On the Topological Fundamental Groups of Quotient Spaces
Let p: X → X/A be a quotient map, where A is a subspace of X. We study the conditions under which p ∗(π 1 qtop (X, x 0)) is dense in π 1 qtop (X/A,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and p * is a continuous homomorphism induced by the...
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| author | Mashayekhy, B. Pakdaman, A. Torabi, H. Машаєхі, Б. Пакдаман, А. Торабі, Х. |
| author_facet | Mashayekhy, B. Pakdaman, A. Torabi, H. Машаєхі, Б. Пакдаман, А. Торабі, Х. |
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| description | Let p: X → X/A be a quotient map, where A is a subspace of X.
We study the conditions under which p ∗(π 1 qtop (X, x 0)) is dense in π 1 qtop (X/A,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and p * is a continuous homomorphism induced by the quotient map p. In addition, we present some applications in order to determine the properties of π 1 qtop (X/A,∗). In particular, we establish conditions under which π 1 qtop (X/A,∗) is an indiscrete topological group. |
| first_indexed | 2026-03-24T02:25:32Z |
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UDC 517.91
H. Torabi (Center of Excellence in Analysis on Algebraic Structures, Ferdowsi Univ. Mashhad, Iran),
A. Pakdaman (Golestan Univ. Gorgan, Iran),
B. Mashayekhy (Center of Excellence in Analysis on Algebraic Structures, Ferdowsi Univ. Mashhad, Iran)
ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES
ПРО ТОПОЛОГIЧНI ФУНДАМЕНТАЛЬНI ГРУПИ ФАКТОР-ПРОСТОРIВ
Let p : X → X/A be a quotient map, where A is a subspace of X . We study the conditions under which p∗(π
qtop
1 (X,x0))
is dense in πqtop
1 (X/A, ∗)), where the fundamental groups have the natural quotient topology inherited from the loop
space and p∗ is a continuous homomorphism induced by the quotient map p. In addition, we present some applications to
determine some properties of πqtop
1 (X/A, ∗). In particular, we establish some conditions under which πqtop
1 (X/A, ∗) is
an indiscrete topological group.
Нехай p : X → X/A — фактор-вiдображення, де A — пiдпростiр X . Дослiджуються умови, за яких p∗(π
qtop
1 (X,x0))
є щiльною в πqtop
1 (X/A, ∗)), де фундаментальнi групи надiленi природною фактор-топологiєю, успадковaною вiд
простору петель, а p∗ — неперервний гомоморфiзм, iндукований фактор-вiдображенням p. Крiм того, наведено
деякi застосування з метою визначити деякi властивостi πqtop
1 (X/A, ∗). Наприклад, встановлено умови, за яких
πqtop
1 (X/A, ∗) є недискретною топологiчною групою.
1. Introduction and motivation. Let p : (X,x0)→ (Y, y0) be a continuous map of pointed topolog-
ical spaces. By applying the fundamental group functor on p there exists the induced homomorphism
p∗ : π1(X,x0) −→ π1(Y, y0).
It seems interesting to relate the homology and homotopy groups of X with that of Y using properties
of p. Vietoris first studied the problem with his mapping theorem [18]. Also, Smale first discovered
an analog of Vietoris’s mapping theorem hold for homotopy groups [15]. Recently, Calcut, Gompf,
and Mccarthy [6] proved a generalization of Smale’s theorem as follows:
Let p : (X,x0) → (Y, y0) be a quotient map of topological spaces, where X is locally path
connected and Y is semilocally simply connected. If each fiber p−1(y) is connected, then the induced
homomorphism p∗ : π1(X,x0)→ π1(Y, y0) is surjective.
For a pointed topological space (X,x0) by πqtop1 (X,x0) we mean the topological fundamen-
tal group endowed with the quotient topology inherited from the loop space under the natural map
Ω(X,x0) −→ π1(X,x0) that makes it a quasitopological group. A quasitopological group G is
a group with a topology such that inversion g −→ g−1 and all translations are continuous. For
more details, see [2, 3, 5]. It is known that this construction gives rise a homotopy invariant functor
πqtop1 : hTop∗ −→ qTopGrp from the homotopy category of based spaces to the category of qua-
sitopological groups and continuous homomorphisms [3]. Also, πτ1 (X,x0) is the fundamental group
endowed with another topology introduced by Brazas [4]. In fact, the functor πτ1 removes the smallest
number of open sets from the topology of πqtop1 (X,x) so that makes it a topological group.
Let X be a topological space and A1, A2, . . . , An be a finite collection of its subsets. The quotient
space X/(A1, . . . , An) is obtained from X by identifying each of the sets Ai to a point. Now, let
(A, a) be a pointed subspace of (X, a) and p : (X, a) −→ (X/A, ∗) be the associated quotient map.
In this paper, first we prove that if A is an open subset of X such that the closure of A, A, is path
connected, then the image of p∗ is dense in πqtop1 (X/A, ∗). Then by this fact, we show that the
c© H. TORABI, A. PAKDAMAN, B. MASHAYEKHY, 2013
1700 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES 1701
image of p∗ is dense in πqtop1 (X/(A1, A2, . . . , An), ∗), where the Ai are open subsets of X with
path connected closures and p : X −→ X/(A1, A2, . . . , An) is the associated quotient map. Second,
we prove that if A is a closed subset of a locally path connected and first countable space X, then
the image of p∗ is also dense in πqtop1 (X/A, ∗). By the two previous results we can show that the
image of p∗ is dense in πqtop1 (X/(A1, A2, . . . , An), ∗), where X is first countable, connected, locally
path connected and the Ai are open or closed subsets of X with disjoint path connected closures.
