], while on [e 0, e ], L equals 0. We define a function Φ n,n : X Y by Φ n,n (t, s, x) = f n,n (L(t, s), x). Then Φ n,n : X Y is st(v n )-continuous Φ n,n (e 0, x) = f n (x), Φ n,n (e, x) = f n (x), Φ n,n (e, x) = f n (x). Moreover, Φ n,n : X Y is V n -continuous at (e 0, x), (e, x), (e, x) for any x X. For example, since f n,n : I X Y is V n -continuous at (0, x), there exist [0, ε) I, U a neighborhood of x in X, V V n such that f n,n ([0, ε) U) V. Then L ([0, ε)) is a neighborhood of e 0, Φ n,n (L [0, ε) U) = f n,n (LL [0, ε) U) f n,n ([0, ε) U) V, i.e., is V n -continuous at (e 0, x). Also, we define a function Φ : X Y by n,n Then Φ n,n Φ (t, s, x) = f n,n n,n (L(t, s), x). : X Y is st(v n )-continuous Φ n,n (e 0, x) = f n (x), Φ (e, x) = f n,n n (x), Φ (e, x) = f n,n n (x). Finally, if is the half square, = (t, s) t s} I I, n n n are three indices n < n, we put v = ( n n n n, n n n n ) we define a function f nn n : X Y by Φ f nn n (t, s, x) = nn (t, s, x), (t, s) [(, 0), v, (0, 0)] (t, s, x), (t, s) [v, (, 0), (, )]. Φ n n Theorem If (f n, f n,n+ ) is a strong proximate sequence over (V n ) (g k, g k,k+ ) is a subsequence, then (g k, g k,k+ ) is also a strong proximate sequence over (V n ). The strong proximate sequences (f n, f n,n+ ) (g k, g k,k+ ) are homotopic. Proof. We have to define a homotopy (H k, H k,k+ ) between strong proximate sequences (f k, f k,k+ ) (g k, g k,k+ ). First, we define a homotopy H k : I X Y by H k (t, x) = f k,nk (t, x). Now, we have to define H k,k+ : I I X Y. Case : If n k = k n k+ = k +, then (f k, f k,k+ ) = (g k, g k,k+ ). Case : n k = k n k+ > k +. We put p = n k+ k, T = (t, s) 0 t + ( ) s} I I, p define H k,k+ : I I X Y by p fk,k+ ( ( s)p+s t, x), (t, s, x) T X H k,k+ (t, s, x) = Φ k+,nk+ (t, s, x), (t, s) [(, p ), (, 0), (, )].

9 34 N. SHEKUTKOVSKI The function is well defined along the line t = + ( p ) s. The two points of this line which belong to (I I) are (, 0) ( p, ). The function Φ k+,nk+ is V k -continuous at (, 0, x) ( p,, x). We have to p prove that the function defined by the expression f k,k+ ( ( s)p+s t, x) is also V k -continuous at these points. Then, by Theorem., H k,k+ : I I X Y will be st(v k )-continuous at these points, condition () for a homotopy of strong proximate sequences will be satisfied. If we p put h(t, s) = ( s)p+s t, h : T I, then h(, 0) = h( p, ) = p f k,k+ (h(t, s), x) = f k,k+ ( t, x). ( s)p + s Since f k,k+ : I X Y is V k -continuous at (, 0, x), there exist ( ε, ] I, a neighborhood U of x in X, V V k such that f k,k+ (( ε, ] U) V. Then h (( ε, ]) is a neighborhood of (, 0), f k,k+ (h(h ( ε, ]) U) f k,k+ (( ε, ] U) V ; i.e., the function defined by the expression f k,k+ ( ( s)p+s t, x) is V k- continuous at (, 0, x). Case 3: n k > k n k+ > k+. In order to define H k,k+ : I I X Y, we decompose I I = P, where are the simplexes, =[(, 0), (0, n k k ), (0, 0)] =[(0, ), (, n k k ), (, )] Π is the parallelogram defined by the vertices (0, n k k ), (0, ), (, 0), (, n k k ). We put Φ k,k+ (t, s, x), (t, s, x) X t H k,k+ (t, s, x) = f k,nk ( n k k + s, x), (t, s, x) Π X Φ k+,nk+ (t, s, x), (t, s, x) X. Proposition If f : X Y is W-continuous, then id f : K X K Y is K W-continuous, where K is compact K W = K W W W}. Theorem Let G : I Y Z be a st(w)-continuous function let there exist a neighborhood N of I, such that G N Y is W-continuous. Then there exists a finite covering V of Y, such that for each V-continuous function f : X Y, G(id f) : I X Z is st(w)-continuous, G(id f) N X is W-continuous. Proof. Choose a point y Y. For any s I, there exist an interval Js y (a neighborhood of s in I ) a neighborhood Vy s of y such that G(Js y Vy s ) W y s for some W s y st(w), 0 < s <, G(J y 0 V 0 y ) W 0 y, G(J y V y ) W y for some W 0 y, W y W. p

