Topological Coarse Shape Homotopy Groups
Fateme Ghanei, Hanieh Mirebrahimi, Behrooz Mashayekhy, Tayyebe Nasri
aa r X i v : . [ m a t h . A T ] A p r Topological Coarse Shape Homotopy Groups
Fateme Ghanei a , Hanieh Mirebrahimi a, ∗ , Behrooz Mashayekhy a , Tayyebe Nasri b a Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures,Ferdowsi University of Mashhad,P.O.Box 1159-91775, Mashhad, Iran. b Department of Pure Mathematics, Faculty of Basic Sciences, University of Bojnord,Bojnord, Iran.
Abstract
Cuchillo-Ibanez et al. introduced a topology on the sets of shape morphisms betweenarbitrary topological spaces in 1999. In this paper, applying a similar idea, we in-troduce a topology on the set of coarse shape morphisms Sh ∗ ( X, Y ), for arbitrarytopological spaces X and Y . In particular, we can consider a topology on the coarseshape homotopy group of a topological space ( X, x ), Sh ∗ (( S k , ∗ ) , ( X, x )) = ˇ π ∗ k ( X, x ),which makes it a Hausdorff topological group. Moreover, we study some proper-ties of these topological coarse shape homotopoy groups such as second countability,movability and in particullar, we prove that ˇ π ∗ top k preserves finite product of com-pact Hausdorff spaces. Also, we show that for a pointed topological space ( X, x ),ˇ π topk ( X, x ) can be embedded in ˇ π ∗ top k ( X, x ). Keywords:
Topological coarse shape homotopy group, Coarse shape group, Shapegroup, Topological group, Inverse limit.
1. Introduction and Motivation
Suppose that (
X, x ) is a pointed topological space. We know that π k ( X, x ) hasa quotient topology induced by the natural map q : Ω k ( X, x ) → π k ( X, x ), whereΩ k ( X, x ) is the k th loop space of ( X, x ) with the compact-open topology. With this ∗ Corresponding author
Email addresses: [email protected] (Fateme Ghanei), [email protected] (Hanieh Mirebrahimi), [email protected] (Behrooz Mashayekhy), [email protected] (Tayyebe Nasri)
Preprint submitted to Topology and its Applications September 24, 2018 opology, π k ( X, x ) is a quasitopological group, denoted by π qtopk ( X, x ) and for somespaces it becomes a topological group (see [4, 5, 6, 15]).Calcut and McCarthy [7] proved that for a path connected and locally path con-nected space X , π qtop ( X ) is a discrete topological group if and only if X is semilocally1-connected (see also [5]). Pakdaman et al. [24] showed that for a locally ( n − X , π qtopn ( X, x ) is discrete if and only if X is semilocally n-connectedat x (see also [15]). Fabel [12, 13] and Brazas [5] presented some spaces for which theirquasitopological homotopy groups are not topological groups. Moreover, despite Fa-bel’s result [12] that says the quasitopological fundamental group of the Hawaiianearring is not a topological group, Ghane et al. [16] proved that the topological n thhomotopy group of an n -Hawaiian like space is a prodiscrete metrizable topologicalgroup, for all n ≥ X , Y , Sh ( X, Y ). Moszy´nska [21] showed thatfor a compact Hausdorff space (
X, x ), the k th shape group ˇ π k ( X, x ), k ∈ N , is iso-morphic to the set Sh (( S k , ∗ ) , ( X, x )) and Bilan [2] mentioned that the result can beextended for all topological spaces. The authors [22], considering the latter topologyon the set of shape morphisms between pointed spaces, obtained a topology on theshape homotopy groups of arbitrary spaces, denoted by ˇ π topk ( X, x ) and showed thatwith this topology, the k th shape group ˇ π topk ( X, x ) is a Hausdorff topological group,for all k ∈ N . Moreover, they obtained some topological properties of these groupsunder some conditions such as movability, N -compactness and compactness. In par-ticular, they proved that ˇ π topk commutes with finite product of compact Hausdorffspaces. Also, they presented two spaces X and Y with the same shape homotopygroups such that their topological shape homotopy groups are not isomorphic.The aim of this paper is to introduce a topology on the coarse shape homotopygroups ˇ π ∗ k ( X, x ) and to provide some topological properties of these groups. First,similar to the techniques in [9], we introduce a topology on the set of coarse shapemorphisms Sh ∗ ( X, Y ), for arbitrary topological spaces X and Y . Several propertiesof this topology such continuity of the map Ω : Sh ∗ ( X, Y ) × Sh ∗ ( Y, Z ) −→ Sh ∗ ( X, Z )given by the composition Ω( F ∗ , G ∗ ) = G ∗ ◦ F ∗ and the equality Sh ∗ ( X, Y ) =lim ← Sh ∗ ( X, Y µ ), for an HPol-expansion q : Y → ( Y µ , q µµ ′ , M ) of Y , are proved whichare usefull to hereinafter results. Moreover, we show that this topology can also beinduced from an ultrametric similar to the process in [8].By the above topology, we can consider a topology on the coarse shape homo-topy group ˇ π ∗ top k ( X, x ) = Sh ∗ (( S k , ∗ ) , ( X, x )) which makes it a Hausdorff topologicalgroup, for all k ∈ N and any pointed topological space ( X, x ). It is known thatif X and Y are compact Hausdorff spaces, then X × Y is a product in the coarse2hape category [23, Theorem 2.2]. In this case, we show that the k th topologicalcoarse shape group commutes with finite product, for all k ∈ N . Also, we prove thatmovability of ˇ π ∗ top k ( X, x ) can be concluded from the movability of (
X, x ), for topo-logical space (
X, x ) with some conditions. As previously mentioned, ˇ π k ( X, x ) withthe topology defined by Cuchillo-Ibanez et al. [9] on the set of shape morphisms,is a topological group. We show that this topology also coinsides with the topologyinduced by ˇ π ∗ top k ( X, x ) on the subspace ˇ π k ( X, x ).