Moreover, we give some conditions in which p∗ is an epimorphism. Also, by some examples, we
show that p∗ is not necessarily onto. Finally, we give some applications of the above results to find out
some properties of the topological fundamental group of the quotient space X/(A1, A2, . . . , An). In
particular, we prove that with the recent assumptions on X and the Ai, π
qtop
1 (X/(A1, A2, . . . , An), ∗)
is an indiscrete topological group when X is simply connected. It should be mentioned that since
the topology of πτ1 (X,x0) is coarser than πqtop1 (X,x0), the above results can be obtained when we
replace πqtop1 with πτ1 .
2. Notations and preliminaries. For a topological space X, by a path in X we mean a continu-
ous map α : [0, 1] −→ X. The points α(0) and α(1) are called the initial point and the terminal point
of α, respectively. A loop α is a path with α(0) = α(1). For 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, 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 = 1,
α(0) = β(0), t = 0,
α(1) = β(1), t = 1.
The homotopy is an equivalent relation and 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(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(2t),
for all 0 ≤ t ≤ 1/2 and (α ∗ β)(t) = β(2t− 1), for all 1/2 ≤ t ≤ 1.
For a pointed topological 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. When X
is path connected and the basepoint is clear, we just write Ω(X) and we will consistently denote
the constant path at x by ex. The topological fundamental group of a pointed space (X,x) can be
described as the usual fundamental group π1(X,x) with the quotient topology with respect to the
canonical map Ω(X,x) −→ π1(X,x) identifying homotopy classes of loops, denoted by πqtop1 (X,x).
A basic account of topological fundamental groups may be found in [2], [5] and [3]. For undefined
notation, see [12].
Definition 2.1 [1]. A quasitopological group G is a group with a topology such that inversion
G −→ G, g 7→ g−1, is continuous and multiplication G×G −→ G is continuous in each variable.
A morphism of quasitopological groups is a continuous homomorphism.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1702 H. TORABI, A. PAKDAMAN, B. MASHAYEKHY
Theorem 2.1 [3]. πqtop1 is a functor from the homotopy category of based topological spaces to
the category of quasitopological groups.
A space X is called 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 trivial homomorphism
i∗ : π1(U, x) −→ π1(X,x) or equivalently a loop in U can be contracted inside X.
Theorem 2.2 [3]. Let X be a path connected space. If πqtop1 (X,x) is discrete for some x ∈ X,
then X is semi-locally simply connected. If X is locally path connected and semi-locally simply
connected, then πqtop1 (X,x) is discrete for all x ∈ X.
3. Main results. In this section, (A, a) is a pointed 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 πqtop1 on p we have a continuous homomorphism p∗ : πqtop1 (X, a) −→
−→ πqtop1 (X/A, ∗).
Lemma 3.1. If A is an open subset of X, then any loops α : I −→ {∗} ⊆ X/A based at ∗ is
nullhomotopic.
Proof. Define F : I × I −→ X/A by
F (t, s) =
α(t), s = 0,
∗, s > 0.
If we prove that F is continuous, then F is a homotopy between α and e∗. For this, let U be an open
set in X/A. We show 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)
which is open in I × I.
Case 2. If ∗ /∈ U, then U ∩ ∂{∗} = ∅ since if there exists x ∈ ∂{∗} such that x ∈ U, then
{∗}∩U 6= ∅ which is a contradiction. Since U∩{∗} = (U∩{∗})∪(U∩∂{∗}) = ∅ and α(I) ⊆ {∗},
we have F−1(U) = ∅.
Lemma 3.1 is proved.
Theorem 3.1. Let A be an open subset of X such that A is path connected, then for each a ∈ A
the image of p∗ is dense in πqtop1 (X/A, ∗), i.e.,
p∗(π
qtop
1 (X, a)) = πqtop1 (X/A, ∗).
Proof. Step 1. First, we show that for every loop α : (I, ∂I) −→ (X/A, ∗) such that α−1({∗}c)
is connected, we have [α] ∈ Im(p∗). By assumption and openness of {∗} in X/A, there exist s1, s2 ∈
∈ (0, 1) such that α−1({∗}c) = [s1, s2]. Since α−1({∗}c) is a compact subset of I, it suffices to let
s1 = inf{α−1({∗}c)} and s2 = sup{α−1({∗}c)}. Let α : [s1, s2] −→ X by α(t) = q−1(α(t)), then
α(s1), α(s2) ∈ A. Since if G is an open neighborhood of α(s1) and G ∩A = ∅, then q(G) = p(G)
is an open neighborhood of α(s1). Using continuity of α, there exists an open neighborhood J of
s1 in I such that α(J) ⊆ G. On the other hand, by definition of s1, for all s < s1, α(s) = ∗
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES 1703
which implies that ∗ ∈ q(G) which is a contradiction since ∗ /∈ Im(q). Similarly α(s2) ∈ A.
Since A is path connected, there exist two paths λ1 : [0, s1] −→ A and λ2 : [s2, 1] −→ A such that
λ1(0) = λ2(1) = a, λ1(s1) = α(s1) and λ2(0) = α(s2). Define α̃ : I −→ X by
α̃(t) =
λ1(t), 0 ≤ t ≤ s1,
α(t), s1 ≤ t ≤ s2,
λ2(t), s2 ≤ t ≤ 1.
By gluing lemma α̃ is continuous, so it remains to show that p ◦ α̃ ' α rel{∗}. Put α1 = α|[0,s1],
α2 = α|[s2,1] and let ϕ1 : [0, 1] −→ [0, s1] and ϕ2 : 0, 1] −→ [s2, 1] be linear homeomorphisms such
that ϕ1(0) = 0 and ϕ2(0) = s2, then p◦λi ◦ϕi ' αi ◦ϕi, rel{0, 1} since the (p◦λi ◦ϕi)◦ (αi ◦ϕi)−1
are loops in {∗} which by Lemma 3.1 are nullhomotopic.
