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Spanier spaces and covering theory of non-homotopically path

Hausdorff spaces

Ali Pakdaman, Hamid Torabi, Behrooz Mashayekhy∗

Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures,

Ferdowsi University of Mashhad,

P.O.Box 1159-91775, Mashhad, Iran.

Abstract

H. Fischer et al. (Topology and its Application, 158 (2011) 397-408.) introduced

the Spanier group of a based space (X,x) which is denoted by πsp

Spanier space we mean a space X such that πsp

In this paper, first we give some examples of Spanier spaces. Then we study the

influence of the Spanier group on covering theory and introduce Spanier coverings

which are universal coverings in the categorical sense. Second, we give a necessary

and sufficient condition for the existence of Spanier coverings for non-homotopically

path Hausdorff spaces. Finally, we study the topological properties of Spanier groups

and prove that the topological fundamental groups of spaces with trivial Spanier

groups have T1topology.

1(X,x). By a

1(X,x) = π1(X,x), for every x ∈ X.

Keywords:

Hausdorff.

2010 MSC: 57M10, 57M05, 55Q05, 57M12

Covering space, Spanier group, Spanier space, Homotopically path

1. Introduction and motivation

A continuous map p :? X −→ X is a covering of X, and? X is called a covering space

is evenly covered by p, that is, p−1(U) is a disjoint union of open subsets of? X each

connected and semi-locally simply connected space X, it is well known that there

of X, if for every x ∈ X there exists an open subset U of X with x ∈ U such that U

of which is mapped homeomorphically onto U by p. For a connected, locally path

∗Corresponding author

Email addresses: Alipaky@yahoo.com (Ali Pakdaman), hamid−torabi86@yahoo.com

(Hamid Torabi), bmashf@um.ac.ir (Behrooz Mashayekhy)

Preprint submitted toMay 23, 2011

arXiv:1105.4010v1 [math.AT] 20 May 2011

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is a 1-1 correspondence between its connected covering spaces and the subgroups of

its fundamental group π1(X,x), for a point x ∈ X [8]. E. H. Spanier [8] classified

connected covering spaces of the space X using some subgroups of the fundamental

group of X, recently named Spanier groups. If U is an open cover of X, the subgroup

of π1(X,x) consisting of the homotopy classes of loops that can be represented by a

product of the following type:

n

?

where the uj’s are arbitrary paths starting at the base point x and each vjis a loop

inside one of the neighborhoods Uj∈ U. This group is called the unbased Spanier

group with respect to U, denoted by π(U,x) [8, 5]. The following theorem is an

interesting result on the above notion.

j=1

ujvju−1

j,

Theorem 1.1. ([8]) For a connected, locally path connected space X, if H is a

subgroup of π1(X,x) for x ∈ X and there exists an open cover U of X such that

π(U,x) ≤ H, then there exists a covering p :? X −→ X such that p∗π1(? X, ˜ x) = H.

there exists an open cover U such that π(U,x) = 1, for a point x ∈ X, the existence

of simply connected universal covering follows from the above theorem. But without

locally path connectedness, these results fail since there exists a semi-locally simply

connected space with nontrivial Spanier groups corresponding to every its open cover.

H. Fischer, D. Repovs, Z.Virk, A. Zastrow [5] proposed a modification of Spanier

groups so that the corresponding results will be correct for all spaces. In order to

do this, they instead of open sets U also considered “pointed open sets”, i.e. pairs

(U,x), where x ∈ U and U is open. Let U = {Ui|i ∈ I} be a cover of X by open sets.

For each Ui∈ U take |Ui| copies into V and define each of those copies as (Ui,p), i.e.

use the same set Uias first entry, and let the second entry run over all points p ∈ Ui.

Definition 1.2. ([5]) Let X be a space, x ∈ X, and V = {(Vi,xi)|i ∈ I} be a cover

of X by open neighborhood pairs. Then let π∗(V,x) be the subgroup of π1(X,x) which

contains all homotopy classes having representatives of the following type:

Since for a locally path connected and semi-locally simply connected space X

n

?

j=1

ujvju−1

j,

where the uj’s are arbitrary paths that run from x to some point xiand each vjthen

must be a closed path inside the corresponding Vi. This group is called the based

Spanier group with respect to V, denoted by π∗(V,x).