2. Preliminaries
Recall from [1] some of the main notions concerning the coarse shape category andpro ∗ -category. Let T be a category and let X = ( X λ , p λλ ′ , Λ) and Y = ( Y µ , q µµ ′ , M )be two inverse systems in the category T . An S ∗ -morphism of inverse systems,( f, f nµ ) : X → Y , consists of an index function f : M → Λ and of a set of T -morphisms f nµ : X f ( µ ) → Y µ , n ∈ N , µ ∈ M , such that for every related pair µ ≤ µ ′ in M , there exist a λ ∈ Λ, λ ≥ f ( µ ) , f ( µ ′ ), and an n ∈ N so that for every n ′ ≥ n , q µµ ′ f n ′ µ ′ p f ( µ ′ ) λ = f n ′ µ p f ( µ ) λ . If M = Λ and f = 1 Λ , then (1 λ , f nλ ) is said to be a level S ∗ -morphism . The compo-sition of S ∗ -morphisms ( f, f nµ ) : X → Y and ( g, g nν ) : Y → Z = ( Z ν , r νν ′ , N ) is anS ∗ -morphism ( h, h nν ) = ( g, g nν )( f, f nµ ) : X → Z , where h = f g and h nν = g nν f ng ( ν ) , forall n ∈ N . The identity S ∗ -morphism on X is an S ∗ -morphism (1 Λ , nX λ ) : X → X ,where 1 Λ is the identity function and 1 nX λ = 1 X λ in T , for all n ∈ N and every λ ∈ Λ.An S ∗ -morphism ( f, f nµ ) : X → Y is said to be equivalent to an S ∗ -morphism( f ′ , f ′ nµ ) : X → Y , denoted by ( f, f nµ ) ∼ ( f ′ , f ′ nµ ), provided every µ ∈ M admits a λ ∈ Λ and n ∈ N such that λ ≥ f ( µ ) , f ′ ( µ ) and for every n ′ ≥ n , f n ′ µ p f ( µ ) λ = f ′ n ′ µ p f ′ ( µ ) λ . The relation ∼ is an equivalence relation among S ∗ -morphisms of inverse systemsin T . The category pro ∗ - T has as objects all inverse systems X in T and as morphismsall equivalence classes f ∗ = [( f, f nµ )] of S ∗ -morphisms ( f, f nµ ). The composition inpro ∗ - T is well defined by putting g ∗ f ∗ = h ∗ = [( h, h nν )] , where ( h, h nν ) = ( g, g nν )( f, f nµ ) = ( f g, g nν f ng ( ν ) ). For every inverse system X in T , theidentity morphism in pro ∗ - T is ∗ X = [(1 Λ , nX Λ )].3n particular, if ( X ) and ( Y ) are two rudimentary inverse systems in HTop, thenevery set of mappings f n : X → Y , n ∈ N , induces a map f ∗ : ( X ) → ( Y ) inpro ∗ -HTop.A functor J = J T : pro − T → pro ∗ − T is defined as follows: For every inversesystem X in T , J ( X ) = X and if f ∈ pro − T ( X , Y ) is represented by ( f, f µ ), then J ( f ) = f ∗ = [( f, f nµ )] ∈ pro ∗ − T ( X , Y ) is represented by the S ∗ -morphism ( f, f nµ ),where f nµ = f µ for all µ ∈ M and n ∈ N . Since the functor J is faithful, we mayconsider the category pro- T as a subcategory of pro ∗ - T .Let P be a subcategory of T . A P - expansion of an object X in T is a morphism p : X → X in pro- T , where X belongs to pro- P characterised by the following twoproperties:(E1) For every object P of P and every map h : X → P in T , there is a λ ∈ Λ anda map f : X λ → P in P such that f p λ = h ;(E2) If f , f : X λ → P in P satisfy f p λ = f p λ , then there exists a λ ′ ≥ λ such that f p λλ ′ = f p λλ ′ .The subcategory P is said to be pro-reflective ( dense ) subcategory of T providedthat every object X in T admits a P -expansion p : X → X .Let P be a pro-reflective subcategory of T . Let p : X → X and p ′ : X → X ′ betwo P -expansions of the same object X in T , and let q : Y → Y and q ′ : Y → Y ′ betwo P -expansions of the same object Y in T . Then there exist two natural (unique)isomorphisms i : X → X ′ and j : Y → Y ′ in pro- P with respect to p , p ′ and q , q ′ ,respectively. Consequently J ( i ) : X → X ′ and J ( j ) : Y → Y ′ are isomorphismsin pro ∗ - P . A morphism f ∗ : X → Y is said to be pro ∗ - P equivalent to a morphism f ′∗ : X ′ → Y ′ , denoted by f ∗ ∼ f ′∗ , provided that the following diagram in pro ∗ - P commutes: X J ( i ) −−−→ X ′ y f ∗ f ′∗ y Y J ( j ) −−−→ Y ′ . (1)This is an equivalence relation on the appropriate subclass of Mor(pro ∗ - P ). Now,the coarse shape category Sh ∗ ( T , P ) for the pair ( T , P ) is defined as follows: The objectsof Sh ∗ ( T , P ) are all objects of T . A morphism F ∗ : X → Y is the pro ∗ - P equivalenceclass < f ∗ > of a mapping f ∗ : X → Y in pro ∗ - P . The composition of F ∗ = < f ∗ > : X → Y and G ∗ = < g ∗ > : Y → Z is defined by the representatives, i.e., G ∗ F ∗ = < g ∗ f ∗ > : X → Z . The identity coarse shape morphism on an object X ,1 ∗ X : X → X , is the pro ∗ - P equivalence class < X ∗ > of the identity morphism X ∗ in pro ∗ - P . 4he faithful functor J = J ( T , P ) : Sh ( T , P ) → Sh ∗ ( T , P ) is defined by keeping objectsfixed and whose morphisms are induced by the inclusion functor J = J T : pro −P → pro ∗ − P . Remark 2.1.
Let p : X → X and q : Y → Y be P -expansions of X and Y respectively. For every morphism f : X → Y in T , there is a unique morphism f : X → Y in pro- P such that the following diagram commutes in pro- P : X ←−−− p X y f f y Y ←−−− q Y. (2) If we take other P -expansions p ′ : X → X ′ and q ′ : Y → Y ′ , we obtain anothermorphism f ′ : X ′ → Y ′ in pro- P such that f ′ p ′ = q ′ f and so we have f ∼ f ′ andhence J ( f ) ∼ J ( f ′ ) in pro ∗ - P . Therefore, every morphism f ∈ T ( X, Y ) yields anpro ∗ - P equivalence class < J ( f ) > , i.e., a coarse shape morphism F ∗ : X → Y ,denoted by S ∗ ( f ) . If we put S ∗ ( X ) = X for every object X of T , then we obtain afunctor S ∗ : T → Sh ∗ , which is called the coarse shape functor. Since the homotopy category of polyhedra HPol is pro-reflective (dense) in the ho-motopy category HTop [19, Theorem 1.4.2], the coarse shape category Sh ∗ ( HT op,HP ol ) =Sh ∗ is well defined.
