On locally 1-connectedness of quotient spaces and its applications to fundamental groups

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1 Filomat xx (yyyy), zzz zzz DOI (will be added later) Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On locally -connectedness of quotient spaces and its applications to fundamental groups Ali Pakdaman a, Hamid Torabi b, Behrooz Mashayekhy b arxiv: v [math.at] 3 Apr 204 a Department of Mathematics, Faculty of Sciences, Golestan University, P.O.Box 55, Gorgan, Iran. b Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box , Mashhad, Iran. Abstract. Let X be a locally -connected metric space and A, A 2,..., A n be connected, locally path connected and compact pairwise disjoint subspaces of X. In this paper, we show that the quotient space X/(A, A 2,..., A n ) obtained from X by collapsing each of the sets A i s to a point, is also locally -connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space X/(A, A 2,..., A n ).. Introduction and Motivation Let be an equivalent relation on a topological space X. Then one can consider the quotient topological space X/ and the quotient map p : X X/. In general, quotient spaces are not well behaved and it seems interesting to determine which topological properties of the space X may be transferred to the quotient space X/. For example, a quotient space of a simply connected or contractible space need not share those properties. Also, a quotient space of a locally compact space need not be locally compact. Let A, A 2,..., A n be a finite collection of pairwise disjoint subspaces of a topological space X. Then by X/(A, A 2,..., A n ) we mean the quotient space obtained from X by identifying each of the sets A i s to a point. The main result of this paper is as follows: Theorem A. If X is a locally -connected metric space and A, A 2,..., A n are connected, locally path connected and compact pairwise disjoint subspaces of X, then the quotient space X/(A, A 2,..., A n ) is locally -connected. The quasitopological fundamental group π qtop (X, x) of a based space (X, x) is the familiar fundamental group π (X, x) endowed with the quotient topology with respect to the canonical function Ω(X, x) π (X, x) identifying path components on the loop space Ω(X, x) with the compact-open topology (see [3]). It is known that this construction gives rise a homotopy invariant functor π qtop : htop qtopgrp from the homotopy category of based spaces to the category of quasitopological group and continuous homomorphism [4]. By 200 Mathematics Subject Classification. Primary 55P65; Secondary 55Q52, 55Q70 Keywords. Locally -connected, Quotient space, Quasitopological fundamental group Received: dd Month yyyy; Revised: dd Month yyyy; Accepted: dd Month yyyy Communicated by Ljubisa Kocinac addresses: a.pakdaman@gu.ac.ir (Ali Pakdaman), hamid torabi86@yahoo.com (Hamid Torabi), bmashf@um.ac.ir (Behrooz Mashayekhy)

2 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 2 applying the above functor on the quotient map p : X X/, we have a continuous homomorphism p : π qtop (X, x) π qtop (X/, ). Recently, the authors [0] proved that if X is a first countable, connected, locally path connected space and A, A 2,..., A n are disjoint path connected, closed subspaces of X, then the image of p is dense in π qtop (X/(A, A 2,..., A n )). By using the results of [0] and Theorem A we have the primary application of the main result of this paper as follows: Theorem B. If X is a locally -connected metric space and A, A 2,..., A n are disjoint connected, locally path connected and compact subspaces of X, then for each a n i= A i the continuous homomorphism is an epimorphism. p : π qtop (X, a) π qtop (X/(A, A 2,..., A n ), ) Also, by some examples, we show that p is not necessarily onto if the hypotheses of the above theorem do not hold. Finally, we give some applications of the above results to explore the properties of the fundamental group of the quotient space X/(A, A 2,..., A n ). In particular, we prove that if X is a separable, connected, locally -connected metric space and A,..., A n are connected, locally path connected and compact subsets of X, then π (X/(A, A 2,..., A n ), ) is countable. Moreover, π (X/(A, A 2,..., A n ), ) is finitely presented. Note that with the recent assumptions on X and the A i s, X/(A, A 2,..., A n ) is simply connected when X is simply connected.