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Let U be a refinement of an open covering V. Then π(U) ≤ π(V) and π∗(U) ≤

π∗(V) if U and V are open coverings of pointed sets. By these inclusions, there

exist inverse limits of these Spanier groups, defined via the directed system of all

coverings with respect to refinement. H. Fisher et al. [5] called them the unbased

Spanier group and the based Spanier group of the space X which we denote them by

πsp

are realized by intersections as follows:

1(X,x) and πbsp

1(X,x), respectively. They also mentioned that these inverse limits

πsp

1(X,x) =

?

?

open covers U

π(U,x),

πbsp

1(X,x) =

open covers V by pointed sets

π∗(V,x).

For the spaces that are not locally nice, classification of covering spaces is not

as pleasant. H. Fischer and A. Zastrow in [6] defined a generalized regular covering

which enjoys most of the usual properties of classical coverings, with the possi-

ble exception of evenly covered neighborhoodness. If X is connected, locally path-

connected and semi-locally simply connected, then the generalized universal covering

p :? X −→ X agrees with the classical universal covering. While semi-local simple

covering space theory mainly considered the condition called “homotopically Haus-

dorf”. A space X is homotopically Hausdorff if given any point x in X and any

nontrivial homotopy class [α] ∈ π1(X,x), then there is a neighborhood U of x which

contains no representative for [α].

H. Fischer et al. [5] proved that triviality of based Spanier group implies the

existence of generalized universal covering. In fact, they proved that for a space X,

if πbsp

of generalized universal covering.

Although all homotopically path Hausdorff spaces are homotopically Hausdorff,

but there exist non-homotopically path Hausdorff spaces which are homotopically

Hausdorff [5, Proposition 3.4]. Also, the authors [7, 10] studied the covering theory

of non-homotopically Hausdorff spaces. Accordingly, we would like to study coverings

of non-homotopically path Hausdorff spaces and investigate the topology type of their

fundamental group and their universal covering spaces.

In Section 2, we introduce Spanier spaces and based Spanier spaces which are

the spaces that their fundamental group is equal to their Spanier groups and based

Spanier groups, respectively. By some examples we show that these notions are

different, although for locally path connected spaces they are the same since for

connectivity is a crucial condition in classical covering space theory, the generalized

1(X,x) = 1, then X is homotopically path Hausdorff which implies the existence

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locally path connected spaces we have πbsp

that small generated spaces in the sense of [10] are based Spanier spaces and hence

Spanier spaces but the converse is not true in general.

In Section 3, we prove that for every covering p : ? X −→ X of a space X,

covering space if it is non-homotopically path Hausdorff. Then, we introduce Spanier

coverings that are universal coverings in the categorical sense. Also, in this case

πsp

In Section 4, we present the main result of this article which states that a con-

nected and locally path connected space X has a Spanier covering if and only if X is

a semi-locally Spanier space, that is, for every x ∈ X there exists an open neighbor-

hood U such that the homotopy class of every loop in U belongs to πsp

we prove that for a connected and locally path connected space X which admits a

Spanier covering, for every subgroup H of π1(X,x) which contains πsp

exists a covering p :? XH−→ X such that p∗π1(? XH, ˜ x) = H, where ˜ x ∈ p−1({x}).

topological fundamental groups. We prove that in connected and locally path con-

nected spaces, πsp

topology, then X is a Spanier space. Using this, we show that the topological funda-

mental group of a connected and locally path connected space with trivial Spanier

group has T1topology.

Throughout this article, all the homotopies between two paths are relative to end

points, X is a path connected space with the base point x ∈ X, and p :? X −→ X

by the Spanier group we mean the unbased once and since in locally path connected

spaces based and unbased cases are coincide, we remove prefix unbased.

1(X,x) = πsp

1(X,x) [5]. Also, we show

p∗π1(? X, ˜ x) contains πsp

1(X,x) = p∗π1(? X, ˜ x).