3. A topology on the set of coarse shape morphisms
Similar to the method of [9], we can define a topology on the set of coarse shapemorphisms. Let X and Y be topological spaces. Assume X = ( X λ , p λλ ′ , Λ) is aninverse system in pro-HPol and p : X → X is an HPol-expansion of X . For every λ ∈ Λ and F ∗ ∈ Sh ∗ ( Y, X ) put V F ∗ λ = { G ∗ ∈ Sh ∗ ( Y, X ) | p λ ◦ F ∗ = p λ ◦ G ∗ } . First,we prove the following results. Proposition 3.1.
The family { V F ∗ λ | F ∗ ∈ Sh ∗ ( Y, X ) and λ ∈ Λ } is a basis fora topology T p on Sh ∗ ( Y, X ) . Moreover, if p ′ : X → X ′ = ( X ν , p νν ′ , N ) is anotherHPol-expansion of X , then the identity map ( Sh ∗ ( Y, X ) , T p ) −→ ( Sh ∗ ( Y, X ) , T p ′ ) isa homeomorphism which shows that this topology depends only on X and Y .Proof. We know that F ∗ ∈ V F ∗ λ for every λ ∈ Λ and every F ∗ ∈ Sh ∗ ( Y, X ). Suppose F ∗ , G ∗ ∈ Sh ∗ ( Y, X ) and λ , λ ∈ Λ and H ∗ ∈ V F ∗ λ ∩ V G ∗ λ . Since H ∗ ∈ V F ∗ λ , then p λ ◦ F ∗ = p λ ◦ H ∗ . We show that V F ∗ λ = V H ∗ λ . Suppose K ∗ ∈ V F ∗ λ , so p λ ◦ K ∗ = p λ ◦ ∗ = p λ ◦ H ∗ . Therefore K ∗ ∈ V H ∗ λ and hence V F ∗ λ ⊆ V H ∗ λ . Conversely, if K ∗ ∈ V H ∗ λ ,then we have p λ ◦ K ∗ = p λ ◦ H ∗ = p λ ◦ F ∗ . So K ∗ ∈ V F ∗ λ and hence V H ∗ λ ⊆ V F ∗ λ .Similarly, since H ∗ ∈ V G ∗ λ , we have V G ∗ λ = V H ∗ λ and so H ∗ ∈ V H ∗ λ ∩ V H ∗ λ . We knowthat there exists a λ ∈ Λ such that λ ≥ λ , λ . We show that H ∗ ∈ V H ∗ λ ⊆ V H ∗ λ ∩ V H ∗ λ which completes the proof of the first assertion.Given K ∗ ∈ V H ∗ λ . We have p λ λ p λ = p λ and p λ λ p λ = p λ . Since K ∗ ∈ V H ∗ λ , so p λ ◦ K ∗ = p λ ◦ H ∗ and therefore p λ ◦ K ∗ = p λ ◦ H ∗ . Hence K ∗ ∈ V H ∗ λ . Similarly K ∗ ∈ V H ∗ λ and so K ∗ ∈ V H ∗ λ ∩ V H ∗ λ .Now, suppose that p ′ : X → X ′ is another HPol-expansion of X . Then thereexists a unique isomorphism i : X −→ X ′ given by ( i ν , φ ) such that i ◦ p = p ′ . Let V F ∗ ν be an arbitrary element in the basis of T p ′ , where ν ∈ N and F ∗ ∈ Sh ∗ ( Y, X ).Then φ ( ν ) ∈ Λ. We show that V F ∗ ν ∈ T p and this follows that T p ′ ⊆ T p . Given G ∗ ∈ V F ∗ ν , thus V G ∗ ν = V F ∗ ν . For each H ∗ ∈ V G ∗ φ ( ν ) , we have p φ ( ν ) ◦ G ∗ = p φ ( ν ) ◦ H ∗ and so p ′ ν ◦ G ∗ = p ′ ν ◦ H ∗ . Hence H ∗ ∈ V G ∗ ν and therefore G ∗ ∈ V G ∗ φ ( ν ) ⊆ V G ∗ ν = V F ∗ ν .Similarly, one can show that T p ⊆ T p ′ . Corollary 3.2.
Let X ∈ Obj ( HP ol ) . Then Sh ∗ ( Y, X ) is discrete, for every topo-logical space Y . Example 3.3.
Let P = {∗} be a singleton and Q = {∗} ˙ ∪{∗} (disjoint union). Then card ( Sh ( P, Q )) = 2 while card ( Sh ∗ ( P, Q )) = 2 ℵ (see [1, Example 7.4]). It showsthat Sh ( P, Q ) is a countable discrete space while Sh ∗ ( P, Q ) is an uncountable discretespace. Theorem 3.4.
The map
Ω : Sh ∗ ( X, Y ) × Sh ∗ ( Y, Z ) −→ Sh ∗ ( X, Z ) given by thecomposition Ω( F ∗ , G ∗ ) = G ∗ ◦ F ∗ is continuous, for arbitrary topological spaces X, Y and Z .Proof. Consider HPol-expansions p : X → X = ( X λ , p λλ ′ , Λ), q : Y → Y =( Y µ , q µµ ′ , M ) and r : Z → Z = ( Z ν , r νν ′ , N ) of X, Y and Z , respectively. Let F ∗ ∈ Sh ∗ ( X, Y ) and G ∗ ∈ Sh ∗ ( Y, Z ) given by ( f nµ , f ) and ( g nν , g ), respectively. Let ν ∈ N and G ∗ ◦ F ∗ ∈ V G ∗ ◦ F ∗ ν . We show that Ω( V F ∗ g ( ν ) × V G ∗ ν ) ⊆ V G ∗ ◦ F ∗ ν . To do this,we must show that for any F ∗ ∈ V F ∗ g ( ν ) and G ∗ ∈ V G ∗ ν , r ν ◦ G ∗ ◦ F ∗ = r ν ◦ G ∗ ◦ F ∗ .Since F ∗ ∈ V F ∗ g ( ν ) , we have q g ( ν ) ◦ F ∗ = q g ( ν ) ◦ F ∗ and since G ∗ ∈ V G ∗ ν , we have r ν ◦ G ∗ = r ν ◦ G ∗ . Note that r ν ◦ G ∗ is an S ∗ -morphism given by ( g nν , gα ν ), where α ν : { ν } −→ N is the inclusion map. Define α : Y g ( ν ) −→ Z ν as an S ∗ -morphismgiven by ( g nν , β ν ), where β ν : { ν } −→ { g ( ν ) } . We have r ν ◦ G ∗ = α ◦ q g ( ν ) and so r ν ◦ G ∗ ◦ F ∗ = α ◦ q g ( ν ) ◦ F ∗ = α ◦ q g ( ν ) ◦ F ∗ = r ν ◦ G ∗ ◦ F ∗ = r ν ◦ G ∗ ◦ F ∗ .6he following corollary is an immediate consequence of the above theorem. Corollary 3.5.