3 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 3 2. Notations and preliminaries For a pointed topological space (X, x), by a path we mean a continuous map α : [0, ] X. The points α(0) and α() are called the initial point and the terminal point of α, respectively. A loop α is a path with α(0) = α(). For a path α : [0, ] X, α denotes a path such that α(t) = α( t), for all t [0, ]. Denote [0, ] by I, two paths α, β : I X with the same initial and terminal points are called homotopic relative to end points if there exists a continuous map F : I I X such that F(t, s) = α(t) s = 0 β(t) s = α(0) = β(0) t = 0 α() = β() t =. The homotopy class containing a path α is denoted by [α]. Since most of the homotopies that appear in this paper have this property and end points are the same, we drop the term relative homotopy for simplicity. For paths α, β : I X with α() = 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(2t), for 0 t /2 and (α β)(t) = β(2t ), for /2 t. For a space (X, x), let Ω(X, x) be the space of based maps from I to X with the compact-open topology. A subbase for this topology consists of neighborhoods of the form K, U = γ Ω(X, x) γ(k) U}, where K I is compact and U is open in X. We will consistently denote the constant path at x by e x. The quasitopological fundamental group of a pointed space (X, x) may be described as the usual fundamental group π (X, x) with the quotient topology with respect to the canonical map Ω(X, x) π (X, x) identifying homotopy classes of loops, denoted by π qtop (X, x). A basic account of quasitopological fundamental groups may be found in [3], [4] and [5]. For undefined notation, see [9]. Definition 2.. [] A quasitopological group G is a group with a topology such that inversion G G,, is continuous and multiplication G G G is continuous in each variable. A morphism of quasitopological groups is a continuous homomorphism. Theorem 2.2. [4] π qtop is a functor from the homotopy category of based topological spaces to the category of quasitopological groups. A space X is semi-locally simply connected if for each point x X, there is an open neighborhood U of x such that the inclusion i : U X induces the zero map i : π (U, x) π (X, x) or equivalently a loop in U can be contracted inside X. Theorem 2.3. [4] Let X be a connected and locally path connected topological space. The quasitopological fundamental group π qtop (X, x) is discrete for some x X if and only if X is semi-locally simply connected. Theorem 2.4. [0] Let A, A 2,..., A n be disjoint path connected, closed subsets of a first countable, connected, locally path connected space X such that X/(A, A 2,..., A n ) is semi-locally simply connected, then for each a n i= A i, p : π (X, a) π (X/(A, A 2,..., A n, ) is an epimorphism. In this note all homotopies are relative. Also, when α : [a, b] X is a path, for brevity by ˆα we mean α ϕ where ϕ : I [a, b] is a suitable linear homeomorphism.

4 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 4 3. Constructing homotopies This section is dedicated to some technical lemmas about homotopy properties of loops. Assume that α is a path in X and t 0, t,..., t n are real numbers such that 0 = t 0 < t <... < t n =. Let α i = α [ti,t i+ ], for 0 i n. Then we know that ([9, Theorem 5.3]) α ˆα 0 ˆα... ˆα n. Corollary 3.. Let α be a path in X and I, I 2,...I n be closed disjoint subintervals of I. If α j = α Ij and β j : I j X are homotopic relative to end points for j =,..., n, then α β, where βj (t) t I β(t) = j α(t) otherwise. If A is a subset of a topological space X, then we denote the complement of A in X by A c = X A. Lemma 3.2. Let α be a loop in X based at x 0. Let t i } (0, ) be an increasing sequence such that t i and V i } form a nested neighborhood basis at x 0 such that α(t i ) V i for every integer i. If β i : [t i, ] V i is a path from α(t i ) to x 0 such that ˆβ i ˆα [ti,t i+ ] ˆβ i+ is a null loop in V i, for every integer i, then α α i, where α(t) t ti α i (t) = β i (t) t t