1(X,x) as a subgroup and so X has no simply connected

1(X,x). Also,

1(X,x) there

Finally in Section 5, we study the topological properties of the Spanier group in

1(X,x) is a closed subgroup and hence if πtop

1(X,x) has indiscrete

is a path connected covering of X with ˜ x ∈ p−1({x}) as the base point of? X. Also,

2. Spanier spaces

Definition 2.1. We call a topological space X the unbased Spanier space if π1(X,x) =

πsp

x ∈ X.

Since for locally path connected spaces the based and unbased Spanier groups

coincide [5], we use the terminology the Spanier space but for general spaces we omit

the prefix unbased for brevity. If a space X is not homotopically Hausdorff, then

there exists x ∈ X and a nontrivial loop in X based at x which is homotopic to a

loop in every neighborhood U of x. Z. Virk [11] called these loops as small loops and

showed that for every x ∈ X they form a subgroup of π1(X,x) which is named small

1(X,x), and the based Spanier space if π1(X,x) = πbsp

1(X,x), for an arbitrary point

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loop group and denoted by πs

generated) subgroup, denoted by πsg

following set

{[α ∗ β ∗ α−1] | [β] ∈ πs

where P(X,x) is the set of all paths from I into X with initial point x [11]. Also, Virk

[11] introduced small loop spaces which are spaces with all loops as small loops and

the authors [10] introduced small generated spaces which their SG subgroup is their

fundamental group. The following proposition easily comes from the definitions.

1(X,x). In general, these loops make the SG (small

1(X,x), which is the subgroup generated by the

1(X,α(1)), α ∈ P(X,x)},

Proposition 2.2. For a topological space X and every x ∈ X, πs

πbsp

1(X,x) ≤ πsg

1(X,x) ≤

1(X,x) ≤ πsp

Example 2.3. Small loop spaces and small generated spaces are based Spanier spaces

and therefore by the are Spanier spaces.

1(X,x)

In the following we construct a Spanier space which is not small generated.

Example 2.4. Let X = {(x,y) ∈ R2|1

Consider the loops αi: I −→ X by αi(t) = rie2πit, where −1

for i ∈ N. Put X0 = X and let Xi = Xi−1∪αiCi, where Ci is a cone over S1

with height 1, be the space obtained by attaching the cone Ci to Xi−1 via αi, for

all i ∈ N and let Y =?

by α(t) =

λ(0) = (1,0). Since [λi∗ αi∗ λ−1

We show that [λ ∗ α ∗ λ−1] ∈ πsp

therefore Y is a Spanier space. Given any open cover U of α([0,1]) choose a finite

refinement by balls U0,...,Uk so that Ui∩ Uj ?= ∅, for i,j ∈ Zk+1 if and only if

|i − j| ≤ 1. Choose m big enough so that αm([0,1]) is covered by U0,...,Ukas well.

Fix points xi∈ Ui−1∩ Ui∩ Imα and yi∈ Ui−1∩ Ui∩ Imαm and let ηi be a path

from xito yi. Note that for every i ∈ Zk+1the loop θi= α|i∗ ηi∗ αm|−1

contained in Uias suggested by Figure 1, where α|iis a path from bito bi+1which

is the restriction of α to an appropriate interval and αm|iis a path from aito ai+1

which is the restriction of αmto an appropriate interval. Without lost of generality,

we can assume that x0= α(0),y0= αm(0) ∈ U0. Define a path hibetween x0and

xito be the restriction of α to an appropriate interval and cito be the path λ ∗ hi.

Then (c0∗θ0∗c−1

π(U,y) and

(c0∗ θ0∗ c−1

4≤ x2+y2≤ 1}\{(−1

2−

2−

1

i+1,0) ∈ R2| i ∈ N}.

1

i+1< ri< −1

2−

1

i+2,

i∈NXi. Define λi : I −→ [1

2+1

i, α : I −→ Y be the loop

i,1] × {0} be the linear

homeomorphism such that λi(0) = (1,0), βi= λi∗ αi∗ λ−1

1

2e2πitand λ : I −→ [1

2,1] × {0} be the linear homeomorphism such that

i] = 1, π1(Y,y) = ?[λ ∗ α ∗ λ−1]?, where y = (1,0).