Let X and Y be topological spaces and let F ∗ : X −→ Y be an S ∗ -morphism. Let Z be a topological space and consider F ∗ : Sh ∗ ( Y, Z ) −→ Sh ∗ ( X, Z ) and F ∗ : Sh ∗ ( Z, X ) −→ Sh ∗ ( Z, Y ) to be defined by F ∗ ( H ∗ ) = H ∗ ◦ F ∗ and F ∗ ( G ∗ ) = F ∗ ◦ G ∗ .(i) F ∗ and F ∗ are continuous, ( G ∗ ◦ F ∗ ) = G ∗ ◦ F ∗ , ( G ∗ ◦ F ∗ ) = F ∗ ◦ G ∗ and Id ∗ and Id ∗ are the corresponding identity maps.(ii) Assume Sh ∗ ( X ) ≥ Sh ∗ ( Y ) . Then Sh ∗ ( Y, Z ) is homeomorphic to a retract of Sh ∗ ( X, Z ) and Sh ∗ ( Z, Y ) is homeomorphic to a retract of Sh ∗ ( Z, X ) , for every topo-logical space Z .(iii) Assume Sh ∗ ( X ) = Sh ∗ ( Y ) . Then Sh ∗ ( Y, Z ) is homeomorphic to Sh ∗ ( X, Z ) and Sh ∗ ( Z, Y ) is homeomorphic to Sh ∗ ( Z, X ) , for every topological space Z . Now, we want to prove the following theorem which is usefull to study the topo-logical properties of the space of coarse shape morphisms.
Theorem 3.6.
Let X and Y be topological spaces and let p : X −→ X = ( X λ , p λλ ′ , Λ) and q : Y −→ Y = ( Y µ , q µµ ′ , M ) be HPol-expansions of X and Y , respectively. Take Sh ∗ ( X, Y ) = ( Sh ∗ ( X, Y µ ) , ( q µµ ′ ) ∗ , M ) and consider the morphism q ∗ : Sh ∗ ( X, Y ) −→ Sh ∗ ( X, Y ) induced by q . Then q ∗ is an inverse limit of Sh ∗ ( X, Y ) in Top.Proof. Let Z be a topological space and let g : Z −→ ( Sh ∗ ( X, Y µ ) , ( q µµ ′ ) ∗ , M ) bea morphism in pro-Top. We must show that there is a unique continuous map α : Z −→ Sh ∗ ( X, Y ) such that q ∗ ◦ α = g in pro-Top. We know that g µ ( z ) ∈ Sh ∗ ( X, Y µ ), for every z ∈ Z and µ ∈ M . Suppose g µ ( z ) = < [( g nµ,z , λ µ,z )] > anddefine h z : M −→ Λ by h z ( µ ) = λ µ,z . We define α ( z ) = < [( g nµ,z , h z )] > . Since( q µµ ′ ) ∗ ◦ g µ ′ = g µ , so for every z ∈ Z , (( q µµ ′ ) ∗ ◦ g µ ′ )( z ) = g µ ( z ). Thus, there is a λ ≥ λ µ,z , λ µ ′ ,z and n ∈ N such that for every n ′ ≥ n , q µµ ′ ◦ g n ′ µ ′ ,z ◦ p λ µ ′ ,z λ = g n ′ µ,z ◦ p λ µ,z λ .It follows that α ( z ) is an S ∗ -morphism. It is clear that q ∗ ◦ α = g . To complete theproof, we show that α is continuous. Let z ∈ Z , µ ∈ M and F ∗ = α ( z ) ∈ V F ∗ µ . wehave α − ( V F ∗ µ ) = { z ′ ∈ Z : α ( z ′ ) ∈ V F ∗ µ } = { z ′ ∈ Z : ( q µ ) ∗ ◦ α ( z ′ ) = ( q µ ) ∗ ◦ α ( z ) } = { z ′ ∈ Z : g µ ( z ) = g µ ( z ′ ) } = g − µ ( g µ ( z )). Since Sh ∗ ( X, Y µ ) is discrete, { g µ ( z ) } is anopen subset of Sh ∗ ( X, Y µ ) and since g µ is continuous, we have g − µ ( g µ ( z )) is opensubset of Z . It follows that α is continuous. Corollary 3.7.
Let X and Y be two topological spaces. Then Sh ∗ ( X, Y ) is a Ty-chonoff space. M, ≤ ) is a directed set. From [8], we denote by L ( M ) the set of alllower classes in M ordered by inclusion, in which ∆ ⊆ M is called a lower class if forevery δ ∈ ∆ and µ ∈ M with µ ≤ δ , then µ ∈ ∆. Moreover, for any two lower classes∆ , ∆ ′ ∈ L ( M ), we say that ∆ ≤ ∆ ′ if and only if ∆ ⊃ ∆ ′ . Then ( L ( M ) , ≤ ) is apartially ordered set with the least element M which is denoted by 0. Furthermore, L ( M ) ∗ = L ( M ) − X be a set and (Γ , ≤ ) be a partial ordered set with a least element 0. Recallfrom [17] that an ultrametric on X is a map d : X × X → Γ such that for all x, y ∈ X and γ ∈ Γ, the following hold:1) d ( x, y ) = 0 ⇐⇒ x = y .2) d ( x, y ) = d ( y, x ).3) if d ( x, y ) ≤ γ and d ( y, z ) ≤ γ , then d ( x, z ) ≤ γ .Now, using the same idea as in [8], we can prove the following theorem: Theorem 3.8.
