Internat. J. Math. & Math. Sci.
(1989) 89-98
89
VOL. 12 NO.
ABELIAN GROUPS IN A TOPOS OF SHEAVES: TORSION AND ESSENTIAL EXTENSIONS
KIRAN R. BHUTANI
Department of Mathematics
The Catholic University of America
Washington, D.C. 20064
(Received September 24, 1987)
We
ABSTRACT.
investigate
the
of
properties
torsion
groups
and
their
essential
in the category AbShL of Abellan groups in a topos of sheaves on a
We show that every torsion group is a direct sum of its p-prlmary components
and for a torsion group A, the group [A,B] is reduced for any BeAbShL.. We give an
extensions
locale.
example to show that in AbShL the torsion subgroup of an inJectlve group need not be
inJectlve. Further we prove that if the locale is Boolean or finite then essential
extensions of torsion groups are torsion.
Finally we show that for a first countable
hausdorff space X essential extensions of torsion groups in AbSh0(X) are torsion iff X
is discrete.
KEYWORDS AND PHRASES.
Locale, Abellan groups in a topos, Sheaves on a locale.
1980 AMS CLASSIFICATION CODE.
1.
18E15
INTRODUCTION.
In
[I]
the author discusses the notion of inJectlvlty and
abellan groups
in a
topos of sheaves on a locale where as in
inJectlve hulls of
[2] the notion of
inJectlvlty, injectlve hulls and the role played by the in[tlal Boolean algebra in a
topos
is
discussed.
Our purpose here is
to
show how torsion groups and their
essential extensions behave in the category AbShL of abellan groups in the topos ShL
We show that torsion is a local property (Theorem 3.1) but
not a global one (3.2), that is, A torsion in AbShL does not necessarily imply that AE
of sheaves on a locale L.
is
a
However if L has ACC, then torsion implies global
group in Ab.
We prove number of results about torsion groups in AbShL which are analogous
torsion
torsion.
to their counterparts in Ab,
in particular,
we show that every torsion group is a
3.5), and for a torsion group A the
Recall that in the
.I0.
We show by
subgroup of an inJectlve group is [njective.
direct sum of its p-primary components (Theorem
group [A,B]
category Ab,
is
reduced for all B e AbShL (Proposition
the torsion
giving an example that this does not hold in AbShL for an arbitrary L (3.11).
In section 4 we show that in AbShL, essential extensions of torsion groups are
torsion iff every injective group splits into a direct sum of a torsion group and a
torsion free group (Proposition 4.2).
For a Boolean locale and any lignite locale the
above result holds (4.3) and (4.4) respectively).
We also give an example to show
that the converse of (4.3) does not hold.
In (4.7) we give an example of a space X
and a torsion group in AbShX with a non torlson essential extension.
After proving
K.R. BHUTANI
90
some more results about essential extensions of torsion groups, we conclude our paper
by
for a first countable Hausdorff space X,
that
showing
essential extensions of
(Theorem 4.8). For basic
facts about about abelian groups with which this paper is concerned see [3] and [4].
Details concerning presheaves and sheaves on a locale can be found in [5], category
torsion groups in AbShX are torsion groups iff X is discrete
theory in [6] and topos theory in [7].
BACKGROUND
2.
(2.1) Recall that a locale denoted by L is a complete lattice satisfying the
following distribution law;
u
^
^
for all U, and any family (Ui}is in L.
The zero (= bottom) of L will be denoted by
I
O, and the unit (= top) of L by E. A m0.ghism of locales h: L + M (also called
local lattice homomorphism) is a map which preserves arbitrary Joins and finite meets
(hence preserves the zero and the unit).
An obvious example of a locale is the topology OX (that is the lattice of open
sets) of any topolocial space X with Joins as unions and meets as intersections.
REMARKS. A locale L satisfies both the Ascending and Descending Chain Conditions
iff L is finite.
[8]
spatial
To prove the non-trivial implication
and
L
if
0(X)
and X
order g given, such that x g y(x,ysX) iff
{yly
/x
x}.
by AbE one means
a
category
(1) The diagonal map
A
n
A
1A
(if) The sum
+
A
n
denoted by
is the
n
It follows that L is also finite.
n
pf: A
/
qi
A is the unique map such that
fth injection
and the kernel of
monomorphfsm for all 0
(I)
A
for f=l,2...n.