i. Proof. It is obvious that α i α j, for every integers i, j and hence it suffices to prove α α. Let α α by F(t, s) = α (t). Since ([t, ] 0}) (} I) is compact and V is an open neighborhood of F([t, ] 0} } I), there exist ε (0, ) such that F(B ) V, where B = (x, y) I I ( t )y ε (x t )}. Let η : I V be the path defined by η (t) = F(t(t, 0) + ( t)(, ε )). By definition of η and since F(t, 0) = β (t) for t t, η ˆβ e x0 by a homotopy in V. Therefore η ˆα [t,t 2 ] ˆβ 2 η ˆβ ˆβ ˆα [t,t 2 ] ˆβ 2 e x0 by a homotopy in V and hence there exists a continuous map H : B V such that (i) H (t(t, 0) + ( t)(, ε )) = η (t) for t I, (ii) H (t, 0) = α(t) for t [t, t 2 ], (iii) H (t, 0) = β 2 (t) for t t 2, (iv) H (, t) = x 0. Since ([t 2, ] 0}) (} I) is compact and V 2 is an open neighborhood of H ([t 2, ] 0} } I), there exist ε 2 < ε (0, ) such that H (B 2 ) V 2, where B 2 = (x, y) I I ( t 2 )y ε 2 (x t 2 )}. Let η 2 : I V 2 be the path defined by η 2 (t) = H (t(t 2, 0) + ( t)(, ε 2 )). Then η ˆα [t,t 2 ] η is null in V 2 by the homotopy K = H B B 2 (note that B B 2 is homeomorphic to I I). Also, η 2 ˆβ 2 is null in V 2 because H (B 2 ) V 2 and hence η 2 ˆα [t2,t 3 ] ˆβ 3 e x0 by homotopy H 2 : B 2 V 2. By the similar way, for every n N we have the path η n : I V n by η n (t) = H n (t(t n, 0) + ( t)(, ε n )) for which η n α [tn,t n ] η n is null in V n by a homotopy K n = H n Bn B n+, where B n = (x, y) I I ( t n )y ε n (x t n )} and ε n 0 (see Figure ). Define G : I I X by Kn (t, s) (t, s) B G(t, s) = n B n+ F(t, s) o.w. Then G(t, 0) = α(t) and G(t, ) = α (t). Let (r n, s n ) (, 0) and U be an open neighborhood of x 0. There exist n 0 N such that V n0 U. By the construction of B n s, there exist n > n 0 such that (r n, s n ) B n for each n > n which implies that G(r n, s n ) V n U for each n > n. Hence G is continuous(preserving the other convergent sequences by G is obvious).

5 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 5 ε ε 2 ε 3 t t 2 t 3 Figure : In the following examples we show that the above lemma does not hold ifβ i ˆα [ti,t i+ ] β i+ is only null in X. Example 3.3. Let X be the Griffiths space: the one-point union C(HE) C(HE) of two copies of the cone over the Hawaiian earring as described in [8]. Let θ i s and λ i s be the consecutive loops of the right and left Hawaiian earring, respectively. Then all loops θ i and λ i are null by homotopies in X. Let I j = [, ], for every j N and 2 j 2 j define α : I X by θj ϕ α(t) = 2j (t) t I 2j, λ j ϕ 2j (t) t I 2j, where ϕ j s are suitable linear homeomorphisms from I j to I. Note that α is not null in X [3, Example 2.5.8]. Let t i =, β 2 i 2i = θ i ϕ 2i and β 2i = λ i ϕ 2i. Then β i ˆα [ti,t i+ ] β i+ is homotopic to a null loop θ j or λ j (depended on i is even or odd). Also, every loop α i defined as like as in Lemma 3.3 is null and hence α is not homotopic to α i, for every i N. Lemma 3.4. Let α be a loop in X based at x 0, x 0 } be closed and V i } form a nested neighborhood basis at x 0. If α (x 0 } c ) = n= (a n, b n ) and ˆα [an,b n ] e x0 by a homotopy in V n (α([a n, b n ]) V n ), then α is nullhomotopic. Proof. Let I n = [a n, b n ] and α n = α In. Assume ˆα n e x0 by a homotopy F n : I n I V n. Define F : I I V by Fn (t, s) t I F(t, s) = n, f or n N otherwise x 0 Since F(t, 0) = α(t) and F(t, ) = x 0, it remains to show that F is continuous. Let (t n, s n ) (t, s). If F(t, s) x 0, then t n (a k, b k ), for a k N and every n > k. Continuity of F k implies that F(t n, s n ) = F k (t n, s n ) F k (t, s) = F(t, s). If F(t, s) = x 0, then the construction of F, F n s and nested property of V n s imply that F(t n, s n ) U, for an open neighborhood U of x 0 and sufficiently large n. By the next example we show that Lemma 3.4 does not hold if ˆα [an,b n ] is a null loop in X instead of V n. Example 3.5. Let X, α, θ i and λ i be defined as in Example 3.3 and x 0 be the common point of two cones. We have all the hypothesis of the lemma except θ i and λ i are null in X and α is not null.