Let X and Y be topological spaces. Assume q : Y −→ Y =( Y µ , q µµ ′ , M ) is an HPol-expansion of Y . For every F ∗ , G ∗ ∈ Sh ∗ ( X, Y ) take d ( F ∗ , G ∗ ) = { µ ∈ M : q µ ◦ F ∗ = q µ ◦ G ∗ } . Then we have an ultrametric d : Sh ∗ ( X, Y ) × Sh ∗ ( X, Y ) → ( L ( M ) , ≤ ) .Proof. First, we show that d ( F ∗ , G ∗ ) is a lower class. Suppose µ ∈ d ( F ∗ , G ∗ ) and µ ′ ∈ M such that µ ′ ≤ µ . Then q µ ′ = q µ ′ µ q µ and we have q µ ′ ◦ F ∗ = q µ ′ µ ◦ q µ ◦ F ∗ = q µ ′ µ ◦ q µ ◦ G ∗ = q µ ′ ◦ G ∗ . It follows that µ ′ ∈ d ( F ∗ , G ∗ ). Now, let F ∗ , G ∗ ∈ Sh ∗ ( X, Y ) such that d ( F ∗ , G ∗ ) = 0.It is equivalent to q µ ◦ F ∗ = q µ ◦ G ∗ , for every µ ∈ M or equivalently F ∗ = G ∗ . Otherconditions can also be proved easily.Let ( M, ≤ ) be a directed set and ( L ( M ) , ≤ ) be the corresponding ordered set oflower classes in M . For every µ ∈ M , consider { µ ′ ∈ M : µ ≥ µ ′ } as the lower classgenerated by µ , which is denote by [ µ ] and define φ : ( M, ≤ ) → ( L ( M ) , ≤ ) that maps µ to [ µ ]. If µ ≥ µ ′ , then [ µ ] ≤ [ µ ′ ] and ( φ ( M ) , ≤ ) is a partial ordered set and also isdownward directed in L ( M ) (see [8]).Now, we have: 8 roposition 3.9. Let X and Y be topological spaces. Suppose q : Y −→ Y =( Y µ , q µµ ′ , M ) is an HPol-expansion of Y . For every µ ∈ M and F ∗ ∈ Sh ∗ ( X, Y ) take B [ µ ] ( F ∗ ) = { G ∗ ∈ Sh ∗ ( X, Y ) : d ( F ∗ , G ∗ ) ≤ [ µ ] } . Then the family { B [ µ ] ( F ∗ ) : F ∗ ∈ Sh ∗ ( X, Y ) , µ ∈ M } is a basis for a topology in Sh ∗ ( X, Y ) . Moreover, this topology is independent of the fixed HPol-expansion of Y and it coinsides with the topology defined previously .Proof. It is obvious that F ∗ ∈ B [ µ ] ( F ∗ ), for all µ ∈ M . Suppose F ∗ , G ∗ ∈ Sh ∗ ( X, Y )and µ , µ ∈ M and H ∗ ∈ B [ µ ] ( F ∗ ) ∩ B [ µ ] ( G ∗ ). Therefore d ( H ∗ , F ∗ ) ≤ [ µ ]and d ( H ∗ , G ∗ ) ≤ [ µ ]. Let K ∗ ∈ B [ µ ] ( H ∗ ), then we have d ( K ∗ , H ∗ ) ≤ [ µ ] and d ( H ∗ , F ∗ ) ≤ [ µ ] and so by the definition of an ultrametric, d ( K ∗ , F ∗ ) ≤ [ µ ]. Itshows that K ∗ ∈ B [ µ ] ( F ∗ ) and B [ µ ] ( H ∗ ) ⊆ B [ µ ] ( F ∗ ). Conversely, we can show that B [ µ ] ( F ∗ ) ⊆ B [ µ ] ( H ∗ ) and hence we have B [ µ ] ( H ∗ ) = B [ µ ] ( F ∗ ). Similarly, we canconclude that B [ µ ] ( H ∗ ) = B [ µ ] ( G ∗ ). Hence, to prove the first assertion, it is enoughto consider µ ∈ M such that µ ≥ µ , µ , then [ µ ] ≤ [ µ ] , [ µ ] and this easily impliesthat H ∗ ∈ B [ µ ] ( H ∗ ) ⊆ B [ µ ] ( H ∗ ) ∩ B [ µ ] ( H ∗ ) = B [ µ ] ( F ∗ ) ∩ B [ µ ] ( G ∗ ).Now, suppose that q ′ : Y → Y ′ = ( Y ν , q νν ′ , N ) is another HPol-expansion of Y . Then there exists a unique isomorphism j : Y −→ Y ′ given by ( j ν , φ ) suchthat j ◦ q = q ′ (we can assume that φ is an increasing map). Let ν ∈ N and F ∗ ∈ Sh ∗ ( X, Y ), then φ ( ν ) ∈ M . Given G ∗ ∈ B [ ν ] ( F ∗ ), so by the above argument B [ ν ] ( F ∗ ) = B [ ν ] ( G ∗ ). For each H ∗ ∈ B [ φ ( ν )] ( G ∗ ), we have d ( G ∗ , H ∗ ) ≤ [ φ ( ν )], i.e., if µ ≤ φ ( ν ), then q µ ◦ G ∗ = q µ ◦ H ∗ . Given ν ′ ∈ N such that ν ′ ≤ ν , then φ ( ν ′ ) ≤ φ ( ν ) and so q φ ( ν ′ ) ◦ G ∗ = q φ ( ν ′ ) ◦ H ∗ . It implies that q ′ ν ′ ◦ G ∗ = q ′ ν ′ ◦ H ∗ and thus H ∗ ∈ B [ ν ] ( G ∗ ). Therefore G ∗ ∈ B [ φ ( ν )] ( G ∗ ) ⊆ B [ ν ] ( G ∗ ) = B [ ν ] ( F ∗ ) and it followsthat the topology corresponding to HPol-expansion q is stronger than the topologycorresponding to HPol-expansion q ′ . Similarly, we can prove that the converse istrue.Finally, we want to show that the topology induced by the ultrametric d coinsideswith the topology T q studied in Proposition 3.1. It is easy to see that V F ∗ µ = B [ µ ] ( F ∗ ),for every µ ∈ M and F ∗ ∈ Sh ∗ ( X, Y ) which completes the proof.
4. The topological coarse shape homotopy groups
Let X be a topological space and p : X → X = ( X λ , p λλ ′ , Λ) be an HPol-expansion of X . We know that the k th coarse shape group ˇ π ∗ k ( X, x ), k ∈ N , is theset of all coarse shape morphisms F ∗ : ( S k , ∗ ) → ( X, x ) with the following binaryoperation F ∗ + G ∗ = < f ∗ > + < g ∗ > = < f ∗ + g ∗ > = < [( f nλ )] + [( g nλ )] > = < [( f nλ + g nλ )] >, F ∗ and G ∗ are represented by morphisms f ∗ = [( f, f nλ )]and g ∗ = [( g, g nλ )] : ( S k , ∗ ) → ( X , x ) in pro ∗ -HPol ∗ , respectively (see [2]).Now, we show that ˇ π ∗ k ( X, x ) = Sh ∗ (( S k , ∗ ) , ( X, x )) with the above topology is atopological group which is denoted by ˇ π ∗ top k ( X, x ), for all k ∈ N . Theorem 4.1.