A
+
A shall be denoted by
is
An
+A
+
A
The composition
n
AbE
fth
A fs the
+A
A
DEFINITION 2.3.
in E
abelian groups
as
maps as
and
For A AbE and 0nN,
n
A + A is a unique map such that
for all f=l,2,, ,n, where
An
/x is finite,
E
Pf
A
qi A
AA:
W and for the partial
x
If E is any finitely complete category then
with objects
homomorphisms between them [9].
/
following observations
t4reover DCC for L then implies that
(2.2) ABELIAN GROUPS IN A CATEGORY
A
the
0(x)0(y) (hence iff
and since X is compact by ACC, X itself is finite.
A
note that such an L is
one has
Each x s X has a smallest open neighbourhood
concerning X:
WyEWx), Wx
TO,
is
:n"
called a t_pr_s_fon
proJectfon
1A where
+A AA:
A
!i
A
n
A
A fs
A.
fiE
n
A
fs
a
n s N.
(2) A s AbE is called a torsion group_ fff all
k
0
is, for any two homomorphisms f and g with domain A, if
n
N are jointly epic, that
fk =gk for all 0nsN then
n
n
f;g.
2.4.
Recall that by AbPShL and AbShL one means the categories of Abelfan groups
fn the. topos PShL and ShL of presheaves and sheaves, respectively, on a locale L with
ABELIAN GROUPS IN A TOPOS OF SHEAVES
the component of A at U and if
written as
a
denoted by
A=)
alV.
U the restriction map
V
BU.
then we shall write AU
BU.
AU
Ab
IXI.
Further AbSh3 for the three-element locale is
the Sierpinski space S with points 0 and
Further AbSh3 is also AbShS for
,
#/
case we get for each U
Eu:AbSh+U
,
L
a pair of
pp
AbSh+U and
U
0 if V
U
exact
to
.
We shall also denote
RuA by
pp
AbShL, for which [A,B]U
[A,B]V (V U), given by h
x AbShL
[A,B]U
HL:
H
Ab, AbShL also has an internal hom-functor [-,-]:
x AbShL
H+u(A[U,B[U) with the restriction
(hw)wU hlV (hw)wv [I0].
In (2.3) we described what we mean by torsion free and torsion groups in
2.7.
For the case E
ShL, we have the following:
(I) A
AbShL is a torsion free group iff each AU is torsion free in Ab.
(2) A
AbShL is a torsion group iff
A
t Kern
0#hEN
exists a cover
U
PROPOSITION 2.8.
ViiUi,
and 0
m.1
For any Ue L
e Z,
A
That is for a
such that
the functors
Let A e AbShL which is a torsion group.
PROOF.
mialU
i,
and
E,,
Then
A
Ru(Ker A)
0#neN
OneN
argument it can be shown that
nAIU,
E
U
AIU
preserves torsion groups.
AU, there
0 for all iel.
p erve torsion
groups.
it Kern since
A
On:N
R preserves all co-limlts and limits (2.5),
U
is torsion in AbSh +U.
t Ker
hence
t
n
it follows
RuA
As a special
A(WAU) and
V
left
#,.
to
:AbShL
functors
Besides the obvious external Ab valued hom-functor
2.6.
AbE.
adjoint
adJolnt
preserves all limits and co-llmlts.
Further
maps
left
=I
adJolnt
(A)W
,AV if
(EuA)V
is left exact, left
#
Then
AbShL defined by
is
(#,A)U
AbShL where
(W e L).
tCW
(w)v
(# C)V
AbShL
#: L M we get a pair of
L, and for any V e M
A(#(U)) for U
,
/,
#
AbShL
{I}.
and non-trivlal open set
Recall that for any local lattice homomorphism
2.5.
adjoint functors AbshM
U
AV will be
Ab for the two-element locale 2 and if X is a discrete
AbSh2
topological space then AbShX
E
/
B is a morphism in AbShL
Also if h: A
the same as AbPSh2 that is the arrow category of Ab.