1(Y,y) which implies that π1(Y,y) = πsp

1(Y,y) and

i

∗ η−1

i−1is

0)∗(c1∗θ1∗c−1

1)∗...(ck∗θk∗c−1

k) is a loop with homotopy class in

0) ∗ (c1∗ θ1∗ c−1

1) ∗ ...(ck∗ θk∗ c−1

k) ? (λm∗ αm∗ λ−1

m) ∗ (λ ∗ α−1∗ λ−1),

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Figure 1:

which implies that [λ ∗ α−1∗ λ−1] ∈ π(U,y) since [λm∗ αm∗ λ−1

not depend to cover U, [λ ∗ α ∗ λ−1] ∈ πsp

The following example shows that every Spanier space is not necessarily a based

Spanier space.

m] = 1. Since α does

1(Y,y), as desired.

Example 2.5. Consider the space Y introduced in [5, Fig. 1] which is obtained by

taking the surface obtained by rotating the topologists sine curve about its limiting

arc and then adding a single arc C. This arc C can be easily embedded into R3, so

as not to intersect the surface portion or the central axis at any other points than its

endpoints (see Fig. 1). By [5, Proposition 3.1], the space Y is semi-locally simply

connected which implies that πsg

Y . Let ρrdenote a simple path on the surface starting at x0, contained in the plane

determined by x0and the central axis, with endpoint at distance r from the central

axis. Let αrbe the simple loop with radius 0 < r < 1 on the surface. Obviously αris

not null homotopic and any neighborhood of a point of the central axis contains such

a loop. For every 0 < r < 1 the loops ρ ∗ αr∗ ρ−1(with αrappropriately based) are

homotopic to each other and non-trivial and hence πsp

the space Y is not locally path connected and πbsp

πsp

1(X,x) = 1. Fix a point x0on the surface portion of

1(Y,x0) = π1(Y,x0). Note that

1(Y,x0) = 1 and hence the equality

1(X,x) = πbsp

1(X,x) does not hold in general.

3. Spanier coverings

The importance of Spanier groups was pointed out by Conner, Meilstrup, Repovs,

Zastrow and Zeljko [4] and by Fischer, Repoves, Virk and Zastrow [5]. In this

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section, we study some basic properties of Spanier groups and their relations to the

covering spaces. By convention, the term universal covering will always mean a

categorical universal object, that is, a covering p :? X −→ X with the property that

r :? X −→?Y such that q ◦ r = p. Also, we denote by COV(X) the category of all

Theorem 3.1. For every covering p :? X −→ X and x ∈ X the following relations

πbsp

for every covering q :?Y −→ X with a path connected space?Y there exists a covering

coverings of X as objects and covering maps between them as morphisms.

hold:

1(X,x) ≤ πsp

1(X,x) ≤ p∗π1(? X, ˜ x).

Proof. Let U be a cover of X by evenly covered open subsets. It suffices to show

that π(U,x) ≤ p∗π1(? X, ˜ x). For this, let [α] ∈ π(U,x). Then there are open subsets

i = 1,2,...,n, such that

Ui ∈ U, paths αi from x to xi ∈ Ui and loops βi : I −→ Ui based at xi, for

α ? (α1∗ β1∗ α−1

1) ∗ (α2∗ β2∗ α−1

2) ∗ ... ∗ (αn∗ βn∗ α−1

α−1

i

be the lift of α−1

n).

Let ? αibe the lift of α with initial point ˜ x,?

˜ xi= ? αi(1). Now define

? α = (?

We know that the image subgroup p∗π1(? X, ˜ x) in π1(X,x) consists of the homotopy

following result holds.

i

with initial point

? αi(1) and?βi = (p|Vi)−1◦ βi where Vi is the homeomorphic copy of Ui containing

α1∗?β1∗?

α−1

1) ∗ (?

