On topological fundamental groups of quotient spaces

ON TOPOLOGICAL FUND AMENT AL GR OUPS OF QUOTIENT SP A CES HAMID TORABI † , ALI P AKDAMAN ‡ AND BEHROOZ MASHA YEKHY † , ∗ † Dep artment of Pur e Mathematics, Center of Exc el lenc e in Analysis on Algebr aic Structur es, F er dowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Ir an. ‡ Dep artment of Mathematics, F aculty of Scienc e, Golestan University, P.O.Box 155, Gor gan, Ir an. Abstract. Let p : X → X/ A b e a quotient map, where A is a subspace of X . W e explore conditions under which p ∗ ( π qtop 1 ( X, x 0 )) is dense in π qtop 1 ( X/ A, ∗ )), where the fundamental groups enjoy the natural quotient top ology inherited from the lo op space and p ∗ is the induced contin uous homomorphism by the quotient map p . Also, w e giv e some applications to find out some properties for π qtop 1 ( X/ A, ∗ ). In particular, we give some conditions in which π qtop 1 ( X/ A, ∗ ) is an indiscrete topological group. 1. Introduction and Motiv a tion Let p : ( X , x 0 ) → ( Y , y 0 ) b e a con tinuous map of p ointed top ological spaces. By applying the fundamen tal group functor on p there exists the induced homomor- phism p ∗ : π 1 ( X, x 0 ) − → π 1 ( Y , y 0 ) . It seems in teresting to relate the homology and homotop y groups of X with that of Y using properties of p . Vietoris first studied the problem with his mapping theorem [18]. Also, Smale first discov ered an analog of Vietoris’s mapping theorem hold for homotop y groups [15]. Recen tly , Calcut, Gompf, and Mccarthy [6] prov ed a generalization of Smale’s theorem as follows: L et p : ( X , x 0 ) → ( Y , y 0 ) b e a quotient map of top olo gic al sp ac es, wher e X is lo c al ly p ath c onne cte d and Y is semilo c al ly simply c onne cte d. If e ach fib er p − 1 ( y ) is c onne cte d, then the induc e d homomorphism p ∗ : π 1 ( X, x 0 ) → π 1 ( Y , y 0 ) is surje ctive . F or a p ointed top ological space ( X , x 0 ) by π q top 1 ( X, x 0 ) we mean the top ological fundamen tal group endow ed with the quotient top ology inherited from the lo op space under the natural map Ω( X , x 0 ) − → π 1 ( X, x 0 ) that mak es it a quasitop olo gic al gr oup . A quasitop olo gic al gr oup G is a group with a top ology suc h that in v ersion g − → g − 1 and all translations are con tinuous. F or more details, see [2, 3, 5]. It Key wor ds and phr ases. T op ological fundamen tal group, Quasitop ological fundamental group, Quotient map, Dense subgroup. 2010 Mathematics Subject Classific ation : 55P65; 55Q52; 55Q70. ∗ Corresponding author. E-mail addresses: hamid − torabi86@yahoo.com; a.pakdaman@gu.ac.ir and bmashf@um.ac.ir. 1 2 H. TORABI, A. P AKDAMAN AND B. MASHA YEKHY is known that this construction gives rise a homotopy inv ariant functor π q top 1 : hT op ∗ − → q T opGrp from the homotopy category of based spaces to the category of quasitop ological groups and contin uous homomorphisms [3]. Also, π τ 1 ( X, x 0 ) is the fundamental group endow ed with another top ology introduced by Brazas [4]. In fact, the functor π τ 1 remo ves the smallest num ber of op en sets from the top ology of π q top 1 ( X, x ) so that makes it a topological group. Let X b e a top ological space and A 1 , A 2 , ..., A n b e a finite collection of its sub- sets. The quotien t space X/ ( A 1 , ..., A n ) is obtained from X by identifying each of the sets A i to a p oin t. No w, let ( A, a ) b e a p oin ted subspace of ( X , a ) and p : ( X , a ) − → ( X/ A, ∗ ) b e the asso ciated quotient map. In this pap er, first we pro ve that if A is an open subset of X such that the closure of A , A , is path con- nected, then the image of p ∗ is dense in π q top 1 ( X/ A, ∗ ). Then by this fact, we sho w that the image of p ∗ is dense in π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ), where the A i ’s are op en subsets of X with path connected closures and p : X − → X/ ( A 1 , A 2 , ..., A n ) is the asso ciated quotient map. Second, we prov e that if A is a closed s ubset of a lo cally path connected and first coun table space X , then the image of p ∗ is also dense in π q top 1 ( X/ A, ∗ ). By the tw o previous results we can show that the image of p ∗ is dense in π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ), where X is first countable, connected, lo cally path connected and the A i ’s are open or closed subsets of X with disjoin t path con- nected closures. Moreov er, we give some conditions in whic h p ∗ is an epimorphism. Also, by some examples, w e show that p ∗ is not necessarily onto. Finally , w e give some applications of the abov e results to find out some prop erties of the topological fundamen tal group of the quotien t space X/ ( A 1 , A 2 , ..., A n ). In particular, we prov e that with the recen t assumptions on X and the A i ’s, π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is an indiscrete topological group when X is simply connected. It should b e men- tioned that since the topology of π τ 1 ( X, x 0 ) is coarser than π q top 1 ( X, x 0 ), the ab o v e results can b e obtained when w e replace π q top 1 with π τ 1 . 