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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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