6 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 6 4. The main result This section dedicated to prove Theorem A by introducing notions π -contained for subspaces of a given space X and nested basis for a subspace A X. Definition 4.. Let X be a topological space, Z, K be path connected subset of Y X. We say that Z is π -contained in K at Y if i π (Z, x) j π (K, x) for every x Z K, where i : Z Y, j : K Y are inclusion maps. Equivalently, every loop in Z is homotopic to a loop in K by a relative homotopy in Y. For a topological space X and open subsets G G X, we say G is simply connected in G if every loop in G is nullhomotopic in G. If A is a subset of a topological space X, then we denote the boundary of A in X by A. Also, in a metric space (X, d), by S r (a) we mean the open ball x X d(x, a) < r}. Lemma 4.2. Let X be a metric space and A be a locally path connected, closed subset of X. Let U X be an open neighborhood of a A which is simply connected in an open subset W of X. Then there exists an open neighborhood V of the point a such that V U and A V is π -contained in A at A W. Proof. Since X is metric, there exists U U such that U U. Since A is locally path connected, there exists a neighborhood V U of a such that V A is path connected. There exists r > 0 such that the closed ball B r (a) is contained in V. Let V be the open ball S r (a), then we show that A V is π -contained in A at A W. Let α be a loop in A V with base point in A. If (r, s) is a component of α (V A c ) = α (A c ), then α(r), α(s) V A V A. Since V A is path connected, there exists a path in V A from α(r) to α(s). Case : Let α (A c ) = N n= (a n, b n ). If for every n N, β n : [a n, b n ] V A is a path from α(a n ) to α(b n ), then α [an,b n ] β n by a homotopy in W since ˆα [an,b n ] ˆβ n is a loop in U. Using Corollary 3., α is homotopic to a loop in A by a homotopy in A W. Case 2: Let α (A c ) = n= (a n, b n ). Since α is uniformly continuous, there exists δ > 0 such that if t s < δ, then d(α(t), α(s)) < r/2. Also, there exists n 0 N such that for every n > n 0, b n a n < δ. Similar to Case, there are paths β i : [a i, b i ] V A such that β i α [ai,b i ], for i n 0 by homotopies in W. Therefore α is homotopic to a loop α : I A V by a homotopy in A W and α (A c ) = n=n 0 + (a n, b n ). Let F = a n n > n 0 } and E = b n n > n 0 }. If t = in f α (A c ) F} and s = sup[t, t + δ] E}, then by definition of E and F, t, s exist and s t < δ which implies that α [t,s ] S r (a) V. Since V A is path connected, there exists a path λ : I V A with λ (0) = α (t ) and λ () = α (s ) such that ˆα [t,s ] λ by a homotopy in W because ˆα [t,s ] λ is a loop in V U. Now, let t n = in f [s n, ] α (A c ) F} and s n = sup[t n, t n + δ] E}. Similarly, α ([t n, s n ]) S r (a) V and there exists a path λ n : I V A with λ n (0) = α (t n ) and λ n () = α (s n ) such that ˆα [tn,s n ] λ n by a homotopy in W because ˆα [tn,s n ] λ n is a loop in V U. Since I \ i n0 (a i, b i ) is compact and δ > 0, there exists n > 0 such that [s n, ] α (A c ) F =. Hence, we can replace ˆα [ti,s i ], by the path λ i, for i n, with homotopies in W. Therefore α is homotopic to a loop in A by a homotopy in A W, as desired. Definition 4.3. For a topological space X, a family U = U n } n N of open neighborhoods of a subset A is called a nested basis of A if U n+ U n for all n N and for every open set G containing A, there is n 0 N such that U n0 G. Let a nested basis V n } n N of A be a refinement of a nested basis U n } n N of A i.e V n U n for each n. Then we say that V n } n N is π -refinement in A at U n } n N if V n is π -contained in A at U n for each n. There is two natural questions that whether a subset of a topological space X has a nested basis and whether every nested basis has a π -refinement? Obviously, compact subsets of metric spaces have nested basis and for the second question we have the following theorem. We recall that a topological space X is said to be locally -connected if it is locally path connected and locally simply connected. Theorem 4.4. Let X be a locally -connected metric space and A be a locally path connected compact subset of X. Then every nested basis U n } n N of A has a π -refinement in A at U n } n N.