α2∗?β2∗?

α−1

2) ∗ ... ∗ (?

αn∗?

βn∗?

α−1

n)

which is a loop in? X and p ◦ ? α ? α which implies that [α] ∈ p∗π1(? X, ˜ x).

classes of loops in X based at x whose lifts to? X starting at ˜ x are loops. Hence the

Corollary 3.2. If p :? X −→ X is a covering and [α] ∈ πsp

Corollary 3.3. If the space X is not homotopically path Hausdorff, then X does not

admit a simply connected covering space.

1(X,x), then every lift of

α in? X is a loop.

Corollary 3.4. Let X be a connected, locally path connected and simply connected

space. If the action of a group G on X is properly discontinuous, then πsp

and therefore it is homotopically path Hausdorff.

1(X/G) = 1

Corollary 3.5. For every covering p :? X −→ X, πsp

1(X,x) acts trivially on p−1({x}),

1(X,x). that is, ˆ x.[α] = ˆ x, for all ˆ x ∈ p−1({x}) and [α] ∈ πsp

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Theorem 3.6. Every covering space of a (based or unbased) Spanier space X is

homeomorphic to X.

Proof. Let p :? X −→ X be a covering of a Spanier space X. Then by Theorem 3.1

p :? X −→ X is a one sheeted covering of X. Hence? X is homeomorphic to X.

Definition 3.7. By a (based) Spanier covering of a topological space X we mean a

covering p :? X −→ X such that? X is a (based) Spanier space.

The following proposition comes from definitions and Theorem 3.1.

πsp

1(X,x) ≤ p∗π1(? X, ˜ x) ≤ π1(X,x) = πsp

1(X,x), for each x ∈ X which implies that

Proposition 3.8. For a space X the following statements hold.

(i) Every based Spanier covering is a Spanier covering.

(ii) If X has a based Spanier covering, then πbsp

1(X,x) = πsp

1(X,x).

Lemma 3.9. If p : ? X −→ X is a covering and [α] belongs to πbsp

Proof. Let U be a cover of X by evenly covered open neighborhoods and then V =

p−1(U) is an open cover of? X. Since πbsp

from ˜ x and loops βi: I −→ Vibased at αi(1) = yi, for i = 1,2,...,n, such that

α ? (α1∗ β1∗ α−1

If λi= p ◦ αiand θi= p ◦ βi, for i = 1,2,...,n, then the θi’s are loops in (Ui,xi) =

p((Vi,yi)) ∈ U based at λi(1) = xi. Therefore

p ◦ α ? (λ1∗ θ1∗ λ−1

which implies that [p ◦ α] ∈ π(U,x). Since every open cover of X has a refinement

by evenly covered open subset, [p ◦ α] ∈ πbsp

Theorem 3.10. (i) A covering p :? X −→ X is a based Spanier covering if and only

(ii) A covering p : ? X −→ X is a Spanier covering if and only if πsp

1(? X, ˜ x) (or

πsp

1(? X, ˜ x)), then [p ◦ α] belongs to πbsp

1(X,x) (or πsp

1(X,x)).

1(? X, ˜ x) ⊆ π(V, ˜ x), [α] ∈ π(U, ˜ x) and hence

there are pointed open subsets (V1,y1),(V2,y2),...,(Vn,yn) ∈ V, the paths αiinitiated

1) ∗ (α2∗ β2∗ α−1

2) ∗ ... ∗ (αn∗ βn∗ α−1

n).

1) ∗ (λ2∗ θ2∗ λ−1

2) ∗ ... ∗ (λn∗ θn∗ λ−1

n)

1(X,x).

if πbsp

1(X,x) = p∗π1(? X, ˜ x).

p∗π1(? X, ˜ x).