Step 2. By continuity of α, α−1({∗}c) is a closed subset of I. Connected subsets of I are intervals
or one point sets, also connected components of α−1({∗}c) are closed in α−1({∗}c) and so they are
compact in I. Therefore a component of α−1({∗}c) is either closed interval or singleton. Given
[α] ∈ π1(X/A, ∗), we show that there exists a sequence of homotopy classes of loops {[αn]}n∈N in
Im (p∗) such that [αn] −→ [α] in πqtop1 (X/A, ∗).
We claim that the number of non-singleton components of α−1({∗}c) is countable. Let S be the
union of singleton components of α−1({∗}c) and for each n ∈ N, Bn be the set of non-singleton
components of α−1({∗}c) with length at least 1/n. Each Bn is finite since if Bn is infinite, then it
has at least n+ 1 members. Therefore
⋃
C∈Bn
C ⊆ α−1({∗}c) ⊆ I which implies that
(n+ 1)× 1/n ≤
∑
C∈Bn
diam (C) ≤ diam (I) = 1
which is a contradiction. Thus each Bn is finite which implies that B =
⋃
n∈NBn is countable.
Rename elements of B by Ii = [ai, bi], i ∈ J = {1, 2, . . . , s}, where s = |B| if B is finite and
i ∈ N = J if B is infinite. For every n ∈ J define
αn(t) =
α(t), t ∈
⋃n
i=1[ai, bi],
∗, otherwise.
If B is finite, put αn = αs, for every n > s. We claim that the αn are continuous. For, if V ⊆ X/A
is open, then
(i) If ∗ ∈ V, then
α−11 (V ) = [0, a1) ∪ (b1, 1] ∪ α|−1[a1,b1]
(V )
which by continuity of α is open in I.
(ii) If ∗ /∈ V, then we show that α−11 (V ) = α|−1[a1,b1]
(V ) = α|−1(a1,b1)
(V ) which guaranties α−1(V )
is open. For this it suffices to show that α1(a1), α1(b1) /∈ V.
For each n ∈ N, α(an), α(bn) ∈ {∗} and {α(a)| a ∈ S} ⊆ {∗}. For, if G is an open
neighborhood of α(an), then α−1(G) is an open neighborhood of an, so there exists ε > 0
such that (an − ε, an + ε) ⊆ α−1(G) or equivalently α((an − ε, an + ε)) ⊆ G. If ∗ /∈ G, then
(an − ε, an + ε) ⊆ α−1({∗}c) which is a contradiction since [an, bn] is a connected component of
α−1({∗}c). Similarly α(bn) ∈ {∗} for each n ∈ N and α(S) ⊆ {∗}. Thus if α1(b1) = α(b1) ∈ V,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1704 H. TORABI, A. PAKDAMAN, B. MASHAYEKHY
then V must meet {∗} which is a contradiction since ∗ /∈ V. Therefore α1 is a continuous loop such
that [α1] ∈ Im(p∗). Similarly, all the αn are continuous. Also, for every n ∈ N, [αn] is a product
of n homotopy classes of loops which are similar to loops introduced in Step 1. This implies that
[αn] ∈ Im(p∗) since Im(p∗) is a subgroup.
Now we show that the sequence {αn} converges to α. Let α ∈ 〈K,U〉, 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, αn ∈ 〈K,U〉 since for each t ∈ K, αn(t) = α(t) or αn(t) = ∗
which in both 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 open. Thus ∗ ∈ U and by (i), for each n, αn(K) ⊆ U.
(iii) If K ⊆
⋃∞
n=1 In and there exist n1, n2, . . . , ns such that K ⊆
⋃s
i=1 Ini , then by definition
of the αn, for each n ≥ max{n1, . . . , ns} we have αn(K) ⊆ U.
(vi) If K ⊆
⋃∞
n=1 In and there exists an infinite subsequence {Inr} such that K ∩ Inr 6= ∅,
then there is a sequence {xnr |xnr ∈ K ∩ Inr} such that it has a subsequence {xnrs
} converges 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 s0 such that for each s ≥ s0, xnrs
∈ (b−ε/2, b+ε/2) since
xnrs
−→ b. Since diam(In) −→ 0, there is n0 such that for each n ≥ n0, diam(In) < ε/2. Choose
s1 ∈ N such that s1 ≥ s0 and nrs1 ≥ n0, then xnrs1
∈ (b−ε/2, b+ε/2). Also xnrs1
∈ K∩Inrs1
and
diam(Inrs1
) < ε/2, so anrs1
∈ Inrs1
⊆ (b − ε, b + ε) ⊆ α−1(U) which implies that α(anrs1
) ∈ U
and therefore ∗ ∈ U since α(anrs1
) ∈ ∂({∗}). Using the last (i) we have αn(K) ⊆ U, for each
n ∈ N.
Theorem 3.1 is proved.
Definition 3.1. Let X be a topological space and A1, A2, . . . , An be any subsets of X, n ∈ N.
By the quotient space X/(A1, . . . , An) we mean the quotient space obtained from X by identi-
fying each of the sets Ai to a point. Also, we denote the associated quotient map by p : X −→
−→ X/(A1, A2, . . . , An).
Corollary 3.1. If A1, A2 are open subsets of a path connected space X such that A1, A2 are
path connected. Then for every a ∈ A1 ∪A2 the following equality holds:
p∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2), ∗).
Proof. We can assume that the Ai are disjoint. If they are not disjoint, the result follows from
Theorem 3.1. Let p1 : X −→ X/A1, p2 : X/A1 −→ X/(A1, A2) be associated quotient maps and
a1 = a ∈ A1. By Theorem 3.1, (p1)∗(π
qtop
1 (X, a1)) = πqtop1 (X/A1, ∗1), where ∗1 = p1(a1).