2. Not a tions and preliminaries F or a top ological space X , by a path in X we mean a con tinuous map α : [0 , 1] − → X . The p oin ts α (0) and α (1) are called the initial p oint and the terminal p oin t of α , resp ectively . A lo op α is a path with α (0) = α (1). F or a path α : [0 , 1] − → X , α − 1 denotes a path such that α − 1 ( t ) = α (1 − t ), for all t ∈ [0 , 1]. Denote [0 , 1] by I , tw o paths α, β : I − → X with the same initial and terminal p oin ts are called homotopic relativ e to end p oin ts if there exists a contin uous map F : I × I − → X suc h that F ( t, s ) =        α ( t ) s = 0 β ( t ) s = 1 α (0) = β (0) t = 0 α (1) = β (1) t = 1 . The homotop y is an equiv alent relation and the homotopy class containing a path α is denoted b y [ α ]. Since most of the homotopies that app ear in this paper ha ve this prop ert y and end p oints are the same, we drop the term “relative homotop y” for simplicity . F or paths α , β : I − → X with α (1) = β (0), α ∗ β denotes the concatenation of α and β that is a path from I to X such that ( α ∗ β )( t ) = α (2 t ), for all 0 ≤ t ≤ 1 / 2 and ( α ∗ β )( t ) = β (2 t − 1), for all 1 / 2 ≤ t ≤ 1. F or a pointed top ological space ( X , x ), let Ω( X, x ) b e the space of based maps from I to X with the compact-op en top ology . A subbase for this top ology consists ON TOPOLOGICAL FUNDAMENT AL GROUPS OF QUOTIENT SP ACES 3 of neigh b orhoo ds of the form h K , U i = { γ ∈ Ω( X , x ) | γ ( K ) ⊆ U } , where K ⊆ I is compact and U is open in X . When X is path connected and the basep oin t is clear, we just write Ω( X ) and we will consistently denote the constant path at x by e x . The top ological fundamental group of a pointed space ( X , x ) can be describ ed as the usual fundamental group π 1 ( X, x ) with the quotien t top ology with resp ect to the canonical map Ω( X, x ) − → π 1 ( X, x ) identifying homotop y classes of lo ops, denoted by π q top 1 ( X, x ). A basic account of top ological fundamental groups may b e found in [2], [5] and [3]. F or undefined notation, see [12]. Definition 2.1. ( [1] ). A quasitop olo gic al gr oup G is a gr oup with a top olo gy such that inversion G − → G , g 7→ g − 1 , is c ontinuous and multiplic ation G × G − → G is c ontinuous in e ach variable. A morphism of quasitop olo gic al gr oups is a c ontinuous homomorphism. Theorem 2.2. ( [3] ). π q top 1 is a functor fr om the homotopy c ate gory of b ase d top o- lo gic al sp ac es to the c ate gory of quasitop olo gic al gr oups. A space X is called semi-lo c al ly simply c onne cte d if for each p oin t x ∈ X , there is an op en neigh borho o d U of x such that the inclusion i : U  → X induces the trivial homomorphism i ∗ : π 1 ( U, x ) − → π 1 ( X, x ) or equiv alently a lo op in U can b e con tracted inside X . Theorem 2.3. ( [3] ). L et X b e a p ath c onne cte d sp ac e. If π q top 1 ( X, x ) is discr ete for some x ∈ X , then X is semi-lo c al ly simply c onne cte d. If X is lo c al ly p ath c onne cte d and semi-lo c al ly simply c onne cte d, then π q top 1 ( X, x ) is discr ete for al l x ∈ X . 3. Main Resul ts In this section, ( A, a ) is a p oin ted subspace of ( X, a ), p : ( X, a ) − → ( X/ A, ∗ ) is the canonical quotient map so that q := p | X − A : X − A − → X/ A − {∗} is a homeomorphism. Also, by applying the functor π q top 1 on p we ha v e a contin uous homomorphism p ∗ : π q top 1 ( X, a ) − → π q top 1 ( X/ A, ∗ ). Lemma 3.1. If A is an op en subset of X , then any lo ops α : I − → {∗} ⊆ X/ A b ase d at ∗ is nul lhomotopic. Pr o of. Define F : I × I − → X/ A by F ( t, s ) =  α ( t ) s = 0 ∗ s > 0 . If we pro v e that F is contin uous, then F is a homotopy b et ween α and e ∗ . F or this, let U b e an op en set in X/ A . W e sho w that F − 1 ( U ) is open in I × I . Case 1 : If ∗ ∈ U , then F − 1 ( U ) = F − 1 ( {∗} ) ∪ ( α − 1 ( U ) × { 0 } ) = (( I × (0 , 1]) ∪ ( α − 1 ( U ) × { 0 } ) = ( I × (0 , 1]) ∪ ( α − 1 ( U ) × I ) whic h is op en in I × I . Case 2 : If ∗ / ∈ U , then U ∩ ∂ {∗} = ∅ since if there exists x ∈ ∂ {∗} such that x ∈ U , then {∗}∩ U 6 = ∅ which is a con tradiction. Since U ∩{∗} = ( U ∩{∗} ) ∪ ( U ∩ ∂ {∗} ) = ∅ and α ( I ) ⊆ {∗} , w e hav e F − 1 ( U ) = ∅ .  