7 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 7 Proof. Let U n = S (A) be the open ball with radius /n of A, then U n } n N is a nested basis for A. Since X is n locally -connected, for every n N and a A there is an open neighborhood G n (a) U n at a such that G n (a) is simply connected in U n. By Lemma 4.2, there exists G n(a) G n (a) such that A G n(a) is π -contained in A at U n. By local 0-connectivity of X we can assume every G n(a) is path connected. Since A is compact, for every n N there exists m n N and a n, a 2 n,..., a m n n such that A int(a) ( m n i= G i n(a i n)). We claim for every n N there exists integer k n n such that for every m > k n and every couple G i m(a i m), G j m(a j m) with nonempty intersection, A G i m(a i m) G j m(a j m) is π -contained in A at U n. By contradiction assume there exists no such k n0 for an n 0 N, then there are a sequence s n } n N, a couple G i s n (a i s n ), G j s n (a j s n ) and loops α n : I A G i s n (a i s n ) G j s n (a j s n ) such that there is no loop in A homotopic to α n by a homotopy in U n0. Let x n Imα n A. Since A is compact, there is a subsequence x nk } k N converging to x A. Let G be a simply connected neighborhood of x in U n0. By Lemma 4.2 there exists an open neighborhood x G G such that A G is π -contained in A at U n0. There exists k 0 N such that for each k > k 0, G i s nk (a i s nk ) G j s nk (a j s nk ) G A which implies Imα nk A G, for k > k 0. Therefore α nk is homotopic to a loop in A by a homotopy in U n0 which is a contradiction. Let V n := A ( m kn i= G i k n (a i k n )). We show that V n is π -contained in A at U n. Consider α : I V n as a loop at a A. Since Imα int(a) ( m kn i= G i k n (a i k n )), there is the Lebesque number δ for this open cover. Let n 0 N such that n 0 < δ, then there is G, G 2,..., G n0 } int(a), G k n (a k n ), G 2 k n (a 2 k n ),..., G m kn (a m kn )} such that k n k n α([ i i n 0, n 0 ]) G i. Let α i = α [ i n, i 0 n ] ϕ i, where ϕ i : [0, ] [ i i 0 n 0, n 0 ] is linear homeomorphism for i =,..., n 0. Since a = α(0) G and α( n 0 ) G G 2, if β : I A G 2 is a path from α( n 0 ) to a (G i s and A are path connected), then α β is a loop in A G G 2 which implies that α β is homotopic to a loop γ : I A by a homotopy in U n. Also, there exists β 2 : I A G 3 such that β α 2 β 2 is homotopic to a loop in A by a homotopy in U n and similarly there are β i s and γ i s for i =, 2,..., n 0 such that α (α β ) (β α 2 β 2 )... (β n0 2 α n 0 β n0 ) (β n0 α n 0 ) γ γ 2... γ n0 which implies that α is homotopic to a loop in A by a homotopy in U n, as desired. Note that if V n } n N is not decreasing, then we can put V n be the path component of V V 2... V n that contains A. Let U n} n N be an arbitrary nested basis of A. Since U n } n N is a nested basis of A, there exists a subsequence k n } n N of N such that U kn U n for every n N. Let V kn } be the π -refinement of U kn }. Then V n} n N is a π -refinement of U n} n N, where V n = V kn for each n. Remark 4.5. If X is a metric space with a metric d, then for the quotient space X/A, the map d : X/A X/A [0, + ) defined by mind(x, y), d(x, A) + d(y, A)} x, y d(x, y) = d(a, x) x y = 0 x = y =, where = p(a) and p : X X/A is the natural quotient map, is a metric and the topology induced by this metric on X/A is equivalent to the quotient topology on X/A since A is compact (see [2, Section 3.]). For a topological space X, a loop α in X based at x is called semi-simple if α (x} c ) = (0, ) and is called geometrically simple if α (x} c ) = (a, b) for a, b [0, ]. Also, every geometrically simple loop is homotopic to a semi-simple loop [0]. Theorem 4.6. Let X be a locally -connected metric space and A be a connected, locally path connected and compact subset of X. Then X/A is locally -connected space. Proof. Since X is locally path connected and A is connected, X/A is locally path connected. Clearly X A is an open subset of X and hence it is locally -connected. Since q := p X A : X A (X/A) } is homeomorphism, (X/A) } is locally -connected.