Since X is path connected, so is X/A1. Also, p1(A2) is an open subset of X/A1 and the clo-
sure of p1(A2) in X/A1 is path connected. Let a2 ∈ p1(A2), then (p2)∗(π
qtop
1 (X/A1, a2)) =
= πqtop1 (X/(A1, A2), ∗2), where ∗2 = p2(a2). Since X/A1 is path connected, there exists a home-
omorphism ϕ1 : πqtop1 (X/A1, ∗1) −→ πqtop1 (X/A1, a2) by ϕ1([α]) = [γ ∗ α ∗ γ−1], where γ is a
path from a2 to ∗1. We have (p2)∗ ◦ ϕ1 ◦ (p1)∗(π
qtop
1 (X, a)) ⊇ ((p2)∗ ◦ ϕ1)((p1)∗(π
qtop
1 (X, a))) =
= ((p2)∗◦ϕ1)(π
qtop
1 (X/A1, ∗1)) = (p2)∗(π
qtop
1 (X/A1, a2)) = Im(p2)∗ which implies that Im(p2)∗◦
ϕ1 ◦ (p1)∗ is dense in π1(X/(A1, A2), ∗2). If γ′ = p2 ◦ γ, then ϕ2 : πqtop1 (X/(A1, A2), ∗2) −→
−→ πqtop1 (X/(A1, A2), ∗1) by ϕ2([α]) = [γ′−1 ∗α∗γ′] is a homeomorphism. Hence Im(ϕ2 ◦ (p2)∗ ◦
ϕ1 ◦ (p1)∗) is dense in πqtop1 (X/(A1, A2), ∗1). Moreover ϕ2 ◦ (p2)∗ ◦ϕ1 ◦ (p1)∗ = p∗ which implies
that Im(p∗) is dense in πqtop1 (X/(A1, A2), ∗), as desired.
Corollary 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES 1705
By induction and Corollary 3.1, we have the following results.
Corollary 3.2. Let A1, A2, . . . , An be open subsets of a path connected space X such that the
Ai are path connected for each i = 1, 2, . . . , n. Then for any a ∈
⋃n
i=1Ai the following equality
holds:
p∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2, . . . , An), ∗).
Corollary 3.3. Let A1, A2, . . . , An be open subsets of a connected, locally path connected space
X such that the Ai are path connected for every i = 1, 2, . . . , n. If X/(A1, A2, . . . , An) is semi-
locally simply connected, then for each a ∈
⋃n
i=1Ai, p∗ : πqtop1 (X, a) −→ πqtop1 (X/(A1, A2, . . .
. . . , An), ∗) is an epimorphism.
Proof. Let U be an open neighborhood of x̄ ∈ (X/A1) \ {∗}. Since X is locally path connected,
there is a path connected open neighborhood Ũ ⊆ p−1(U) of x = q−1(x̄) such that Ũ ∩ A1 = ∅.
Then V := p(Ũ) = q(Ũ) ⊆ U is a path connected open neighborhood of x̄. X/A1 is locally
path connected at ∗ since {∗} is an open subset of X/A1. Let U be an open neighborhood of
x̄ ∈ ∂({∗}), then there exists a path connected open neighborhood Ũ ⊆ p−1(U) of x. Since
p−1(p(Ũ)) = Ũ ∪ A1, p(Ũ) is a path connected open neighborhood of x̄ in U. Therefore X/A1
is locally path connected. Similarly X/(A1, A2, . . . , An) is connected, locally path connected. Since
X/(A1, A2, . . . , An) is a connected, semi-locally simply connected and locally path connected space,
by Theorem 2.2, πqtop1 (X/(A1, A2, . . . , An), ∗) is a discrete topological group which implies that
Im(p∗) = πqtop1 (X/(A1, A2, . . . , An), ∗) by Corollary 3.2.
Corollary 3.3 is proved.
In the following example we show that with the assumptions of Theorem 3.1, p∗ is not necessarily
onto.
Example 3.1. Let An = {1/(2n− 1), 1/2n} × [0, 1 + 1/2n]
⋃
[1/2n, 1/2n− 1]× {1 + 1/2n}
for each n ∈ N. Consider X =
(⋃
n∈NAn
)⋃
{0} × [0, 1]
⋃
[0, 1] × {0} with a = (0, 0) as the base
point and A = {(x, y) ∈ X | y < 1} (see Figure). A is an open subset of X with path connected
closure. Assume In = (1/2 + 1/2(n + 1), 1/2 + 1/2n] and fn be a homeomorphism from In to
An − {(1/2n, 0)} for every n ∈ N. Define f : I −→ X by
f(t) =
the point (0, 2t), t ∈ [0, 1/2],
fn(t), t ∈ In.
We claim that α = p◦f is a loop in X/A at ∗. It suffices to show that α is continuous on t = 1/2 and
boundary points of In since f is continuous on [0, 1/2) and by gluing lemma on
⋃
int(In). Since
α is locally constant at t = 1/2 + 1/2n for each n ∈ N, α is continuous at boundary points of In.
For each open neighborhood G of f(1/2) = (0, 1) in X, there exists n0 ∈ N such that G contains
An
⋂
Ac for n > n0. Therefore continuity at t = 1/2 follows from α(1/2) ∈ {∗}. Now let B ⊆ N
and define
gB(t) =
(p ◦ f)(t), t ∈
⋃
m∈B Im,
∗, otherwise.
Then gB is continuous and for B1, B2 ⊆ N such that B1 6= B2, [gB1 ] 6= [gB2 ] which implies that
π1(X/A, ∗) is uncountable. But by compactness of I, a given path in X can traverse finitely many
of the An and therefore π1(X, a) is a free group on countably many generators which implies that p
does not induce a surjection of fundamental groups.