Then
AU
If A is the sheaf reflection of the given presheaf B (also
then its component at U e L is denoted by
NOTE.
L and A e AbShL,AU will
For any U, V
values in the category Ab of abelian groups.
denote
91
By a similar
92
K.R. BHUTANI
TORSION GROUPS.
3.
A AbShL is a torsion group iff there is a cover E
THEOREM 3.1.
AIU
i,
that
is torsion in
AIU
VielU i
such
for all il.
i
(/) Clear by taking the trivial cover of E. On the other hand if sll
AbSh+Ui, we claim A is torsion. So consider any be AU,
PROOF.
are torsion groups in
i
L. Then U
U
AbSh+U
Viel(U
A
UI)
(U A U i) eA(U A
b
and
Ui)= AIUi(UAUi
for all ie I.
all ie I.
But
A[U i
is torsion in
=VjEJi Wji
U
U
b
AU,
i
and
0
njiblWi
0 for all i, J, 0
=JeJi
U
ni
N,
iel
there is a cover
njlb[Wji
N such that
nji
we can find a cover
I,
and so for each i
AbSh+Ui,
0, j e
Ji"
Hence for
such that
Wji
which shows that A is a torsion group in
AbShL.
Proposition 3.1
COUNTEREXAMPLE 3.2.
torsi6n is a local property.
that
shows
However it is not a global property as we shall see from the following counter
+I and A
AbShL given by
example: Consider L
n
Z/nZ
/
>
m
>
z/2z
z/32
3
>
/
z/2z
/
2
>
By Proposition 3.1, A is torsion, since for the cover
Aln
Z/kZ is torsion in
kn
AbSh+n for all
is not torsion in Ab, as the element
DEFINITION 3.3.
n
<
0 (- z/z)
>0
m
m.
(l+nZ)n<m does
0
But
Vn<mn
the group
Am
nmZ/nZ
not have a finite order.
For a given prime p, by the p-primary component of a group A e
n
U
Ker p
We denote the p-primary
0nN
A
A AeAbShL is called a p-primary group if A=A
P
P
AbShL we mean the subgroup of A given by
component of A by
DEFINITION 3.4.
By the torsion subgroup B of an any group AAbShL we mean the
B
n
UonENKer
subgroup of A given by
A.
THEOREM 3.5.
Every torsion group is a direct sum of its p-prlmary components.
Let A be a torsion group and denote by B the presheaf BU
t(AU) the
torsion subgroup of AU.
Then A is the sheaf reflection of ,0".--.>. Now BU
t(AU)-PROOF.
(t(AU))p
where
(t(AU))p
is the subpresheaf
B U
P
denotes the p-prlmary component of t(aU).
(t(AU))
P
then clearly
B
,B
P
in AbPShL.
If
B
P
B
The Sheaf
reflection being a left adjoint preserves co-llmlts, in particular direct sums and so
A
B
(-B)
P
.B
P
But
B
P
A and hence we get
P
A
A
P
93
ABELIAN GROUPS IN A TOPOS OF SHEAVES
_
DEFINTION 3.6. By the torsion t_of a group A we mean the set of all prime
numbers p such that A # 0.
P
A is an essential extension
If A is a torsion group and B
PROPOSITION 3.7.
then B and A have the same torsion type.
Since A
PROOF.
p.
A
B, It follows
c
p
B
p
and therefore
A O B for all
p
A c
p-
Consider any U e L,then
n
"- AU
-"
Hence
A
B
0.
(UonNKer PBU
n
UOnN(AUO Ker PBu
n
"- UOngN(A
Ker
-"
PA)U
Uong N
PB)U
n
(Ker
A U
P
B is an
A
We now want to show that
P- p
B is essential it follows A OC
then since A
c B
is
A N C, thereby showing that App
P
same torsion
the
have
B
A
and
0 which means that
A for all primes p.
P
P
AOC N B
0
This means
B
Hence
0CcB
If
essential extension.
essential.
n
(Uoner PB)U
AU
AUN B U
P
(AO B )U
P
P
A
P
0 iff
P
type.