8 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 8 Let U be an open neighborhood of. By the proof of Theorem 4.4, the nested basis U n = S (A)} n N for n A has a refinement V n } n N as its π -refinement in A at U n } n N such that V n s are path connected. We can assume that U p (U) and show that V := p(v ) is a desired open neighborhood. Let α : I V be a loop at, then we must show that (i V ) ([α]) = 0, where i : V U is the inclusion map. Case : Suppose that α is a semi-simple loop and a 0 = lim t 0 p (α(t)) and a = lim t p (α(t)) exist. Obviously a 0, a A and since A is closed path connected, there exists a loop α in V with base point a A such that p α α by a homotopy in V. Since V is π -contained in A at U, there is a loop β in A such that β α by a homotopy in U. Therefore α p α p β e by homotopies in p(u ) U, as desired. Case 2: Suppose that α is a semi-simple loop and at least one of the above limits does not exist. Let α = q (α (0,) ). Without loss of generality, assume a 0 = lim t 0 α(t) exists and a A is a limit point of α((/2, )). The point a is a limit point and α is continuous, hence there is an increasing sequence t n } n N in (/2, ) such that α([t n, )) V n, for every n N and t n. Since the V n s are path connected, there are paths γ n : [t n, ] V n from α(t n ) to a. If β n := p γ n, then Imβ n p(v n ) and β n ˆα [tn,t n+ ] β n+ is null in p(u n ) since γ n ˆ α [tn,t n+ ] γ n+ is a loop in V n. Let α n (t) = α(t) t tn β n (t) t t n. By Lemma 3.2, α α n for each n and by Case, α n s are nullhomotopic in p(u ) which implies that α is nullhomotopic. Case 3: If α is not a semi-simple loop in V, then α (X/A }) = i N(a i, b i ). Obviously α [ai,b i ] is null in a p(u ni ), where n i = maxn N α([a i, b i ]) p(v n )}. By Lemma 3.4, α is null. Similarly, the result holds if α is a geometrically simple loop. Also, in Case 3, if α (X/A }) = n i= (a i, b i ), by Corollary 3. the result holds. Proof of Theorem A. By Remark 4.5 and Theorem 4.6, X/A is a locally -connected metric space, hence we can use Theorem 4.6 again and by induction the result holds. In the following examples, we show that locally path connectedness and locally simply connectedness of X are necessary conditions in Theorem A. Example 4.7. Let Y = (x, y) R 2 x 2 + y 2 = (/2 + /n) 2, n N}, A = (x, y) R 2 x 2 + y 2 = /4} (x, 0) R 2 /2 x } and X = Y A with a = (, 0) as the base point (see Figure 2). Then X is connected, locally simply connected metric space and A is compact, connected and locally path connected subset of X, but obviously, X is not locally path connected. Since for every open neighborhood U X/A of there exists N 0 N such that for n > N 0, p (U) contains circles with radius /2 + /n, X/A is the union of a null sequence of simple closed curves meeting in a common point that is homeomorphic to the Hawaiian earing space. Hence X/A is not locally simply connected. Example 4.8. Let X = (x, y) R 2 /4 x 2 + y 2 } \ ( /2 /n, 0) R 2 n N} and A = (x, y) R 2 x 2 + y 2 = /4} (x, 0) R 2 /2 x }. Then X is a connected, locally path connected metric space that is not locally simply connected and A is a compact, connected and locally path connected subset of X. Similar to Example 4.7, it can be proved that X/A is homeomorphic to Y = D \ ( /n, 0) n N} which is not locally simply connected.

9 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 9 a Figure 2: 5. Some Applications to fundamental groups In this section, first we give a short proof of Theorem B and then prove some properties of the fundamental group of the quotient space X/(A, A 2,..., A n ). Proof of Theorem B. By Theorem 2.4, it suffices to prove that X/(A, A 2,..., A n ) is semi-locally simply connected. But by Theorem A, we have X/(A, A 2,..., A n ) is locally -connected which is more. Remark 5.. In Examples 4.7 and 4.8, p is not onto since π (X, a) is a free group on countably many generators and π (X/A, ) is uncountable. This shows that the hypothesis locally -connectedness is necessary for Theorem B. We know that any homomorphic image of a finitely generated group is also finitely generated. Hence we have the following consequence of Theorem B. Corollary 5.2. Let X be connected, locally -connected metric space and A, A 2,..., A n be compact, connected, locally path connected subsets of X. If π (X, a) is finitely generated, then so is π (X/(A, A 2,..., A n ), ). By a theorem of Shelah [2], if X is a compact, connected, locally path connected metric space and π (X, x) is countable, then π (X, x) is finitely generated. Using this and Theorem B, we have Corollary 5.3. Let X be compact, connected, locally -connected, metric space and A, A 2,..., A n be closed, connected, locally path connected subsets of X. If π (X/(A, A 2,..., A n ), ) is not finitely generated, then π (X, a) is uncountable. J. Dydak and Z. Virk [7] proved that if X is a connected, locally path connected metric space and π (X, x) is countable, then π (X, x) is a finitely presented group. Note that a homomorphic image of a finitely presented group is not necessarily a finitely presented group. Corollary 5.4. Let X be connected, locally -connected metric space and A, A 2,..., A n be compact, connected, locally path connected subsets of X. If π (X, a) is countable, then π (X/(A, A 2,..., A n ), ) is finitely presented. J.W. Cannon and G.R. Conner [6] proved that if X is a one-dimensional metric space which is connected, locally path connected and semi-locally simply connected (or equivalently if X admits a universal covering space), then π (X, x) is a free group. Therefore by Theorem A we have Corollary 5.5. If X is a one-dimensional metric space which is connected, locally -connected, and A, A 2,..., A n are compact, connected, locally path connected subsets of X, then π (X/(A, A 2,..., A n ), ) is a free group.