Let ( X, x ) be a pointed topological space. Then ˇ π ∗ top k ( X, x ) is a topo-logical group, for all k ∈ N .Proof. First, we show that φ : ˇ π ∗ top k ( X, x ) → ˇ π ∗ top k ( X, x ) given by φ ( F ∗ ) = F ∗ − iscontinuous, where F ∗ and F ∗ − : ( S k , ∗ ) → ( X, x ) are represented by f ∗ = ( f, f nλ ) and f ∗ − = ( f, f n − λ ) : ( S k , ∗ ) → ( X , x ), respectively and f n − λ : ( S k , ∗ ) → ( X λ , x λ ) is theinverse loop of f nλ . Let V F ∗− λ be an open neighbourhood of F ∗ − in ˇ π ∗ top k ( X, x ). Weknow that for any G ∗ = < [( g, g nλ )] > ∈ V F ∗ λ , p λ ◦ G ∗ = p λ ◦ F ∗ . So there is an n ′ ∈ N such that for any n ≥ n ′ , g nλ ≃ f nλ rel {∗} by [1, Claim 1 and Claim 2]. Then forany n ≥ n ′ , g n − λ ≃ f n − λ rel {∗} and so p λ ◦ G ∗ − = p λ ◦ F ∗ − . Thus φ ( G ∗ ) ∈ V F ∗− λ .Therefore, the map φ is continuous.Second, we show that the map m : ˇ π ∗ top k ( X, x ) × ˇ π ∗ top k ( X, x ) → ˇ π ∗ top k ( X, x ) givenby m ( F ∗ , G ∗ ) = F ∗ + G ∗ is continuous, where F ∗ + G ∗ is the coarse shape morphismrepresented by f ∗ + g ∗ = ( f, f nλ + g nλ ) : ( S k , ∗ ) → ( X , x ) and f nλ + g nλ is the concate-nation of paths. Let V F ∗ + G ∗ λ be an open neighbourhood of F ∗ + G ∗ in ˇ π ∗ top k ( X, x ).For any ( K ∗ , H ∗ ) ∈ V F ∗ λ × V G ∗ λ , we have p λ ◦ ( K ∗ + H ∗ ) = ( p λ ◦ K ∗ ) + ( p λ ◦ H ∗ ) =( p λ ◦ F ∗ ) + ( p λ ◦ G ∗ ) = p λ ◦ ( F ∗ + G ∗ ). Hence m ( K ∗ , H ∗ ) ∈ V F ∗ + G ∗ λ and so m iscontinuous.Using Corollary 3.5, we can conclude the following results: Corollary 4.2. If F ∗ : ( X, x ) → ( Y, y ) is a coarse shape morphism, then F ∗ :ˇ π ∗ top k ( X, x ) → ˇ π ∗ top k ( Y, y ) is continuous. Corollary 4.3. If ( X, x ) and ( Y, y ) are two pointed topological spaces and Sh ∗ ( X, x ) = Sh ∗ ( Y, y ) , then ˇ π ∗ top k ( X, x ) ∼ = ˇ π ∗ top k ( Y, y ) as topological groups. Corollary 4.4.
For any k ∈ N , ˇ π ∗ top k ( − ) is a functor from the pointed coarse shapecategory of spaces to the category of Hausdorff topological groups. Corollary 4.5.
Let X be a topological space and p : X → X = ( X λ , p λλ ′ , Λ) be anHPol-expansion of X . By Theorem 3.6, we know that ˇ π ∗ top k ( X, x ) ∼ = lim ← ˇ π ∗ top k ( X λ , x λ ) as topological groups, for all k ∈ N . Since every ˇ π ∗ top k ( X λ , x λ ) is discrete and Haus-dorff, ˇ π ∗ top k ( X, x ) is prodiscrete and Hausdorff, for every topological space ( X, x ) . orollary 4.6. Let ( X, x ) = lim ← ( X i , x i ) , where X i ’s are compact polyhedra. Thenfor all k ∈ N , ˇ π ∗ top k ( X, x ) ∼ = lim ← ˇ π ∗ top k ( X i , x i ) . Proof.
It can be proved similar to the Corollary 3.8 in [22].
Corollary 4.7.
Let p : ( X, x ) → ( X , x ) = (( X λ , x λ ) , p λλ ′ , Λ) be an HPol ∗ -expansionof a pointed topological space ( X, x ) . Then the following statements hold for all k ∈ N :(i) If the cardinal number of Λ is ℵ and ˇ π ∗ top k ( X λ , x λ ) is second countable for every λ ∈ Λ , then ˇ π ∗ top k ( X, x ) is second countable.(ii) If ˇ π ∗ top k ( X λ , x λ ) is totally disconnected for every λ ∈ Λ , then so is ˇ π ∗ top k ( X, x ) .Proof. The results follow from the fact that the product and the subspace topologiespreserve the properties of being second countable and totally disconnected.
Remark 4.8.
The authors proved a similar result to the above corollary for shape ho-motopy groups [22, Corollary 3.9]. Note that in that case, we can omit the assumptionof second coutability of π qtopk ( X λ , x λ ) , for all λ ∈ Λ . Indeed, If X is a polyhedron, so X is second countable and hence Ω k ( X, x ) is second countable, for all x ∈ X and k ∈ N (see [10]). Since π qtopk ( X, x ) is discrete, then the map q : Ω k ( X, x ) → π qtopk ( X, x ) is abi-quotient map and therefore π qtopk ( X, x ) is also second countable, for all k ∈ N (see[20]). Let X be a topological space and let x , x ∈ X . A coarse shape path in X from x to x is a bi-pointed coarse shape morphism Ω ∗ : ( I, , → ( X, x , x ). X is said to be coarse shape path connected, if for every pair x, x ′ ∈ X , there is acoarse shape path from x to x ′ . If X is a coarse shape path connected space, thenˇ π ∗ k ( X, x ) ∼ = ˇ π ∗ k ( X, x ′ ), for any two points x, x ′ ∈ X and every k ∈ N [3, Corollary 1].Now, we show that these two groups are isomorphic as topological groups, if X is a coarse shape path connected, paracompact and locally compact space. Theorem 4.9.