4 H. TORABI, A. P AKDAMAN AND B. MASHA YEKHY Theorem 3.2. L et A b e an op en subset of X such that A is p ath c onne cte d, then for e ach a ∈ A the image of p ∗ is dense in π q top 1 ( X/ A, ∗ ) i.e. p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ A, ∗ ) . Pr o of. Step One : First, we show that for ev ery loop α : ( I , ∂ I ) − → ( X/ A, ∗ ) suc h that α − 1 ( {∗} c ) is connected, w e hav e [ α ] ∈ I m ( p ∗ ). By assumption and op enness of {∗} in X/ A , there exist s 1 , s 2 ∈ (0 , 1) such that α − 1 ( {∗} c ) = [ s 1 , s 2 ]. Since α − 1 ( {∗} c ) is a compact subset of I , it suffices to let s 1 = inf { α − 1 ( {∗} c ) } and s 2 = sup { α − 1 ( {∗} c ) } . Let α : [ s 1 , s 2 ] − → X b y α ( t ) = q − 1 ( α ( t )), then α ( s 1 ) , α ( s 2 ) ∈ A . Since if G is an op en neigh b orhoo d of α ( s 1 ) and G ∩ A = ∅ , then q ( G ) = p ( G ) is an open neighborho od of α ( s 1 ). Using contin uity of α , there exists an open neigh b orhoo d J of s 1 in I such that α ( J ) ⊆ G . On the other hand, by definition of s 1 , for all s < s 1 , α ( s ) = ∗ whic h implies that ∗ ∈ q ( G ) whic h is a contradiction since ∗ / ∈ I m ( q ). Similarly α ( s 2 ) ∈ A . Since A is path connected, there exist t w o paths λ 1 : [0 , s 1 ] − → A and λ 2 : [ s 2 , 1] − → A suc h that λ 1 (0) = λ 2 (1) = a, λ 1 ( s 1 ) = α ( s 1 ) and λ 2 (0) = α ( s 2 ). Define e α : I − → X b y e α ( t ) =    λ 1 ( t ) 0 ≤ t ≤ s 1 α ( t ) s 1 ≤ t ≤ s 2 λ 2 ( t ) s 2 ≤ t ≤ 1 . By gluing lemma e α is contin uous, so it remains to show that p ◦ e α ' α rel {∗} . Put α 1 = α | [0 ,s 1 ] , α 2 = α | [ s 2 , 1] and let ϕ 1 : [0 , 1] − → [0 , s 1 ] and ϕ 2 : [0 , 1] − → [ s 2 , 1] b e linear homeomorphisms such that ϕ 1 (0) = 0 and ϕ 2 (0) = s 2 , then p ◦ λ i ◦ ϕ i ' α i ◦ ϕ i , r el { 0 , 1 } since the ( p ◦ λ i ◦ ϕ i ) ◦ ( α i ◦ ϕ i ) − 1 ’s are lo ops in {∗} which by Lemma 3.1 are nullhomotopic. Step Two : By contin uity of α , α − 1 ( {∗} c ) is a closed subset of I . Connected subsets of I are in terv als or one p oin t sets, also connected comp onen ts of α − 1 ( {∗} c ) are closed in α − 1 ( {∗} c ) and so they are compact in I . Therefore a comp onen t of α − 1 ( {∗} c ) is either closed interv al or singleton. Given [ α ] ∈ π 1 ( X/ A, ∗ ), we show that there exists a sequence of homotopy classes of lo ops { [ α n ] } n ∈ N in I m ( p ∗ ) such that [ α n ] − → [ α ] in π q top 1 ( X/ A, ∗ ). W e claim that the n um b er of non-singleton comp onen ts of α − 1 ( {∗} c ) is coun t- able. Let S be the union of singleton comp onen ts of α − 1 ( {∗} c ) and for eac h n ∈ N , B n b e the set of non-singleton components of α − 1 ( {∗} c ) with length at least 1 /n . Eac h B n is finite since if B n is infinite, then it has at least n + 1 members. Therefore S C ∈ B n C ⊆ α − 1 ( {∗} c ) ⊆ I whic h implies that ( n + 1) × 1 /n ≤ X C ∈ B n diam ( C ) ≤ diam ( I ) = 1 whic h is a contradiction. Th us each B n is finite whic h implies that B = S n ∈ N B n is countable. Rename elements of B b y I i = [ a i , b i ], i ∈ J = { 1 , 2 , ..., s } , where s = | B | if B is finite and i ∈ N = J if B is infinite. F or every n ∈ J define α n ( t ) =  α ( t ) t ∈ S n i =1 [ a i , b i ] ∗ other wise. If B is finite, put α n = α s , for every n > s . W e claim that the α n ’s are contin uous. F or, if V ⊆ X/ A is op en, then i) If ∗ ∈ V , then α − 1 1 ( V ) = [0 , a 1 ) ∪ ( b 1 , 1] ∪ α | − 1 [ a 1 ,b 1 ] ( V ) ON TOPOLOGICAL FUNDAMENT AL GROUPS OF QUOTIENT SP ACES 5 whic h b y contin uit y of α is open in I . ii) If ∗ / ∈ V , then we show that α − 1 1 ( V ) = α | − 1 [ a 1 ,b 1 ] ( V ) = α | − 1 ( a 1 ,b 1 ) ( V ) which guar- an ties α − 1 ( V ) is open. F or this it suffices to sho w that α 1 ( a 1 ) , α 1 ( b 1 ) / ∈ V . F or each n ∈ N , α ( a n ) , α ( b n ) ∈ {∗} and { α ( a ) | a ∈ S } ⊆ {∗} . F or, if G is an open neigh b orhoo d of α ( a n ), then α − 1 ( G ) is an op en neigh b orhoo d of a n , so there exists ε > 0 such that ( a n − ε, a n + ε ) ⊆ α − 1 ( G ) or equiv alently α (( a n − ε, a n + ε )) ⊆ G . If ∗ / ∈ G , then ( a n − ε, a n + ε ) ⊆ α − 1 ( {∗} c ) which is a contradiction since [ a n , b n ] is a connected component of α − 1 ( {∗} c ). Similarly α ( b n ) ∈ {∗} for each n ∈ N and α ( S ) ⊆ {∗} . Thus if α 1 ( b 1 ) = α ( b 1 ) ∈ V , then V must meet {∗} whic h is a con- tradiction since ∗ / ∈ V . Therefore α 1 is a contin uous lo op such that [ α 1 ] ∈ I m ( p ∗ ). Similarly , all the α n ’s are con tin uous. Also, for every n ∈ N , [ α n ] is a product of n homotopy classes of lo ops whic h are similar to loops in tro duced in Step 1. This implies that [ α n ] ∈ I m ( p ∗ ) since I m ( p ∗ ) is a subgroup. No w w e show that the sequence { α n } conv erges to α . Let α ∈ h K, U i , where K is a compact subset of I and U is an open subset of X/ A , then i) If ∗ ∈ U , then for each n ∈ N , α n ∈ < K, U > since for each t ∈ K , α n ( t ) = α ( t ) or α n ( t ) = ∗ which in b oth cases α ( t ) ∈ U . ii) If K ⊆ α − 1 ( {∗} c ) and K ∩ S 6 = ∅ , then there exists a ∈ K ∩ S ⊆ α − 1 ( U ), so α ( a ) ∈ U , but α ( a ) ∈ {∗} and U is op en. Thus ∗ ∈ U and by (i), for eac h n, α n ( K ) ⊆ U . iii) If K ⊆ S ∞ n =1 I n and there exist n 1 , n 2 , ..., n s suc h that K ⊆ S s i =1 I n i , then by definition of the α n ’s, for each n ≥ max { n 1 , ..., n s } we hav e α n ( K ) ⊆ U . vi) If K ⊆ S ∞ n =1 I n and there exists an infinite subsequence { I n r } suc h that K ∩ I n