1(X,x) =

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Proof. (i) By definition of based Spanier coverings, π1(? X, ˜ x) = πbsp

[α] ∈ π1(? X, ˜ x) and V be an open cover of? X such that U = p(V) be an open cover

πbsp

U, the paths αiinitiated from x and loops βi: I −→ Uibased at αi(1) = xi, for

i = 1,2,...,n, such that

1(? X, ˜ x). Using

Lemma 3.9 and Proposition 2.2 we have πbsp

1(X,x) = p∗π1(? X, ˜ x). Conversely, let

of X by evenly covered open subsets of X. Since p∗π1(? X, ˜ x) = πbsp

1(X,x), [p ◦ α] ∈

1(X,x) and hence there are pointed open subsets (U1,x1),(U2,x2),...,(Un,xn) ∈

p ◦ α ? (α1∗ β1∗ α−1

1) ∗ (α2∗ β2∗ α−1

2) ∗ ... ∗ (αn∗ βn∗ α−1

n).

Let ? αibe the lift of αiwith initial point ˜ x and?βi= (p|Vi)−1◦βibe the loop with base

then

[(?

is injective we have

α1∗?β1∗ ?

Lemma 3.11. Let X have a based Spanier covering p :? X −→ X and α be a loop in

if a loop α in X has a closed lift ? α in? X, then every lift of α in? X is also closed.

open neighborhood pairs and let V = p−1(U). Since [? α] ∈ π1(? X, ˜ x) = πbsp

loops βi: I −→ Vibased at αi(1) = yi, for i = 1,2,...,n, such that

? α ? (α1∗ β1∗ α−1

α ? p ◦?(α1∗ β1∗ α−1

which implies that [α] ∈ πbsp

Theorem 3.12. A Spanier covering of a locally path connected space X is the uni-

versal object in the category COV(X).

Proof. Assume that p :? X −→ X is a Spanier covering of X and q :?Y −→ X

q∗π1(?Y ) which implies that there exists f :? X −→?Y such that q ◦ f = p.

9

point ? αi(1), where Viis the homeomorphic copy of Uiin? X which contains ? αi(1),

α1

If?

[α] = [(?

α1∗?β1∗ ?

−1) ∗ ... ∗ (?

αn∗?

βn∗ ?

αn

−1)] ∈ π(V, ˜ x).

α−1

i

= ? αi

βn∗ ?

α−1

i

is the lift of α−1

i

with initial point?βi(1), then?

α1

−1and hence Since p∗

−1) ∗ ... ∗ (?

αn∗?

αn

−1)] ∈ π(V, ˜ x).

(ii) By a similar proof to (i) the result holds.

X which has a closed lift ? α in? X, then [α] ∈ πbsp

Proof. Assume that a lift ? α of α is closed and U is a cover of X by evenly covered

π(V, ˜ x), there are (V1,y1),(V2,y2),...,(Vn,yn) ∈ V, the paths αi from ˜ x to yi and

1(X,x) = πsp

1(X,x). Consequently,

1(? X, ˜ x) ≤

1) ∗ (α2∗ β2∗ α−1

2) ∗ ... ∗ (αn∗ βn∗ α−1

n).

If (Ui,xi) = p((Vi,yi)), then (Ui,xi) ∈ U since V = p−1(U) and hence

1) ∗ (α2∗ β2∗ α−1

1(X,x) = πsp

2) ∗ ... ∗ (αn∗ βn∗ α−1

1(X,x) since U is arbitrary.

n)?∈ π(U,x),

is another covering. By Proposition 2.2 and Theorem 3.10 p∗π1(? X) = πsp

1(X) ≤

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4. Existence

Suppose p :? X −→ X is a Spanier covering. Every point x ∈ X has a neigh-

α in U lifts to a loop ? α in?U, and [? α] ∈ πsp

borhood U having a lift?U ⊆? X projecting homeomorphically to U by p. Each loop

composing this with Lemma 3.9, [α] ∈ πsp

property: Every point x ∈ X has a neighborhood U such that i∗π1(U,x) ≤ πsp

Definition 4.1. We call a space X semi-locally Spanier space if and only if for each

x ∈ X there exists an open neighborhood U of x such that i∗π1(U,x) ≤ πsp

where i : U −→ X is the inclusion map.