Let X be a coarse shape path connected, paracompact and locallycompact space. Then ˇ π ∗ top k ( X, x ) ∼ = ˇ π ∗ top k ( X, x ′ ) , for every pair x, x ′ ∈ X and all k ∈ N .Proof. If X is a topological space admitting a metrizable polyhedral resolution andfor a pair x, x ′ ∈ X there exists a coarse shape path in X from x to x ′ , then ( X, x )and (
X, x ′ ) are isomorphic pointed spaces in Sh ∗ ⋆ (see [3, Theorem 3]). Since coarseshape path connected, paracompact and locally compact spaces satisfy in the abovecondition [25], Sh ∗ ( X, x ) ∼ = Sh ∗ ( X, x ′ ). Hence by Corollary 4.3 we have ˇ π ∗ top k ( X, x ) ∼ =ˇ π ∗ top k ( X, x ′ ), for every pair x, x ′ ∈ X and all k ∈ N .11 . Main results It is well-known that if the cartesian product of two spaces X and Y admits anHPol-expansion, which is the cartesian product of HPol-expansions of these space,then X × Y is a product in the shape category (see [18]). In this case, the authorsshowed that the k th topological shape group commutes with finite products, for all k ∈ N [22, Theorem 4.1].Also, if X and Y admit HPol-expansions p : X → X and q : Y → Y , respectively,such that p × q : X × Y → X × Y is an HPol-expansion, then X × Y is a product in thecoarse shape category [23, Theorem 2.2]. Mardeˇsi´c [18] proved that if p : X → X and q : Y → Y are HPol-expansions of compact Hausdorff spaces X and Y , respectively,then p × q : X × Y → X × Y is an HPol-expansion and so in this case, X × Y is aproduct in the coarse shape category.Now, we show that under the above condition, the k th topological coarse shapegroup commutes with finite products, for all k ∈ N . Theorem 5.1. If X and Y are coarse shape path connected spaces with HPol-expansions p : X → X and q : Y → Y such that p × q : X × Y → X × Y isan HPol-expansion, then ˇ π ∗ top k ( X × Y ) ∼ = ˇ π ∗ top k ( X ) × ˇ π ∗ top k ( Y ) , for all k ∈ N .Proof. Let S ∗ ( π X ) : X × Y → X and S ∗ ( π Y ) : X × Y → Y be the induced coarseshape morphisms of canonical projections and assume that φ X : ˇ π ∗ top k ( X × Y ) → ˇ π ∗ top k ( X ) and φ Y : ˇ π ∗ top k ( X × Y ) → ˇ π ∗ top k ( Y ) are the induced continuous homo-morphisms by S ∗ ( π X ) and S ∗ ( π Y ), respectively. Then the induced homomorphism φ : ˇ π ∗ top k ( X × Y ) → ˇ π ∗ top k ( X ) × ˇ π ∗ top k ( Y ) is continuous. Since X × Y is a productin Sh ∗ , we can define a homomorphism ψ : ˇ π ∗ top k ( X ) × ˇ π ∗ top k ( Y ) → ˇ π ∗ top k ( X × Y )by ψ ( F ∗ , G ∗ ) = ⌊ F ∗ , G ∗ ⌋ , where ⌊ F ∗ , G ∗ ⌋ : S k → X × Y is a unique coarse shapemorphism with S ∗ ( π X )( ⌊ F ∗ , G ∗ ⌋ ) = F ∗ and S ∗ ( π Y )( ⌊ F ∗ , G ∗ ⌋ ) = G ∗ . In fact, if F ∗ = h f ∗ = ( f, f nλ ) i and G ∗ = h g ∗ = ( g, g nµ ) i , then ⌊ F ∗ , G ∗ ⌋ = h⌊ f ∗ , g ∗ ⌋i , where ⌊ f ∗ , g ∗ ⌋ is given by ⌊ f, g ⌋ nλµ = f nλ × g nµ : S k → X λ × Y µ . By the proof of [23, Theorem2.4], the homomorphism ψ is well define and moreover, φ ◦ ψ = id and ψ ◦ φ = id .To complete the proof, it is enough to show that ψ is continuous. Let ⌊ F ∗ , G ∗ ⌋ ∈ V ⌊ F ∗ ,G ∗ ⌋ λµ be a basis open in the topology on ˇ π ∗ top k ( X × Y ). Considering open sets F ∗ ∈ V F ∗ λ and G ∗ ∈ V G ∗ µ , we show that ψ ( V F ∗ λ × V G ∗ µ ) ⊆ V ⌊ F ∗ ,G ∗ ⌋ λµ . Let H ∗ ∈ V F ∗ λ and K ∗ ∈ V G ∗ µ , then p λ ◦ H ∗ = p λ ◦ F ∗ and q µ ◦ K ∗ = q µ ◦ G ∗ . By a straightcomputation, we can conclude that p λ × q µ ( ⌊ H ∗ , K ∗ ⌋ ) = p λ × q µ ( ⌊ F ∗ , G ∗ ⌋ ) whichimplies that ψ ( H ∗ , K ∗ ) = ⌊ H ∗ , K ∗ ⌋ ∈ V ⌊ F ∗ ,G ∗ ⌋ λµ . Theorem 5.2.
Let ( X, x ) be a pointed topological space. Then for all k ∈ N ,(i) If ( X, x ) ∈ HP ol ∗ , then ˇ π ∗ top k ( X, x ) is discrete. ii) If p : ( X, x ) → ( X , x ) = (( X λ , x λ ) , p λλ ′ , Λ) is an HPol ∗ -expansion of ( X, x ) and ˇ π ∗ top k ( X, x ) is discrete, then ˇ π ∗ top k ( X, x ) ≤ ˇ π ∗ top k ( X λ , x λ ) , for some λ ∈ Λ .Proof. (i) This follows from Corollary 3.2.(ii) Since ˇ π ∗ top k ( X, x ) is a discrete group, { E ∗ x } is an open set of identity point ofˇ π ∗ top k ( X, x ). Thus { E ∗ x } = ∪ λ ∈ Λ V F ∗ λ , where Λ ⊆ Λ. Consider the induced ho-momorphism p λ ∗ : ˇ π ∗ top k ( X, x ) → ˇ π ∗ top k ( X λ , x λ ) given by p λ ∗ ( F ∗ ) = p λ ◦ F ∗ . Let G ∗ ∈ kerp λ ∗ , i.e., p λ ◦ G ∗ = E ∗ x λ = p λ ◦ E ∗ x . Thus G ∗ ∈ V E ∗ x λ ⊆ ∪ λ ∈ Λ V F ∗ λ = { E ∗ x } andso G ∗ = E ∗ x . Therefore p λ ∗ is injective, for all λ ∈ Λ and k ∈ N .Recall that an inverse system X = ( X λ , p λλ ′ , Λ) of pro-HTop is said to be movableif every λ ∈ Λ admits a λ ′ ≥ λ such that each λ ′′ ≥ λ admits a morphism r : X λ ′ → X λ ′′ of HTop with p λλ ′′ ◦ r ≃ p λλ ′ . We say that a topological space X is movableprovided that it admits an HPol-expansion p : X → X such that X is a movableinverse system of pro-HPol [19]. We know that under some conditions, movabilitycan be transferred from a pointed topological space ( X, x ) to ˇ π topk ( X, x ) (see [22])and now we show that it can be transferred to ˇ π ∗ top k ( X, x ) too.