r 6 = ∅ , then there is a sequence { x n r | x n r ∈ K ∩ I n r } such that it has a subsequence { x n r s } conv erges to an element of K , b say , by compactness of K . Since b ∈ K ⊆ α − 1 ( U ), there exists ε > 0 such that ( b − ε, b + ε ) ⊆ α − 1 ( U ). Also, there exists s 0 suc h that for each s ≥ s 0 , x n r s ∈ ( b − ε/ 2 , b + ε/ 2) since x n r s − → b . Since diam ( I n ) − → 0, there is n 0 suc h that for each n ≥ n 0 , diam ( I n ) < ε/ 2. Cho ose s 1 ∈ N such that s 1 ≥ s 0 and n r s 1 ≥ n 0 , then x n r s 1 ∈ ( b − ε/ 2 , b + ε/ 2). Also x n r s 1 ∈ K ∩ I n r s 1 and diam ( I n r s 1 ) < ε/ 2, so a n r s 1 ∈ I n r s 1 ⊆ ( b − ε, b + ε ) ⊆ α − 1 ( U ) whic h implies that α ( a n r s 1 ) ∈ U and therefore ∗ ∈ U since α ( a n r s 1 ) ∈ ∂ ( {∗} ). Using the last (i) w e hav e α n ( K ) ⊆ U , for each n ∈ N .  Definition 3.3. L et X b e a top olo gic al sp ac e and A 1 , A 2 , ..., A n b e any subsets of X , n ∈ N . By the quotient sp ac e X/ ( A 1 , ..., A n ) we me an the quotient sp ac e obtaine d fr om X by identifying e ach of the sets A i to a p oint. Also, we denote the asso ciate d quotient map by p : X − → X/ ( A 1 , A 2 , ..., A n ) . Corollary 3.4. If A 1 , A 2 ar e op en subsets of a p ath c onne cte d sp ac e X such that A 1 , A 2 ar e p ath c onne cte d. Then for every a ∈ A 1 ∪ A 2 the fol lowing e quality holds: p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 ) , ∗ ) . Pr o of. W e can assume that the A i ’s are disjoint. If they are not disjoint, the result follo ws from Theorem 3.2. Let p 1 : X − → X/ A 1 , p 2 : X/ A 1 − → X/ ( A 1 , A 2 ) be asso ciated quotient maps and a 1 = a ∈ A 1 . By Theorem 3.2, ( p 1 ) ∗ π q top 1 ( X, a 1 ) = π q top 1 ( X/ A 1 , ∗ 1 ), where ∗ 1 = p 1 ( a 1 ). Since X is path connected, so is X/ A 1 . Also, p 1 ( A 2 ) is an op en subset of X/ A 1 and the closure of p 1 ( A 2 ) in X/ A 1 is path 6 H. TORABI, A. P AKDAMAN AND B. MASHA YEKHY connected. Let a 2 ∈ p 1 ( A 2 ), then ( p 2 ) ∗ π q top 1 ( X/ A 1 , a 2 ) = π q top 1 ( X/ ( A 1 , A 2 ) , ∗ 2 ), where ∗ 2 = p 2 ( a 2 ). Since X/ A 1 is path connected, there exists a homeomor- phism ϕ 1 : π q top 1 ( X/ A 1 , ∗ 1 ) − → π q top 1 ( X/ A 1 , a 2 ) b y ϕ 1 ([ α ]) = [ γ ∗ α ∗ γ − 1 ], where γ is a path from a 2 to ∗ 1 . W e ha v e ( p 2 ) ∗ ◦ ϕ 1 ◦ ( p 1 ) ∗ ( π q top 1 ( X, a )) ⊇ (( p 2 ) ∗ ◦ ϕ 1 )(( p 1 ) ∗ ( π q top 1 ( X, a ))) = (( p 2 ) ∗ ◦ ϕ 1 )( π q top 1 ( X/ A 1 , ∗ 1 )) = ( p 2 ) ∗ π q top 1 ( X/ A 1 , a 2 ) = I m ( p 2 ) ∗ whic h implies that I m ( p 2 ) ∗ ◦ ϕ 1 ◦ ( p 1 ) ∗ is dense in π 1 ( X/ ( A 1 , A 2 ) , ∗ 2 ). If γ 0 = p 2 ◦ γ , then ϕ 2 : π q top 1 ( X/ ( A 1 , A 2 ) , ∗ 2 ) − → π q top 1 ( X/ ( A 1 , A 2 ) , ∗ 1 ) by ϕ 2 ([ α ]) = [ γ 0− 1 ∗ α ∗ γ 0 ] is a homeomorphism. Hence I m ( ϕ 2 ◦ ( p 2 ) ∗ ◦ ϕ 1 ◦ ( p 1 ) ∗ ) is dense in π q top 1 ( X/ ( A 1 , A 2 ) , ∗ 1 ). Moreo ver ϕ 2 ◦ ( p 2 ) ∗ ◦ ϕ 1 ◦ ( p 1 ) ∗ = p ∗ whic h implies that I m ( p ∗ ) is dense in π q top 1 ( X/ ( A 1 , A 2 ) , ∗ ), as desired.  By induction and Corollary 3.4, we hav e the follo wing results. Corollary 3.5. L et A 1 , A 2 , ..., A n b e op en subsets of a p ath c onne cte d sp ac e X such that the A i ’s ar e p ath c onne cte d for e ach i = 1 , 2 , ..., n . Then for any a ∈ S n i =1 A i the fol lowing e quality holds: p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) . Corollary 3.6. L et A 1 , A 2 , ..., A n b e op en subsets of a c onne cte d, lo c al ly p ath c on- ne cte d sp ac e X such that the A i ’s ar e p ath c onne cte d for every i = 1 , 2 , ..., n . If X/ ( A 1 , A 2 , ..., A n ) is semi-lo c al ly simply c onne cte d, then for e ach a ∈ S n i =1 A i , p ∗ : π q top 1 ( X, a ) − → π q top 1 ( X/ ( A 1 , A 2 , ..., A n , ∗ ) is an epimorphism. Pr o of. Let U be an op en neighborho o d of ¯ x ∈ ( X/ A 1 ) \ {∗} . Since X is lo cally path connected, there is a path connected op en neigh b orhoo d e U ⊆ p − 1 ( U ) of x = q − 1 ( ¯ x ) such that e U ∩ A 1 = ∅ . Then V := p ( e U ) = q ( e U ) ⊆ U is a path connected op en neigh borho o d of ¯ x . X/ A 1 is lo cally path connected at ∗ since {∗} is an op en subset of X/ A 1 . Let U b e an op en neigh b orhoo d of ¯ x ∈ ∂ ( {∗} ), then there exists a path connected open neighborho od e U ⊆ p − 1 ( U ) of x . Since p − 1 ( p ( e U )) = e U ∪ A 1 , p ( e U ) is a path connected op en neighborho o d of ¯ x in U . Therefore X/ A 1 is lo cally path connected. Similarly X/ ( A 1 , A 2 , ..., A n ) is connected, lo cally path connected. Since X/ ( A 1 , A 2 , ..., A n ) is a connected, semi-lo cally simply connected and lo cally path connected space, by Theorem 2.3, π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is a discrete top ological group whic h implies that I m ( p ∗ ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) b y Corollary 3.5.  