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1706 H. TORABI, A. PAKDAMAN, B. MASHAYEKHY
a
A1
A2
A3
Let (X,x) be a pointed topological space such that {x} is closed. If α : [0, 1] −→ X is a loop
in X based at x, then α−1({x}) is a closed subset of [0, 1]. Its complement, α−1({x}c) is therefore
the union of a countable collection of disjoint open intervals. We denote this collection of intervals
by Wα.
Definition 3.2. Let (X,x) be a pointed topological space. A loop α in X based at x is called
semi-simple if Wα = {(0, 1)} and is called geometrically simple if Wα has one element. If Wα is
finite, then the loop α is called geometrically finite [11].
Lemma 3.2. Every geometrically simple loop is homotopic to a semi-simple loop.
Proof. Let α be a geometrically simple loop at x ∈ X. Then there are r, s ∈ [0, 1] such that
α−1({x}c) = (r, s) and α(r) = α(s) = x. Let β := α|[r,s] and ϕ : [0, 1] −→ [r, s] be a linear
homeomorphism, then β ◦ ϕ is a semi-simple loop at x and α ' β ◦ ϕ.
Lemma 3.2 is proved.
In the sequel, for a semi-simple loop α : I −→ X/A denote α̃ = q−1◦α|(0,1) : (0, 1) −→ (X−A),
where A is a closed subset of a topological space X.
Lemma 3.3. Let A ⊆ X be a closed subset of X and α be a semi-simple loop at ∗ in X/A. If
limt→0 α̃(t) and limt→1 α̃(t) do not exist, then for each t0 ∈ (0, 1), there are b0, b1 ∈ A such that b0
is a limit point of α̃((0, t0)) and b1 is a limit point of α̃((t0, 1)).
Proof. Let t0 ∈ (0, 1) and by contrary suppose that each b ∈ A has an open neighborhood Gb
such that Gb ∩ α̃((0, t0)) = ∅. Then G =
⋃
b∈AGb is an open neighborhood of A and so p(G) is
an open neighborhood of ∗ (since p−1(p(G)) = G) such that does not intersect α((0, t0)) which is a
contradiction to continuity of α.
Lemma 3.3 is proved.
Theorem 3.2. If A is a closed path connected subset of a locally path connected space X such
that every point of A has a countable local base in X, then for each a ∈ A we have
p∗(π
qtop
1 (X, a)) = πqtop1 (X/A, ∗).
Proof. Step 1. Let [α] ∈ π1(X/A, ∗), where α is a semi-simple loop in X/A at ∗.
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ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES 1707
Case 1. Assume a0 = limt→0 α̃(t) and a1 = limt→1 α̃(t) exist, so a0,a1 ∈ A and we can define
a path α : I −→ X such that α|(0,1) = α̃, α(0) = a0, α(1) = a1. Since A is path connected, there
exist paths λ0, λ1 : I −→ A such that λ0 is a path from a to a0 and λ1 is a path from a1 to a.
Therefore λ0 ∗ α ∗ λ1 is a loop at a such that p∗([λ0 ∗ α ∗ λ1]) = [α].
Case 2. If at least one of the above limits does not exist, then we make a sequence {[αn]}n∈N
in Im(p∗) so that converges to [α]. Without lost of generality, we can assume that a0 = limt→0 α̃(t)
exists and a1 ∈ A is a limit point of α̃((1/2, 1)) by Lemma 3.3. We can define a continuous map
α : [0, 1) −→ X such that α|(0,1) = α̃, α(0) = a0. By hypothesis, there is a countable local base
{Oi}i∈N at a1. Let {Gi}i∈N be a sequence of open neighborhoods of a1 such that Gi = O1∩ . . .∩Oi.
SinceX is locally path connected and theGi are open neighborhoods of a1, there exist path connected
open neighborhoods G′i ⊆ Gi of a1. Since the point a1 is a limit point, there are ti ∈ (1/2, 1) such
that α̃(ti) ∈ G′i, ti < ti+1, tn −→ 1 and there are paths γi : [ti, 1] −→ G′i from α(ti) to a1, for all
i ∈ N. Since A is path connected, there exist paths λ0, λ1 : I −→ A such that λ0 is a path from a to
a0 and λ1 is a path from a1 to a. Let αn := λ0 ∗α|[0,tn] ◦ξn ∗γn ◦ζn ∗λ1, where ξn : [0, 1] −→ [0, tn]
and ζn : [0, 1] −→ [tn, 1] are increasing linear homeomorphisms. Note that every αn is a loop in X
at a and if βn := p ◦ (γn ◦ ζn), then
α′n := p ◦ αn = e∗ ∗ α|[0,tn] ◦ ξn ∗ βn ∗ e∗
is a loop in X at ∗ and p∗([αn]) = [α′n]. Define αn : I −→ X/A by
αn(t) =
α(t), 0 ≤ t ≤ tn,
p ◦ γn(t), tn ≤ t ≤ 1,
which is a loop at ∗ and [α′n] = [αn]. Thus it suffices to prove that αn −→ α. If α ∈ 〈K,U〉, where
K is a compact subset of [0, 1] and U is an open subset of X/A, then
(i) If ∗ /∈ U, then K ∩ α−1(∗) = ∅. Let m ∈ N such that tm ≥ maxK. Since for each t ∈ K,
t ≤ tm, we have αn(t) = α(t) ∈ U, for all n > m which implies that αn(K) = α(K) ⊆ U, for each
n > m.
(ii) If ∗ ∈ U, then p−1(U) is an open neighborhood of A, thus there exists m ∈ N such that
G′n ⊆ p−1(U), for each n ≥ m. Therefore Im(γn) ⊆ G′n which implies that Im(p ◦ λn) ⊆ U. Thus
for all t ∈ K and n ≥ m we have
αn(t) =
α(t) ∈ U, t ∈ [0, tn],
(p ◦ γn)(t) ∈ U, t ∈ [tn, 1].