Lemma 5.3. If ( X , x ) = (( X λ , x λ ) , p λλ ′ , Λ) is a movable (uniformly movable) in-verse system, then Sh ∗ (( S k , ∗ ) , ( X, x )) = ( Sh ∗ (( S k , ∗ ) , ( X λ , x λ )) , ( p λλ ′ ) ∗ , Λ) is also amovable (uniformly movable) inverse system, for all k ∈ N .Proof. Let λ ∈ Λ. Since ( X , x ) is a movable inverse system, there is a λ ′ ≥ λ such thatfor every λ ′′ ≥ λ there is a map r : ( X λ ′ , x λ ′ ) → ( X λ ′′ , x λ ′′ ) such that p λλ ′′ ◦ r ≃ p λλ ′ rel { x λ ′ } . We consider r ∗ : Sh ∗ (( S k , ∗ ) , ( X λ ′ , x λ ′ )) → Sh ∗ (( S k , ∗ ) , ( X λ ′′ , x λ ′′ )). Hence( p λλ ′′ ) ∗ ◦ r ∗ ≃ ( p λλ ′ ) ∗ and so Sh ∗ (( S k , ∗ ) , ( X, x )) is movable.
Remark 5.4.
Let ( X, x ) be a movable space. Then there exists an HPol ∗ -expansion p : ( X, x ) → ( X , x ) such that ( X , x ) is a movable inverse system. Suppose p ∗ : Sh ∗ (( S k , ∗ ) , ( X, x )) → Sh ∗ (( S k , ∗ ) , ( X, x )) is an HPol ∗ -expansion, then using Lemma5.3, we can conclude that ˇ π ∗ top k ( X, x ) is a movable topological group, for all k ∈ N .By Theorem 3.6, if p : ( X, x ) → ( X , x ) is an HPol ∗ -expansion of X , then p ∗ : Sh ∗ (( S k , ∗ ) , ( X, x )) → Sh ∗ (( S k , ∗ ) , ( X, x )) is an inverse limit of Sh ∗ (( S k , ∗ ) , ( X, x )) =( Sh ∗ (( S k , ∗ ) , ( X λ , x λ )) , ( p λλ ′ ) ∗ , Λ) . Now, if Sh ∗ (( S k , ∗ ) , ( X λ , x λ )) is a compact poly-hedron for all λ ∈ Λ , then by [14, Remark 1] p ∗ is an HPol ∗ -expansion of Sh ∗ (( S k , ∗ ) , ( X, x )) and therefore in this case, movability of ( X, x ) implies movabilityof ˇ π ∗ top k ( X, x ) . Remark 5.5.
Suppose ( X, x ) is a topological space and p : ( X, x ) → ( X , x ) =(( X λ , x λ ) , p λλ ′ , Λ) is an HPol ∗ -expansion of ( X, x ) . Consider J : ˇ π topk ( X, x ) −→ π ∗ top k ( X, x ) given by J ( F = < ( f, f λ ) > ) = F ∗ , where F ∗ = < ( f, f nλ = f λ ) > . Then J is an embedding. To prove this, we show that for each λ ∈ Λ and for all F ∈ ˇ π topk ( X, x ) , J ( V Fλ ) = V J ( F ) λ ∩ J (ˇ π topk ( X, x )) . Suppose G = < ( g, g λ ) > ∈ V Fλ , so p λ ◦ G = p λ ◦ F or equivalently g λ ≃ f λ . We know that J ( G ) = < g nλ = g λ > and J ( F ) = < f nλ = f λ > . So for all n ∈ N , g nλ ≃ f nλ and it follows that p λ ◦ J ( G ) = p λ ◦ J ( F ) . Hence J ( G ) ∈ V J ( F ) λ ∩ J (ˇ π topk ( X, x )) . Conversely, suppose that G ∗ =
X, x ) be a topological space. We know that the induced homomorphism φ : π qtopk ( X, x ) → ˇ π topk ( X, x ) is continuous, for all k ∈ N . Consider the composition J ◦ φ : π qtopk ( X, x ) → ˇ π ∗ top k ( X, x ) in which J is the embedding defined in Remark 5.5.If ( X, x ) is shape injective, then the homomorphism φ is an embedding and hencewe have an embedding from π qtopk ( X, x ) to ˇ π ∗ top k ( X, x ).Let X ⊆ Y and r : Y → X be a retraction. Consider the inclusion map j : X → Y . It is known that j ∗ : ˇ π topk ( X, x ) → ˇ π topk ( Y, x ) is a topological em-bedding [22, Theorem 4.2] and similar to the proof of it, we can conclude that theinduced map j ∗ : ˇ π ∗ top k ( X, x ) → ˇ π ∗ top k ( Y, x ) is also a topological embedding.In follow, we present examples whose topological coarse shape homotopoy groupsare not discrete.
Example 5.6.
Let ( HE, p = (0 , ← ( X i , p i ) be the Hawaiian Earring where X j = ∨ ji =1 S i . The first shape homotopy group ˇ π top ( HE, p ) is not discrete (see [22,Example 4.5]. So the above Remark follows that ˇ π ∗ top ( HE, p ) is not discrete. Example 5.7.
Let k ∈ N and let X = ( X n , p nn +1 , N ) , where X n = Q nj =1 S kj is theproduct of n copies of k -sphere S k , for all n ∈ N and the bonding morphisms of X arethe projection maps. Put X = lim ← X n . Refer to [22], ˇ π topk ( X ) ∼ = lim ← π qtopk ( X n ) ∼ = Y Z is not discrete. Since ˇ π topk ( X ) is a subspace of ˇ π ∗ top k ( X ) and it is not discete, then ˇ π ∗ top k ( X ) is not discrete. eferencesReferences [1] N. K. Bilan, N. Uglesic , The coarse shape,
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