In the following example we show that with the assumptions of Theorem 3.2, p ∗ is not necessarily onto. Example 3.7. L et A n = { 1 / (2 n − 1) , 1 / 2 n } × [0 , 1 + 1 / 2 n ] S [1 / 2 n, 1 / 2 n − 1] × { 1 + 1 / 2 n } for e ach n ∈ N . Consider X = ( S n ∈ N A n ) S { 0 } × [0 , 1] S [0 , 1] × { 0 } with a = (0 , 0) as the b ase p oint and A = { ( x, y ) ∈ X | y < 1 } (se e Figur e 1). A is an op en subset of X with p ath c onne cte d closur e. Assume I n = (1 / 2 + 1 / 2( n + 1) , 1 / 2 + 1 / 2 n ] and f n b e a home omorphism fr om I n to A n − { (1 / 2 n, 0) } for every n ∈ N . Define f : I − → X by f ( t ) =  the point (0 , 2 t ) t ∈ [0 , 1 / 2] , f n ( t ) t ∈ I n . We claim that α = p ◦ f is a lo op in X/ A at ∗ . It suffic es to show that α is c ontinuous on t = 1 / 2 and b oundary p oints of I n ’s sinc e f is c ontinuous on [0 , 1 / 2) ON TOPOLOGICAL FUNDAMENT AL GROUPS OF QUOTIENT SP ACES 7 a A 1 A 2 A 3 Figure 1. and by gluing lemma on S int ( I n ) . Sinc e α is lo c al ly c onstant at t = 1 / 2 + 1 / 2 n for e ach n ∈ N , α is c ontinuous at b oundary p oints of I n . F or e ach op en neighb orho o d G of f (1 / 2) = (0 , 1) in X , ther e exists n 0 ∈ N such that G c ontains A n T A c for n > n 0 . Ther efor e c ontinuity at t = 1 / 2 fol lows fr om α (1 / 2) ∈ {∗} . Now let B ⊆ N and define g B ( t ) =  ( p ◦ f )( t ) t ∈ S m ∈ B I m , ∗ other wise. Then g B is c ontinuous and for B 1 , B 2 ⊆ N such that B 1 6 = B 2 , [ g B 1 ] 6 = [ g B 2 ] which implies that π 1 ( X/ A, ∗ ) is unc ountable. But by c omp actness of I , a given p ath in X c an tr averse finitely many of the A n ’s and ther efor e π 1 ( X, a ) is a fr e e gr oup on c ountably many gener ators which implies that p do es not induc e a surje ction of fundamental gr oups. Let ( X , x ) b e a p oin ted top ological space such that { x } is closed. If α : [0 , 1] − → X is a lo op in X based at x , then α − 1 ( { x } ) is a closed subset of [0 , 1]. Its com- plemen t, α − 1 ( { x } c ) is therefore the union of a countable collection of disjoint op en in terv als. W e denote this collection of in terv als by W α . Definition 3.8. L et ( X, x ) b e a p ointe d top olo gic al sp ac e. A lo op α in X b ase d at x is c al le d semi-simple if W α = { (0 , 1) } and is c al le d geometrically simple if W α has one element. If W α is finite, then the lo op α is c al le d geometrically finite [11] . Lemma 3.9. Every ge ometric al ly simple lo op is homotopic to a semi-simple lo op. Pr o of. Let α be a geometrically simple loop at x ∈ X . Then there are r , s ∈ [0 , 1] suc h that α − 1 ( { x } c ) = ( r, s ) and α ( r ) = α ( s ) = x . Let β := α | [ r,s ] and ϕ : [0 , 1] − → [ r, s ] b e a linear homeomorphism, then β ◦ ϕ is a semi-simple lo op at x and α ' β ◦ ϕ .  In the sequel, for a semi-simple lo op α : I − → X/ A denote e α = q − 1 ◦ α | (0 , 1) : (0 , 1) − → ( X − A ), where A is a closed subset of a top ological space X . Lemma 3.10. L et A ⊆ X b e a close d subset of X and α b e a semi-simple lo op at ∗ in X/ A . If l im t → 0 e α ( t ) and l im t → 1 e α ( t ) do not exist, then for e ach t 0 ∈ (0 , 1) , 8 H. TORABI, A. P AKDAMAN AND B. MASHA YEKHY ther e ar e b 0 , b 1 ∈ A such that b 0 is a limit p oint of e α ((0 , t 0 )) and b 1 is a limit p oint of e α (( t 0 , 1)) . Pr o of. Let t 0 ∈ (0 , 1) and by contrary supp ose that each b ∈ A has an op en neigh- b orhoo d G b suc h that G b ∩ e α ((0 , t 0 )) = ∅ . Then G = S b ∈ A G b is an op en neigh bor- ho od of A and so p ( G ) is an op en neighborho o d of ∗ (since p − 1 ( p ( G )) = G ) such that do es not intersect α ((0 , t 0 )) which is a con tradiction to contin uit y of α .  Theorem 3.11. If A is a close d p ath c onne cte d subset of a lo c al ly p ath c onne cte d sp ac e X such that every p oint of A has a c ountable lo c al b ase in X , then for e ach a ∈ A we have p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ A, ∗ ) . Pr o of. Step One : Let [ α ] ∈ π 1 ( X/ A, ∗ ), where α is a semi-simple lo op in X/ A at ∗ . Case 1 : Assume a 0 = lim t → 0 e α ( t ) and a 1 = lim t → 1 e α ( t ) exist, so a 0 , a 1 ∈ A and we can define a path α : I − → X such that α | (0 , 1) = e α , α (0) = a 0 , α (1) = a 1 . Since A is path connected, there exist paths λ 0 , λ 1 : I − → A suc h that λ 0 is a path from a to a 0 and λ 1 is a path from a 1 to a . Therefore λ 0 ∗ α ∗ λ 1 is a loop at a such that p ∗ ([ λ 0 ∗ α ∗ λ 1 ]) = [ α ]. Case 2 : If at least one of the ab o v e limits do es not exist, then we make a sequence { [ α n ] } n ∈ N in I m ( p ∗ ) so that con verges to [ α ]. Without lost of generality , w e can assume that a 0 = l im t → 0 e α ( t ) exists and a 1 ∈ A is a limit p oint of e α ((1 / 2 , 1)) by Lemma 3.10. W e can define a con tin uous map α : [0 , 1) − → X such that α | (0 , 1) = e α , α (0) = a 0 . By hypothesis, there is a countable lo cal base { O i } i ∈ N at a 1 . Let { G i } i ∈ N b e a sequence of op en neighborho o ds of a 1 suc h that G i = O 1 ∩ ... ∩ O i . Since X is lo cally path