10 A. Pakdaman et al. / Filomat xx (yyyy), zzz zzz 0 Theorem 5.6. If X is a separable, connected, locally -connected metric space and A, A 2,..., A n are compact, connected, locally path connected subsets of X, then π (X/(A, A 2,..., A n ), ) is countable. Moreover, π (X/(A, A 2,..., A n ), ) is finitely presented. Proof. Using Theorem A and [6, Lemma 5.6], we have π (X/(A, A 2,..., A n ), ) is countable and by [7], π (X/(A, A 2,..., A n ), ) is finitely presented. J. Pawlikowski [] proved that compact, connected, locally path connected metric spaces with countable fundamental groups are semi-locally simply connected. Therefore we have Corollary 5.7. If X is a compact, connected, locally path connected metric space and A, A 2,..., A n are connected, locally path connected closed subsets of X, then p : π (X, a) π (X/(A,..., A n ), ) is an epimorphism if one of the following conditions holds (i) π (X/(A, A 2,..., A n ), ) is countable, (ii) π (X/(A, A 2,..., A n ), ) is finitely generated, (iii) π (X/(A, A 2,..., A n ), ) is finitely presented, (iv) X/(A,..., A n ) has universal covering. Proof. By Assumptions, Remark 4.5 and continuity of p : X X/(A, A 2,..., A n )), X/(A, A 2,..., A n ) is a compact, connected, locally path connected metric space and the result follows from Theorem 2.4. Remark 5.8. J. Brazas [4] introduced a new topology on π (X, x) coarser than the topology of π qtop (X, x) and proved that fundamental groups by this new topology are topological groups, denoted by π τ (X, x). Since the topology of π τ (X, x) is coarser than πqtop(x, x), the results of [0] remain true and we have a similar result to Theorem B for π τ (X, x) and πτ (X/(A,..., A n ), ). Acknowledgements The authors would like to thank the referee for the valuable comments and suggestions which improved the manuscript and made it more readable. References [] A. Arhangelskii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics, [2] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, (forth edition), American Math. So., 200 [3] J. Brazas, The topological fundamental group and free topological groups, Topology and its Applications 58 (20) [4] J. Brazas, The fundamental group as topological group, Topology and its Applications 60 (20) [5] J. S. Calcut, J. D. McCarthy, Discreteness and homogeneity of the topological fundamental group, Topology Proc. 34 (2009) [6] J. W. Cannon, G. R. Conner, On the fundamental groups of one-dimensional spaces, Topology and its Applications 53 (2006) [7] J. Dydak, Z. Virk, An alternate proof that fundamental group of a Peano continuum is finitely presented if the group is countable, Glasnik Matematicki 46 (20) [8] H. B. Griffiths, The fundamental group of two spaces with a common point, Q. J. Math. Oxford 5 (954) [9] J. R. Munkres, Topology, (second edition), Prentice-Hall, Upper Saddle River, NJ, [0] H. Torabi, A. Pakdaman, B. Mashayekhy, A dense subgroup of topological fundamental group of quotient space, arxiv: v. [] J. Pawlikowski, The fundamental group of a compact metric space, Proc. of the AMS 26 (998) [2] S. Shelah, Can the fundamental (homotopy) group of a space be the rationals?, Proc. of the AMS 03 (988) [3] H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto, Ont.-London 966.