Therefore for a semi-simple loop α in X/A we have [α] ∈ Im(p∗). Similarly, for every loop α such
that α−1({∗}) is finite, [α] ∈ Im(p∗) which implies that the homotopy class of every geometrically
finite loop belongs to Im(p∗) by Lemma 3.2.
Step 2. If α is not geometrically finite, Wα is countable since every open subset of I is a
countable union of open intervals. Let Ij = Lj where Wα = {Lj |j ∈ N} and let
αj(t) =
α(t), t ∈ I1 ∪ . . . ∪ Ij ,
∗, otherwise,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1708 H. TORABI, A. PAKDAMAN, B. MASHAYEKHY
then [αj ] ∈ Im(p∗) since the αj are geometrically finite.
Since (Im(p∗)) = Im(p∗), it suffices to show that αj −→ α. For, if α ∈ 〈K,U〉 for a compact
subset K of [0, 1] and an open subset U of X/A, then
(i) If ∗ ∈ U, then for each t ∈ K and j ∈ N, αj(t) takes value α(t) or ∗ which in both cases
belongs to U, so αj(K) ⊆ U, for all j ∈ N.
(ii) If ∗ /∈ U, thenK∩α−1({∗}) = ∅, soK ⊆ ∪jLj . By compactness ofK we haveK ⊆ ∪sLjs ,
for s = 1, 2, . . . , nK . Let M = max{js|s = 1, 2, . . . , nK}, then αj(K) = α(K) ⊆ U, for each
j ≥M.
Theorem 3.2 is proved.
Corollary 3.4. Let A1, A2, . . . , An be disjoint path connected, closed subsets of a first countable,
connected, locally path connected space X. Then for every a ∈
⋃n
i=1Ai the following equality holds:
p∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2, . . . , An), ∗).
Proof. Let pi : X/(A1, A2, . . . , Ai−1) −→ X/(A1, A2, . . . , Ai). Since every point of each
pi−1(Ai) has a countable local base in the connected, locally path connected space X/(A1, A2, . . .
. . . , Ai−1), by Theorem 3.2 the result holds.
Corollary 3.5. Let A1, A2, . . . , An be disjoint path connected, closed subsets of a first countable,
connected, locally path connected space X such that X/(A1, A2, . . . , An) is semi-locally simply con-
nected. Then for each a ∈
⋃n
i=1Ai, p∗ : π1(X, a) −→ π1(X/(A1, A2, . . . , An, ∗) is an epimorphism.
Proof. Since X/(A1, A2, . . . , An) is connected, locally path connected and semi-locally simply
connected space, πqtop1 (X/(A1, A2, . . . , An), ∗) is a discrete topological group which implies that
Im(p∗) = π1(X/(A1, A2, . . . , An), ∗) by Corollary 3.4.
In the following example, we show that the condition “path connectedness for A”is necessary in
Theorem 3.2.
Example 3.2. Let A = {(1, 0), (0, 1)} ⊂ X = S1. Clearly X/A is homeomorphic to the Fig-
ure 8 space, S1
∨
S1. Since X and X/A are locally path connected and semi-locally simply con-
nected
p∗ : πqtop1 (X, 0) ∼= Z −→ πqtop1 (X/A, ∗) ∼= Z ∗ Z
is a continuous homomorphism of discrete topological spaces. Since the free product Z ∗ Z is not
abelian, p∗ is not onto and since πqtop1 (X/A, ∗) is discrete, Im(p∗) is not dense in πqtop1 (X/A, ∗).
In the following example, we show that the condition “locally path connectedness for X”is
necessary in Theorem 3.2.
Example 3.3. Let X1 =
{
(x, sin(2π/x)) ∈ R2
∣∣ 0 < x ≤ 1
}
, X2 =
{
(x, y) ∈ R2
∣∣ x2 +
y2
4
=
= 1, y ≤ 0
}
, X3 =
{
(x, 0) ∈ R2
∣∣ −1 ≤ x ≤ 0
}
and A =
{
(0, y) ∈ R2
∣∣ −1 ≤ y ≤ 1
}
. If
X = X1 ∪ X2 ∪ X3 ∪ A, then π1(X,x0) = 0 and π1(X/A, ∗) ∼= Z. Since X/A is a locally path
connected and semi-locally simply connected space, πqtop1 (X/A, ∗) is discrete which implies that
p∗(π
qtop
1 (X,x0)) 6= πqtop1 (X/A, ∗).
In the next example, we show that with the assumptions of Theorem 3.2, p∗ is not necessarily
an epimorphism and hence the hypothesis semi-locally simply connectedness in Corollary 3.5 is
essential.
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ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES 1709
Example 3.4. Let Cn =
{
(x, y) ∈ R2
∣∣∣ (x− 1
n
)2
+ y2 =
1
n2
}
, for n ∈ N, HEo =
=
⋃
n∈NC2n−1, HEe =
⋃
n∈NC2n and X =
(
HE0 × {0}
)
∪
(
HEe × {1}
)
∪ A, where A =
=
(
{(0, 0)} × I
)
. One can easily see that X/A is the Hawaiian Earring space. Let α be the loop in
X/A that traverse p(C1), p(C2), . . . in ascending order. By the structure of the fundamental group of
the Hawaiian Earring [7] we have [α] /∈ Im(p∗) since if p∗([β]) = [α], then the loop β must traverse
infinitely many times A which is a contradiction to the continuity of β.
Corollary 3.6. Let A1, A2, . . . , An be subsets of a first countable, connected, locally path con-
nected space X with disjoint path connected closure such that each Ai is closed or open. Then for
any a ∈
⋃n
i=1Ai we have
p∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2, . . . , An), ∗).