connected and the G i ’s are op en neighborho o ds of a 1 , there exist path connected op en neighborho ods G 0 i ⊆ G i of a 1 . Since the p oin t a 1 is a limit p oin t, there are t i ∈ (1 / 2 , 1) such that e α ( t i ) ∈ G 0 i , t i < t i +1 , t n − → 1 and there are paths γ i : [ t i , 1] − → G 0 i from α ( t i ) to a 1 , for all i ∈ N . Since A is path connected, there exist paths λ 0 , λ 1 : I − → A such that λ 0 is a path from a to a 0 and λ 1 is a path from a 1 to a . Let α n := λ 0 ∗ α | [0 ,t n ] ◦ ξ n ∗ γ n ◦ ζ n ∗ λ 1 , where ξ n : [0 , 1] − → [0 , t n ] and ζ n : [0 , 1] − → [ t n , 1] are increasing linear homeomorphisms. Note that every α n is a lo op in X at a and if β n := p ◦ ( γ n ◦ ζ n ), then α 0 n := p ◦ α n = e ∗ ∗ α | [0 ,t n ] ◦ ξ n ∗ β n ∗ e ∗ is a lo op in X at ∗ and p ∗ ([ α n ]) = [ α 0 n ]. Define α n : I − → X/ A by α n ( t ) =  α ( t ) 0 ≤ t ≤ t n p ◦ γ n ( t ) t n ≤ t ≤ 1 whic h is a lo op at ∗ and [ α 0 n ] = [ α n ]. Thus it suffices to prov e that α n − → α . If α ∈ h K, U i , where K is a compact subset of [0 , 1] and U is an op en subset of X/ A , then i) If ∗ / ∈ U , then K ∩ α − 1 ( ∗ ) = ∅ . Let m ∈ N such that t m ≥ max K . Since for eac h t ∈ K , t ≤ t m , we hav e α n ( t ) = α ( t ) ∈ U , for all n > m which implies that α n ( K ) = α ( K ) ⊆ U , for eac h n > m . ii) If ∗ ∈ U , then p − 1 ( U ) is an op en neighborho o d of A , th us there exists m ∈ N suc h that G 0 n ⊆ p − 1 ( U ), for each n ≥ m . Therefore I m ( γ n ) ⊆ G 0 n whic h implies that I m ( p ◦ λ n ) ⊆ U . Thus for all t ∈ K and n ≥ m we hav e α n ( t ) =  α ( t ) ∈ U t ∈ [0 , t n ] ( p ◦ γ n )( t ) ∈ U t ∈ [ t n , 1] . ON TOPOLOGICAL FUNDAMENT AL GROUPS OF QUOTIENT SP ACES 9 Therefore for a semi-simple lo op α in X/ A we hav e [ α ] ∈ I m ( p ∗ ). Similarly , for ev ery lo op α such that α − 1 ( {∗} ) is finite, [ α ] ∈ I m ( p ∗ ) which implies that the homotop y class of every geometrically finite lo op b elongs to I m ( p ∗ ) by Lemma 3.9. Step Two : If α is not geometrically finite, W α is countable since every op en subset of I is a countable union of op en interv als. Let I j = L j where W α = { L j | j ∈ N } and let α j ( t ) =  α ( t ) t ∈ I 1 ∪ ... ∪ I j ∗ other wise, then [ α j ] ∈ I m ( p ∗ ) since the α j ’s are geometrically finite. Since ( I m ( p ∗ )) = I m ( p ∗ ), it suffices to show that α j − → α . F or, if α ∈ h K , U i for a compact subset K of [0 , 1] and an op en subset U of X/ A , then i) If ∗ ∈ U , then for eac h t ∈ K and j ∈ N , α j ( t ) takes v alue α ( t ) or ∗ which in b oth cases belongs to U , so α j ( K ) ⊆ U , for all j ∈ N . ii) If ∗ / ∈ U , then K ∩ α − 1 ( {∗} ) = ∅ , so K ⊆ ∪ j L j . By compactness of K we ha ve K ⊆ ∪ s L j s , for s = 1 , 2 , ..., n K . Let M = max { j s | s = 1 , 2 , ..., n K } , then α j ( K ) = α ( K ) ⊆ U , for eac h j ≥ M .  Corollary 3.12. L et A 1 , A 2 , ..., A n b e disjoint p ath c onne cte d, close d subsets of a first c ountable, c onne cte d, lo c al ly p ath c onne cte d sp ac e X . Then for every a ∈ S n i =1 A i the fol lowing e quality holds: p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) . Pr o of. Let p i : X/ ( A 1 , A 2 , ..., A i − 1 ) − → X/ ( A 1 , A 2 , ..., A i ). Since ev ery p oin t of eac h p i − 1 ( A i ) has a countable local base in the connected, lo cally path connected space X/ ( A 1 , A 2 , ..., A i − 1 ), by Theorem 3.11 the result holds.  Corollary 3.13. L et A 1 , A 2 , ..., A n b e disjoint p ath c onne cte d, close d subsets of a first c ountable, c onne cte d, lo c al ly p ath c onne cte d sp ac e X such that X/ ( A 1 , A 2 , ..., A n ) is semi-lo c al ly simply c onne cte d. Then for e ach a ∈ S n i =1 A i , p ∗ : π 1 ( X, a ) − → π 1 ( X/ ( A 1 , A 2 , ..., A n , ∗ ) is an epimorphism. Pr o of. Since X/ ( A 1 , A 2 , ..., A n ) is connected, lo cally path connected and semi- lo cally simply connected space, π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is a discrete top ological group whic h implies that I m ( p ∗ ) = π 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) by Corollary 3.12.  In the following example, w e show that the condition “path connectedness for A ” is necessary in Theorem 3.11. Example 3.14. L et A = { (1 , 0) , (0 , 1) } ⊂ X = S 1 . Cle arly X/ A is home omorphic to the Figur e 8 sp ac e, S 1 W S 1 . Sinc e X and X/ A ar e lo c al ly p ath c onne cte d and semi-lo c al ly simply c onne cte d p ∗ : π q top 1 ( X, 0) ∼ = Z − → π q top 1 ( X/ A, ∗ ) ∼ = Z ∗ Z is a c ontinuous homomorphism of discr ete top olo gic al sp ac es. Sinc e the fr e e pr o duct Z ∗ Z is not ab elian, p ∗ is not onto and sinc e π q top 1 ( X/ A, ∗ ) is discr ete, I m ( p ∗ ) is not dense in π q top 1 ( X/ A, ∗ ) . In the follo wing example, we show that the condition “lo cally path connectedness for X ” is necessary in Theorem 3.11. 