Example 4.2. Every Spanier space is a semi-locally Spanier space. Also, the product

X ×Y is a semi-locally Spanier space if X is a Spanier space and Y is either locally

simply connected or locally path connected and semi-locally simply connected. If

(X,x) is a pointed Spanier space and (Y,y) is a locally nice space like the above, then

one point union X ∨ Y is a semi-locally Spanier space.

The following lemma easily comes from definitions.

1(? X, ˜ x) since π1(? X, ˜ x) = πsp

1(? X, ˜ x). So,

1(X,x). Thus the space X has the following

1(X,x).

1(X,x),

Lemma 4.3. Let U be an open cover of a space X and θ be a path in X with θ(0) = x1

and θ(1) = x2. Then π(U,x2) = ϕθπ(U,x1), where ϕθ: π1(X,x1) −→ π1(X,x2) is

the isomorphism given by ϕθ([α]) = [θ−1∗ α ∗ θ].

Corollary 4.4. For a space X and x1,x2∈ X, the homomorphism φθ: πsp

πsp

1(X,x1) −→

1(X,x2) defined as the same as ϕθis an isomorphism.

Theorem 4.5. A connected and locally path connected space X has Spanier covering

if and only if X is a semi-locally Spanier space.

Proof. The discussion at the beginning of the section follows necessity. For suffi-

ciently, we show that there exists an open cover U of X such that π(U,x) = πsp

which implies the existence of a covering p : ? X −→ X such that p∗π1(? X, ˜ x) =

Let U be the open cover of X consisting of the path connected open subsets of X

such that i∗π1(U,y) ≤ πsp

By definition of the Spanier group we have πsp

of U and Corollary 4.5 we have π(U,x) ≤ πsp

as desired.

1(X,x)

πsp

1(X,x), which is a Spanier covering by Theorem 3.10.

1(X,y), for U ∈ U, where i : U −→ X is the inclusion map.

1(X,x) ≤ π(U,x). Also, by the choice

1(X,x). Therefore π(U,x) = πsp

1(X,x),

Using Theorem 1.1 we have the following result.

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Theorem 4.6. Suppose X is a connected, locally path connected and semi-locally

Spanier space. Then for every subgroup H ≤ π1(X,x) containing πsp

exists a covering p :? XH −→ X such that p∗π1(? XH, ˜ x) = H, for a suitably chosen

The following theorem gives a sufficient condition for a fibration with unique path

lifting property to be a covering.

1(X,x), there

base point ˜ x ∈? XH.

Theorem 4.7. Let p :? X −→ X be a fibration with unique path lifting property,

X is a semi-locally Spanier space.

where X is connected and locally path connected. Then p :? X −→ X is a covering if

Proof. By [8, Theorem 2.5.12] p :? X −→ X is a covering if and only if there exists

be chosen as in the proof of Theorem 4.5.

an open cover U of X such that π(U,x) ⊆ p∗π1(? X, ˜ x). Hence it suffices to let U as

5. The topology of Spanier subgroups

In this section, for a space X and x ∈ X, by πtop

fundamental group endowed with the Biss topology that make it a quasitopological

group (a group with a topology that enjoys all properties of topological groups with

the exception continuity of group multiplication). For more details, see [1, 2, 3]. By

[1, Theorem 5.5], the connected coverings of a connected and locally path connected

space X are classified by conjugacy classes of open subgroups of πtop

this and Theorem 1.1, For every open cover U of X, π(U,x) is an open subgroup

of πtop

subgroup of πtop

1(X,x) we mean the famous

1(X,x). Using

1(X,x) and hence is closed subgroup which implies that πsp

1(X,x), by definition.

1(X,x) is closed

Proposition 5.1. For a connected and locally path connected space X, πsp

a closed subgroup of πtop

1(X,x) is

1(X,x), for every x ∈ X.

Using the above proposition the Spanier group of connected and locally path

connected spaces contains the closure of the trivial element of the fundamental group.

Hence we have the following corollary.

Corollary 5.2. Let X be a connected and locally path connected space and x ∈ X.

If πtop

1(X,x) has indiscrete topology, then X is a Spanier space.

Corollary 5.3. Let X be a connected and locally path connected space and x ∈ X.