Proof. By changing the order, we can assume that A1, . . . , Ak are closed and Ak+1, . . . , An are
open, for a 1 ≤ k ≤ n. By applying Corollary 3.4 we have
q∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2, . . . , Ak), ∗),
where q : X → X/(A1, A2, . . . , Ak) is the natural quotient map. Consider the natural quotient map
r : X/(A1, A2, . . . , Ak) → X/(A1, . . . , Ak, . . . , An). Note that p = r ◦ q and since the Aj have
disjoint path connected closures, Aj is also path connected in X/(A1, . . . , Ak), for all j > k. Now,
using Corollary 3.2 the result holds.
Remark 3.1. Note that since the topology of πτ1 (X,x) is coarser than πqtop1 (X,x), the results
of this section can be restated for πτ1 when we replace πqtop1 with πτ1 .
4. Some applications. It seems interesting to investigate on the topology of quasitopologi-
cal fundamental groups and some people have found some properties of this topology (see [2 – 5,
10, 13, 14, 17]). In this section, we intend to give some applications of the results of the previ-
ous section to find out some properties of the topological fundamental group of the quotient space
X/(A1, A2, . . . , An). By (X,A1, A2, . . . , An) we mean an (n + 1)-tuple of spaces with one of the
following conditions (♣):
(i) The Ai are open subsets of X with path connected closures.
(ii) X is a connected, locally path connected, first countable space and the Ai are closed subsets
of X with disjoint path connected closures.
Theorem 4.1. For an (n+ 1)-tuple of spaces (X,A1, A2, . . . , An) with the assumption (♣), if
X is simply connected, then πqtop1 (X/(A1, A2, . . . , An), ∗) is an indiscrete topological group.
Proof. Since X is simply connected, p∗(π
qtop
1 (X, a))={[e∗]}, where e∗ is the constant loop at ∗ in
X/(A1, A2, . . . , An). Then by Corollaries 3.2 and 3.6 {[e∗]}is a dense subset of πqtop1 (X/(A1, A2, . . .
. . . , An), ∗). Since πqtop1 (X/(A1, A2, . . . , An), ∗) is a quasitopological group, for every [α] ∈
∈ πqtop1 (X/(A1, A2, . . . , An), ∗), the left multiplication L[α] : π
qtop
1 (X/(A1, A2, . . . , An), ∗) −→
−→ πqtop1 (X/(A1, A2, . . . , An), ∗) given by L[α]([β]) = [α ∗ β] is a homeomorphism which implies
that {[α]} is also dense in πqtop1 (X/(A1, A2, . . . , An), ∗). Hence every nonempty open subset of
πqtop1 (X/(A1, A2, . . . , An), ∗) contains every element [α] of πqtop1 (X/(A1, A2, . . . , An), ∗) which
implies that πqtop1 (X/(A1, A2, . . . , An), ∗) is an indiscrete topological group.
Theorem 4.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1710 H. TORABI, A. PAKDAMAN, B. MASHAYEKHY
Theorem 4.2. For an (n+ 1)-tuple of spaces (X,A1, A2, . . . , An) with the assumption (♣), if
πqtop1 (X, a) is compact and πqtop1 (X/(A1, A2, . . . , An), ∗) is Hausdorff, then the quasitopological
fundamental group πqtop1 (X/(A1, A2, . . . , An), ∗) is either a discrete topological group or uncount-
able.
Proof. If πqtop1 (X/(A1, A2, . . . , An), ∗) has at least one isolated point, then every singleton is
open since left translations
L[α] : π
qtop
1 (X/(A1, A2, . . . , An), ∗) −→ πqtop1 (X/(A1, A2, . . . , An), ∗)
are homeomorphisms, for every [α] ∈ πqtop1 (X/(A1, A2, . . . , An), ∗). Thus πqtop1 (X/(A1, A2, . . .
. . . , An), ∗) is a discrete topological group. It is a well-known result that a nonempty compact Haus-
dorff space without isolated points is uncountable [12] (Theorem 27.7). Hence if πqtop1 (X/(A1, A2, . . .
. . . , An), ∗) has no isolated points, then in order to show that πqtop1 (X/(A1, A2, . . . , An), ∗) is un-
countable it is enough to show that πqtop1 (X/(A1, A2, . . . , An), ∗) is compact. By Corollaries 3.2
and 3.6,
p∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2, . . . , An), ∗).
Since πqtop1 (X, a) is compact and p∗ is continuous p∗π
qtop
1 (X, a) is compact in πqtop1 (X/(A1, A2, . . .
. . . , An), ∗). Since πqtop1 (X/(A1, A2, . . . , An), ∗) is Hausdorff, p∗(π
qtop
1 (X, a)) is closed in
πqtop1 (X/(A1, A2, . . . , An), ∗) and so p∗(π
qtop
1 (X, a)) = πqtop1 (X/(A1, A2, . . . , An), ∗). Hence
πqtop1 (X/(A1, A2, . . . , An), ∗) is compact and so it is uncountable.
Theorem 4.2 is proved.
Corollary 4.1. For an (n + 1)-tuple of spaces (X,A1, A2, . . . , An) with the assumption (♣),
if πqtop1 (X, a) is a compact, countable quasitopological group, then either X/(A1, A2, . . . , An) is
semi-locally simply connected or πqtop1 (X/(A1, A2, . . . , An), ∗) is not Hausdorff.
Proof. Let πqtop1 (X/(A1, A2, . . . , An), ∗) be Hausdorff, then by a similar proof of Theorem 4.2
p∗ is onto. Therefore πqtop1 (X/(A1, A2, . . . , An), ∗) is countable since πqtop1 (X, a) is countable.
Theorem 4.2 implies that πqtop1 (X/(A1, A2, . . . , An), ∗) is a discrete topological groups. Hence by
Theorem 2.2 X/(A1, A2, . . . , An) is semi-locally simply connected.