10 H. TORABI, A. P AKDAMAN AND B. MASHA YEKHY Example 3.15. L et X 1 = { ( x, sin (2 π /x )) ∈ R 2 | 0 < x ≤ 1 } , X 2 = { ( x, y ) ∈ R 2 | x 2 + y 2 4 = 1 , y ≤ 0 } , X 3 = { ( x, 0) ∈ R 2 | − 1 ≤ x ≤ 0 } and A = { (0 , y ) ∈ R 2 | − 1 ≤ y ≤ 1 } . If X = X 1 ∪ X 2 ∪ X 3 ∪ A , then π 1 ( X, x 0 ) = 0 and π 1 ( X/ A, ∗ ) ∼ = Z . Sinc e X/ A is a lo c al ly p ath c onne cte d and semi-lo c al ly simply c onne cte d sp ac e, π q top 1 ( X/ A, ∗ ) is discr ete which implies that p ∗ ( π q top 1 ( X, x 0 )) 6 = π q top 1 ( X/ A, ∗ ) . In the next example, w e show that with the assumptions of Theorem 3.11, p ∗ is not necessarily an epimorphism and hence the hypothesis semi-lo cally simply connectedness in Corollary 3.13 is essential. Example 3.16. L et C n = { ( x, y ) ∈ R 2 | ( x − 1 n ) 2 + y 2 = 1 n 2 } , for n ∈ N , H E o = S n ∈ N C 2 n − 1 , H E e = S n ∈ N C 2 n and X = ( H E 0 × { 0 } ) ∪ ( H E e × { 1 } ) ∪ A , wher e A = ( { (0 , 0) } × I ) . One c an e asily se e that X/ A is the Hawaiian Earing sp ac e. L et α b e the lo op in X/ A that tr averse p ( C 1 ) , p ( C 2 ) , ... in asc ending or der. By the structur e of the fundamental gr oup of the Hawaiian Earing [7] we have [ α ] / ∈ I m ( p ∗ ) sinc e if p ∗ ([ β ]) = [ α ] , then the lo op β must tr averse infinitely many times A which is a c ontr adiction to the c ontinuity of β . Corollary 3.17. L et A 1 , A 2 , ..., A n b e subsets of a first c ountable, c onne cte d, lo c al ly p ath c onne cte d sp ac e X with disjoint p ath c onne cte d closur e such that e ach A i is close d or op en. Then for any a ∈ S n i =1 A i we have p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) . Pr o of. By changing the order, w e can assume that A 1 , ..., A k are closed and A k +1 , ...A n are op en, for a 1 ≤ k ≤ n . By applying Corollary 3.12 w e ha v e q ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A k ) , ∗ ) , where q : X → X/ ( A 1 , A 2 , ..., A k ) is the natural quotient map. Consider the natural quotien t map r : X/ ( A 1 , A 2 , ..., A k ) → X/ ( A 1 , ..., A k , ..., A n ). Note that p = r ◦ q and since the A j ’s hav e disjoint path connected closures, A j is also path connected in X/ ( A 1 , ..., A k ), for all j > k . Now, using Corollary 3.5 the result holds.  Remark 3.18. Note that sinc e the top olo gy of π τ 1 ( X, x ) is c o arser than π q top 1 ( X, x ) , the r esults of this se ction c an b e r estate d for π τ 1 when we r eplac e π q top 1 with π τ 1 . 4. some applica tions It seems interesting to inv estigate on the top ology of quasitop ological funda- men tal groups and some p eople hav e found some properties of this topology (see [2, 3, 4, 5, 10, 13, 14, 17]). In this section, w e intend to giv e some applications of the results of the previous section to find out some prop erties of the top ological fundamen tal group of the quotient space X/ ( A 1 , A 2 , ..., A n ). By ( X , A 1 , A 2 , ..., A n ) w e mean an ( n + 1)- tuple of spaces with one of the following conditions ( ♣ ): (i): The A i ’s are op en subsets of X with path connected closures. (ii): X is a connected, locally path connected, first countable space and the A i ’s are closed subsets of X with disjoint path connected closures. Theorem 4.1. F or an ( n + 1) - tupl e of sp ac es ( X , A 1 , A 2 , ..., A n ) with the assump- tion ( ♣ ) , if X is simply c onne cte d, then π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is an indiscr ete top olo gic al gr oup. ON TOPOLOGICAL FUNDAMENT AL GROUPS OF QUOTIENT SP ACES 11 Pr o of. Since X is simply connected, p ∗ π q top 1 ( X, a ) = { [ e ∗ ] } , where e ∗ is the con- stan t lo op at ∗ in X/ ( A 1 , A 2 , ..., A n ). Then b y Corollaries 3.5 and 3.17 { [ e ∗ ] } is a dense subset of π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ). Since π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is a quasitop ological group, for every [ α ] ∈ π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ), the left m ultiplication L [ α ] : π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) − → π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) giv en b y L [ α ] ([ β ]) = [ α ∗ β ] is a homeomorphism which implies that { [ α ] } is also dense in π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ). Hence every nonempt y op en subset of π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) con tains every elemen t [ α ] of π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) whic h implies that π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is an indiscrete top ological group.  