If πsp

1(X,x) = 1, then πtop

1(X,x) has T1topology.

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Figure 2:

It is known that πtop

space. In the following we give an easy proof for this fact.

1(HE,0) has T1topology, where HE is the Hawaiian earring

Proposition 5.4. The topological fundamental group of the Hawaiian earing has T1

topology.

Proof. Let Un be an open cover of HE by

is easy to see that?

In the following example we show that the locally path connectedness is a neces-

sary condition for closeness of πsp

1

n-neighborhoods, for every n ∈ N. It

1(HE,0) ≤?

1(X,x).

n∈Nπ(Un,0) = 1 which implies that the Spanier group of the

Hawaiian earing is trivial since πsp

n∈Nπ(Un,0).

Example 5.5. Let the space Z ⊆ R3be consist of a rotated topologists’ sine curve

(as shown in Figure.2), the outer cylinder at radius 1, where this surfaces tends to,

and horizontal segment λ from (0,0,0) to (1,0,0) that attaches them. The segment

λ intersects the inner portion at points (1

segment from (0,0,0) to (1

n ∈ N, and α the simple loop with radius 1. Obviously, λn∗ αn∗ λ−1

in uniform topology which is equivalent to compact open topology in metric space Z.

Since [λn∗αn∗λ−1

if U is an open cover of Z such that for every U ∈ U, diam(U) < 1, then π(U,0) = 1

which implies that πsp

n,0,0). Denote by λn: I −→ Z the line

n,0,0), αn: I −→ Z the simple loop with radius

1

n, for

n → λ ∗ α ∗ λ−1

n] = 1 for all n ∈ N, [λ∗α∗λ−1] ∈ {1}, but [λ∗α∗λ−1] ?= 1. Also,

1(Z,0) = 1.

The authors [9] proved that for a first countable, simply connected and locally

path connected space X, if A ⊆ X is a closed path connected subset, then the

quotient space X/A has indiscrete topological fundamental group. Therefore by

Corollary 5.2 we have the following theorem that gives a family of Spanier spaces.

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Theorem 5.6. For a first countable, simply connected and locally path connected

space X, if A ⊆ X is a closed path connected subset, then the quotient space X/A is

a Spanier space.

Remark 5.7. Note that by assumptions of the above theorem, the quotient space

X/A is locally path connected and so based and unbased Spanier groups coincide.

Acknowledgements

This research was supported by a grant from Ferdowsi University of Mashhad.

References

[1] D. Biss, The topological fundamental group and generalized covering spaces,

Topology and its Applications. 124 (2002) 355–371.

[2] J. Brazas, The topological fundamental group and free topological groups,

Topology and its Applications. 158 (2011) 779–802.

[3] J.S. Calcut, J.D McCarthy, Discreteness and homogeneity of the topo-

logical fundamental group, Topology Proc. Vol. 34 (2009) 339–349.

[4] G. Conner, M. Meilstrup, D. Repovˇ s, A. Zastrow, M.ˇZeljko, On

small homotopies of loops, Topology and its Applications. 155 (2008) 1089–

1097.

[5] H. Fischer, D. Repovs, Z. Virk, A. Zastrow, On semilocally simply

connected spaces, Topology and its Applications. 158 (2011) 397–408.

[6] H. Fischer, A. Zastrow, Generalized universal coverings and the shape

group, Fund. Math. 197 (2007) 167–196.

[7] A. Pakdaman, H. Torabi, B. Mashayekhy, Small loop spaces and cov-

ering theory of non-homotopically Hausdorff spaces, Topology and its Appli-

cation. 158 (2011) 803–809.

[8] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

[9] H. Torabi, A. Pakdaman, B. Mashayekhy, A dense subgroup of topo-

logical fundamental group of quotient space, Preprint.

[10] H. Torabi, A. Pakdaman, B. Mashayekhy, Topological fundamental

groups and small generated coverings, http://arxiv.org/abs/math/1102.1080.

13

Page 14

[11] Z. Virk, Small loop spaces, Topology and its Applications. 157 (2010) 451–

455.

14

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