10 INTRINSIC DEFINITION OF STRONG SHAPE 35 Then Js y s I} is an open covering of the interval I = [0, ]. There is a finite subcovering Js y, Js y,..., Js y n which is minimal. We can choose the indices in such a way that a i+ < b i. If we put J y s = [0, b ), J y s = (a, b ),..., J y s n = (a n, ] V y = V s y then G((a i, b i ) V y ) W si y W s n y. V s y V sn y, G([0, b ) V y ) Wy s, G((a n, ] V y ) There is a finite subcovering V =V z z Z} of the covering V y y Y }. Then Js z i V z i =,..., n} V z V is a finite covering of I Y such that G(Js z i V z ) is contained in some member of st(w), 0 < s i <, G(J0 z V z ) G(J z V z ) are contained in some member of W. Now, if f : X Y is a V-continuous function, then the function H : I X Z, defined by H(t, x) = G(t, f(x)), is st(w)-continuous, G(id f) N X is W-continuous. 3.. Composition of Strong Proximate Sequences. The composition of strong proximate sequences (f n, f n,n+ ) : X Y (g k, g k,k+ ) : Y Z is a strong proximate sequence (h k, h k,k+ ) : X Z defined in the following way: If (g k, g k,k+ ) : Y Z is a strong proximate sequence over W k, then g k,k+ : I Y Z is a homotopy connecting W k -continuous functions g k g k+. () As in Theorem 3.0.5, by putting G = g k,k+, we can construct a cofinal sequence of finite coverings V k of Y, such that for each V k -continuous function f : X Y, G(id f) : I X Z is st(w k )-continuous, G(id f) N X is W k -continuous. We will choose a sequence of integers n < n <.... There exists an integer n such that for n n, the function f n,n is a homotopy connecting W -continuous functions f n f n. There exists an integer n, such that for n n, the function f n,n is a homotopy connecting W -continuous functions f n f n. We proceed in this away construct the sequence of integers n < n <.... From (), it follows that the function g k,k+ ( f nk+ ) : I X Z is a homotopy connecting W k -continuous functions g k f nk+ g k+ f nk+.

11 36 N. SHEKUTKOVSKI From () Proposition 3.0.4, it follows that the function g k f nk,n k+ : I X Z is a homotopy connecting W k -continuous functions g k f nk g k f nk+. By the two previous statements Theorem., it follows that the sequence of pairs of functions (h k, h k,k+ ), defined as follows, is a strong proximate sequence. We put h k = g k f nk h k,k+ (t, x) = gk f nk,n k+ (t, x), 0 t g k,k+ (t, f nk+ (x)), t. Here, f nk,n k+ = f nk,n k +... f nk+,n k+. The function is well defined since g k f nk,n k+ (, x) = g kf nk+ (x) = g k,k+ (, f n k+ (x)) h k,k+ (0, x) = g k f nk,n k+ (0, x) = g k f nk (0, x) = h k (x) h k,k+ (, x) = g k,k+ (, f nk+ (x)) = g k+ f nk+ (x) = h k+ (x). We have to prove that the composition does not depend on the choice of subsequences. Suppose that there is another choice of a subsequence n < n <.... The corresponding strong proximate sequence (h k, h k,k+ ) : X Z is defined by h k = g k f n k h k,k+(t, x) = gk f n k,n k+ (t, x), 0 t g k,k+ (t, f n k+ (x)), t. We have to prove that (h k, h k,k+ ) : X Z (h k, h k,k+ ) : X Z are homotopic. We can construct a sequence n < n <... such that n i, n i < n i for every i, we define a strong proximate sequence (h k, h k,k+ ) : X Z, putting h k = g k f n k h gk f n k k,k+(t, x) = (t, x), 0 t,n k+ g k,k+ (t, f n (x)), k+ t. To prove that (h k, h k,k+ ) : X Z (h k, h k,k+ ) : X Z are homotopic, it is enough to prove that (h k, h k,k+ ) : X Z (h k, h k,k+ ) : X Z are homotopic.

12 INTRINSIC DEFINITION OF STRONG SHAPE 37 We define a homotopy (H k, H k,k+ ) : I X Z by H k (t, x) = g k f nk n (t, x). k This homotopy connects h k = g k f nk h k = g kf n k. We define H k,k+ : I I X Z by H k,k+ (t, s, x) = g k f nk n k n k+ (s, t, x), 0 t s g k f nk n k+ n (t, s, x), s k+ t g k,k+ (t, f nk+ n (s, x)), k+ t. Then (H k, H k,k+ ) : X I Z is a homotopy connecting strong proximate sequences (h k, h k,k+ ) : X Z (h k, h k,k+ ) : X Z. 3.. Composition of Homotopy Classes of Strong Proximate Sequences. In subsection 3., we showed that the definition of the composition (g n, g n,n+ ) (f n, f n,n+ ) : X Z of strong proximate sequences (f n, f n,n+ ) : X Y (g n, g n,n+ ) : Y Z is well defined. Now we define composition of homotopy classes in the stard way by [(g n, g n,n+ )] [(f n, f n,n+ )] = [(g n, g n,n+ ) (f n, f n,n+ )]. Remark 3... By taking a subsequence of the original strong proximate sequence (f n, f n,n+ ), we can suppose that the composition (h n, h n,n+ ) : X Z of (f n, f n,n+ ) : X Y (g n, g n,n+ ) : Y Z is given by h n = g n f n h n,n+ (t, x) = gn f n,n+ (t, x), 0 t g n,n+ (t, f n+ (x)), t. We denote (h n, h n,n+ ) = ((gf) n, (gf) n,n+ ). Theorem 3... Suppose that (f n, f n,n+ ), (f n, f n,n+) : X Y (g n, g n,n+ ), (g n, g n,n+) : Y Z are strong proximate sequences such that (f n, f n,n+ ) (f n, f n,n+) (g n, g n,n+ ) (g n, g n,n+). Then ((gf) n, (gf) n,n+ ) ((g f ) n, (g f ) n,n+ ). Proof. If (f n, f n,n+ ) (f n, f n,n+) (g n, g n,n+ ) (g n, g n,n+) by homotopies (F n, F n,n+ ) (G n, G n,n+ ), respectively, then ((gf) n, (gf) n,n+ ) ((gf ) n, (gf ) n,n+ ) by a homotopy (H n, H n,n+ ), defined by