Theorem 4.2. F or an ( n + 1) - tupl e of sp ac es ( X , A 1 , A 2 , ..., A n ) with the assump- tion ( ♣ ) , if π q top 1 ( X, a ) is c omp act and π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is Hausdorff, then the quasitop olo gic al fundamental gr oup π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is either a discr ete top olo gic al gr oup or unc ountable. Pr o of. If π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) has at least one isolated point, then ev ery singleton is op en since left translations L [ α ] : π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) − → π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) are homeomorphisms, for every [ α ] ∈ π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ). Th us π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is a discrete top ological group. It is a well-kno wn result that a nonempty compact Hausdorff space without isolated points is uncountable [12, Theorem 27.7]. Hence if π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) has no isolated p oin ts, then in order to show that π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is uncoun table it is enough to show that π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is compact. By Corollaries 3.5 and 3.17, p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) . Since π q top 1 ( X, a ) is compact and p ∗ is con tin uous p ∗ π q top 1 ( X, a ) is compact in π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ). Since π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is Hausdorff, p ∗ π q top 1 ( X, a ) is closed in π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) and so p ∗ π q top 1 ( X, a ) = π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ). Hence π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is compact and so it is uncountable.  Corollary 4.3. F or an ( n + 1) - tupl e of sp ac es ( X, A 1 , A 2 , ..., A n ) with the assump- tion ( ♣ ) , if π q top 1 ( X, a ) is a c omp act, c ountable quasitop olo gic al gr oup, then either X/ ( A 1 , A 2 , ..., A n ) is semi-lo c al ly simply c onne cte d or π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is not Hausdorff. Pr o of. Let π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) b e Hausdorff, then by a similar pro of of Theorem 4.2 p ∗ is on to. Therefore π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is countable since π q top 1 ( X, a ) is countable. Theorem 4.2 implies that π q top 1 ( X/ ( A 1 , A 2 , ..., A n ) , ∗ ) is a discrete top ological groups. Hence by Theorem 2.3 X/ ( A 1 , A 2 , ..., A n ) is semi- lo cally simply connected.  If U is an open cov er of a connected and lo cally path connected space X , then the subgroup of π 1 ( X, x ) consisting of all homotop y classes of lo ops that can b e represen ted b y a pro duct of the follo wing type: n Y j =1 u j v j u − 1 j , 12 H. TORABI, A. P AKDAMAN AND B. MASHA YEKHY where the u j ’s are arbitrary paths (starting at the base p oin t x ) and each v j is a lo op inside one of the neigh b orhoo ds U i ∈ U , is called the Spanier group with resp ect to U , denoted by π ( U , x ) [16, 8]. Definition 4.4. [16, 8] The Sp anier gr oup of the sp ac e X which we denote it by π sp 1 ( X, x ) , is define d as fol lows: π sp 1 ( X, x ) = \ open cov ers U π ( U , x ) . The authors [10] introduce Spanier spaces whic h are space s suc h that their Spanier groups are equal to their fundamental groups. Also, the authors prov e that for a connected and lo cally path connected space X , { [ e x ] } ⊆ π sp 1 ( X, x ). Hence, for an ( n + 1)- tupl e of spaces ( X, A 1 , A 2 , ..., A n ) with the assumption ( ♣ ), where X is simply connected, we hav e π q top 1 ( X/ ( A 1 , ..., A n ) , ∗ ) = p ∗ π 1 ( X, x ) = { [ e x ] } ⊆ π sp 1 ( X/ ( A 1 , ..., A n ) , ∗ ) . Clearly simply connected spaces are Spanier spaces which we can call them trivial Spanier spaces. It is in teresting for the authors to obtain some w a ys to construct non trivial Spanier spaces. The follo wing result which is an immediate consequence of the ab o ve argument gives a w a y to construct some Spanier spaces from simply connected spaces. Theorem 4.5. F or an ( n + 1) - tupl e of sp ac es ( X , A 1 , A 2 , ..., A n ) with the assump- tion ( ♣ ) , if X is simply c onne cte d, then X/ ( A 1 , A 2 , ..., A n ) is a Sp anier sp ac e. In the following example, we sho w that there exists a simply connected, locally path connected metric space X with a closed path connected subspace A such that X/ A is not simply connected and by Theorem 4.1 π q top 1 ( X/ A, ∗ ) is an indiscrete top ological group. Hence X/ A is a nontrivial Spanier space. Example 4.6. Using the definitions of Example 3.16, let C H E o and C H E e b e c ones over H E o and H E e with height 1 2 and let X = C H E o ∪ C H E e ∪ A . By the van Kamp en the or em, X is simply c onne cte d, but X/ A is not simply c onne cte d (se e [9] ). Henc e X/ A is a nontrivial Sp anier sp ac e. Ac kno wledgemen ts. The authors would like to thank the referee for the v alu- able commen ts and suggestions whic h impro v ed the man uscript and made it more readable. References [1] A. Arhangelskii, M. Tkachenko , T op ological Groups and Related Structures, Atlantis Studies in Mathematics , 2008. [2] D. 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