If U is an open cover of a connected and locally path connected space X, then the subgroup
of π1(X,x) consisting of all homotopy classes of loops that can be represented by a product of the
following type:
n∏
j=1
ujvju
−1
j ,
where the uj are arbitrary paths (starting at the base point x) and each vj is a loop inside one of the
neighborhoods Ui ∈ U , is called the Spanier group with respect to U , denoted by π(U , x) [8, 16].
Definition 4.1 [8, 16]. The Spanier group of the space X which we denote it by πsp1 (X,x), is
defined as follows:
πsp1 (X,x) =
⋂
open covers U
π(U , x).
The authors [10] introduce Spanier spaces which are spaces such that their Spanier groups are
equal to their fundamental groups. Also, the authors prove that for a connected and locally path
connected space X, {[ex]} ⊆ πsp1 (X,x). Hence, for an (n+ 1)-tuple of spaces (X,A1, A2, . . . , An)
with the assumption (♣), where X is simply connected, we have
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ON TOPOLOGICAL FUNDAMENTAL GROUPS OF QUOTIENT SPACES 1711
πqtop1 (X/(A1, . . . , An), ∗) = p∗(π1(X,x)) = {[ex]} ⊆ πsp1 (X/(A1, . . . , An), ∗).
Clearly simply connected spaces are Spanier spaces which we can call them trivial Spanier spaces.
It is interesting for the authors to obtain some ways to construct nontrivial Spanier spaces. The
following result which is an immediate consequence of the above argument gives a way to construct
some Spanier spaces from simply connected spaces.
Theorem 4.3. For an (n+ 1)-tuple of spaces (X,A1, A2, . . . , An) with the assumption (♣), if
X is simply connected, then X/(A1, A2, . . . , An) is a Spanier space.
In the following example, we show 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 πqtop1 (X/A, ∗) is an indiscrete topological group. Hence X/A is a nontrivial Spanier
space.
Example 4.1. Using the definitions of Example 3.4, let CHEo and CHEe be cones over HEo
and HEe with height
1
2
and let X = CHEo∪CHEe∪A. By the van Kampen theorem, X is simply
connected, but X/A is not simply connected (see [9]). Hence X/A is a nontrivial Spanier space.
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Received 04.04.12,
after revision — 26.07.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
|
| id | umjimathkievua-article-2547 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:32Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f7/dca609d0e74aa0051b502ef7634330f7.pdf |
| spelling | umjimathkievua-article-25472020-03-18T19:26:06Z On the Topological Fundamental Groups of Quotient Spaces Про топологічні фундаментальні групи фактор-просторів Mashayekhy, B. Pakdaman, A. Torabi, H. Машаєхі, Б. Пакдаман, А. Торабі, Х. Let p: X → X/A be a quotient map, where A is a subspace of X. We study the conditions under which p ∗(π 1 qtop (X, x 0)) is dense in π 1 qtop (X/A,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and p * is a continuous homomorphism induced by the quotient map p. In addition, we present some applications in order to determine the properties of π 1 qtop (X/A,∗). In particular, we establish conditions under which π 1 qtop (X/A,∗) is an indiscrete topological group. Нехай p: X → X/A — фактор-відображення, де A — підпростір X. Досліджуються умови, за яких p ∗(π 1 qtop (X, x 0)) є щільною в π 1 qtop (X/A,∗)), де фундаментальні групи наділені природною фактор-топологією, успадковaною від простору петель, а p * — неперервний гомоморфізм, індукований фактор-відображенням p. Крім того, наведено деякі застосування з метою визначити деякі властивості π 1 qtop (X/A,∗). Наприклад, встановлено умови, за яких π 1 qtop (X/A,∗) є недискретною топологічною групою. Institute of Mathematics, NAS of Ukraine 2013-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2547 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 12 (2013); 1700–1711 Український математичний журнал; Том 65 № 12 (2013); 1700–1711 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2547/1854 https://umj.imath.kiev.ua/index.php/umj/article/view/2547/1855 Copyright (c) 2013 Mashayekhy B.; Pakdaman A.; Torabi H. |
| spellingShingle | Mashayekhy, B. Pakdaman, A. Torabi, H. Машаєхі, Б. Пакдаман, А. Торабі, Х. On the Topological Fundamental Groups of Quotient Spaces |
| title | On the Topological Fundamental Groups of Quotient Spaces |
| title_alt | Про топологічні фундаментальні групи фактор-просторів |
| title_full | On the Topological Fundamental Groups of Quotient Spaces |
| title_fullStr | On the Topological Fundamental Groups of Quotient Spaces |
| title_full_unstemmed | On the Topological Fundamental Groups of Quotient Spaces |
| title_short | On the Topological Fundamental Groups of Quotient Spaces |
| title_sort | on the topological fundamental groups of quotient spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2547 |
| work_keys_str_mv | AT mashayekhyb onthetopologicalfundamentalgroupsofquotientspaces AT pakdamana onthetopologicalfundamentalgroupsofquotientspaces AT torabih onthetopologicalfundamentalgroupsofquotientspaces AT mašaêhíb onthetopologicalfundamentalgroupsofquotientspaces AT pakdamana onthetopologicalfundamentalgroupsofquotientspaces AT torabíh onthetopologicalfundamentalgroupsofquotientspaces AT mashayekhyb protopologíčnífundamentalʹnígrupifaktorprostorív AT pakdamana protopologíčnífundamentalʹnígrupifaktorprostorív AT torabih protopologíčnífundamentalʹnígrupifaktorprostorív AT mašaêhíb protopologíčnífundamentalʹnígrupifaktorprostorív AT pakdamana protopologíčnífundamentalʹnígrupifaktorprostorív AT torabíh protopologíčnífundamentalʹnígrupifaktorprostorív |