13 38 N. SHEKUTKOVSKI H n (s, x) = g n F n (s, x) gn F H n,n+ (t, s, x) = n,n+ (t, s, x), 0 t g n,n+ (t, F n+ (s, x)), t. Also, ((gf ) n, (gf ) n,n+ ) ((g f ) n, (g f ) n,n+ ) by a homotopy (K n, K n,n+ ), defined by K n (s, x) = G n (s, f n(x)) Gn (s, f K n,n+ (t, s, x) = n,n+(t, s, x)), 0 t G n,n+ (t, f n+(s, x)), t. Since is an equivalence relation, it follows that((gf) n, (gf) n,n+ ) ((g f ) n, (g f ) n,n+ ). Theorem If (f n, f n,n+ ) : X Y, (g n, g n,n+ ) : Y Z, (h n, h n,n+ ) : Z W are strong proximate sequences, then ((h n, h n,n+ ) (g n, g n,n+ )) (f n, f n,n+ ) (h n, h n,n+ ) ((g n, g n,n+ ) (f n, f n,n+ )). Proof. These two maps are connected by a homotopy (H n, H n,n+ ), defined by H n (s, x) = h n g n f n (x) H n,n+ (t, s, x) = h n g n f n,n+ ( 4t +s, x), 0 t s+ 4 s+ h n g n,n+ (4t s, f n+ (x)), 4 t s+ h n,n+ ( 4t s s, g n+ f n+ (x)), s+ t. The identity map X X is defined by the strong proximate sequence (id X, p) : X X, where p : I X X is defined by p(t, x) = x. Theorem Let (f n, f n,n+ ) : X Y be a strong proximate sequence. Then (id Y, p) (f n, f n,n+ ) (f n, f n,n+ ) (f n, f n,n+ ) (id X, p) (f n, f n,n+ ). Proof. A homotopy (H n, H n,n+ ) connecting strong proximate sequences (id Y, p) (f n, f n,n+ ) (f n, f n,n+ ) is defined by H n,n+ (t, s, x) = H n (s, x) = f n (x) fn,n+ ( t s+ s+, x), 0 t s+ f n+ (x), t.

14 INTRINSIC DEFINITION OF STRONG SHAPE 39 The homotopy (K n, K n,n+ ) connecting strong proximate sequences (f n, f n,n+ ) (id X, p) (f n, f n,n+ ) is defined by K n,n+ (t, s, x) = K n (s, x) = f n (x) fn p(t, x), 0 t s f n,n+ ( t +s s +s, x), t. By theorems 3.., 3..3, 3..4, it follows that compact metric spaces homotopy classes of strong proximate sequences form a category which we call the strong shape category of compact metric spaces. References [] Yuji Akaike Katsuro Sakai, Describing the proper n-shape category by using non-continuous functions, Glas. Mat. Ser. III 33(53) (998), no., [] James E. Felt, ε-continuity shape, Proc. Amer. Math. Soc. 46 (974), [3] R. W. Kieboom, An intrinsic characterization of the shape of paracompacta by means of non-continuous single-valued maps, Bull. Belg. Math. Soc. Simon Stevin (994), no. 5, [4] Ju. T. Lisica S. Mardešić, Coherent prohomotopy strong shape of metric compacta, Glas. Mat. Ser. III 0(40) (985), no., [5] Sibe Mardešić, Strong Shape Homology. Springer Monographs in Mathematics. Berlin: Springer-Verlag, 000. [6] J. Brendan Quigley, An exact sequence from the nth to the (n )-st fundamental group, Fund. Math. 77 (973), no. 3, [7] José M. R. Sanjurjo, A non-continuous description of the shape category of compacta, Quart. J. Math. Oxford Ser. () 40 (989), no. 59, [8] Nikita Shekutkovski, From inverse to coherent systems, Topology Appl. 3 (00), no. 3, Institute of Mathematics; Faculty of Natural Sciences Mathematics; Sts. Cyril Methodius University; 000 Skopje, Republic of Macedonia address: nikita@pmf.ukim.mk