AbShL to be a reduced group if it has no non zero
inJectlve subgroups. Recall that in the category Ab, for any torsion group B the
We shall prove the analogue of thls for
group Hom(B,K) is reduced for all K Ab.
DEFINITION 3.8.
We call an A
[-,-] of AbShL(2.6).
group then H(A,P) is reduced in Ab for all
the Ab-valued hom-functor H and the internal hom-functor
If A
LEMMA 3.9.
P
AbShL is a torsion
AbShL.
Let 0
C, H(&,P) be an injective subgroup. Consider any
O.
0
Since A is
C, then for some U L and A AU,
AU there exists a cover U
N such that
n
torsion and a
and 0
PROOF.
=u(a)
VII Ui
nialU i
I.
I.
0 for all i
Consider now
0
nkE N,
8
that there exists some
But
u(a)
0 which shows that
k(a[Uk)
nk8
0 for some k
.
0, which means
AbShL(A,P)=H(A,P)
Uk(alU )
0, a contradiction,
is reduced in tIe category Ab.
If A Is a torsion group in AbShL,
PROPOSITION 3.10.
AbShL for all P
0 implies that
then C an inJectlve hence divisible group in Ab implies
C such that
Therefore
nkUk (alU k) 8Uk(nkalU k) SUe(0)
hence C
i
then [A,P]
is reduced in
AbShL.
Then for ome U E L, BU
[A,P] be an Injectlve subgroup.
0 is an injectlve subgroup of [A,P]U
Since A is torsion, it
PROOF.
Let
follows
AIU is
in Ab.
Thus BU
0
B
H+u(AIU,PIU).
torsion (2.8) in
0 for all U
g
H+u(AIU,PIU)
AbSh+U.and so by last lemma
is reduced
0 which means [A,P] is reduced in AbShL.
L, hence B
94
K.R. BHUTANI
REMARK.
Recall that in the category Ab, the torsion subgroup of an inJectlve
group is always inJectlve. We show in the following example that, for an arbitrary L,
the torsion subgroup of an inJectlve group need not be inJective, except for some
special locales which we shall discuss in the next section.
EXAMPLE 3. II.
Consider the locale
P
P1
n<
+
where the
>
>...>
n<E (Pn)
E(P I)
denotes the
subgroup of B, then (TB)n
lq:n< E(Pn)
B-
and since A
A,
4.
>
2
>
AbShL given by
t0
0
are finite groups with increasing exponent.
Pi
B
(TB)
P
P2 X P1
n
results ([I], proposition 2.3) the
where
+ 2 and A
L
TB,
is the minimal
d
H
By one of our previous
inJective hull of A is given by the group
+...+
n<E(en)
E(P2
x E(P
PI
fnJective hull of group
,
Bn all n <
but (TB( + 1))
I)
E(P I)
0
If TB is the torsion
in Ab.
and so
T(B( +1))
E(P ).
n.
Hence TB _c B
it follows TB is not inJective since B, being the inJective hull of
inJectlve extension of A, hence the result.
ESSENTIAL EXTENSIONS OF TORSION GROUPS.
AbShL, all essential extensions of A are torsion,
If for any torsion group A
The followlng
then we say that essential extensions in AbShL preserve torsion.
proposition shows "essential extensions preserve torsion" is a local property.
PROPOSITION 4.1. Essentlal extenslons preserve torslon in AbShL iff there exists
such that essential extensions preserve torsion in AbSh/U I for
a cover E
=VlIUi
I.
all i
For the converse, consider
Clear by taking the trivial cover of E.
I the
i
any essential extension B of the torsion group A in AbShL. Since for each
preserves essential extensions and torsion [I] it
AbShL / AbSh+U
functor
PROOF.
(+)
i:
follows
i,
BIU I
is an essential extension of the torsion group
By hypothesls,
BIU i
is torsion in
AbSh+U
i
all i
e
AIUi
in AbSh+U I.
I, hence by Theorem 2.1, B
is
torsion in AbShL.
For any L, essenltal extensions in AbShL preserve torsion iff
PROPOSITION 4.2.
free
every injectlve group splits into a direct sum of a torsion group and a torsion
group.
PROOF.
AbShL.
Let B denote the torsion subgroup of an injectlve group A
A
Since
group.
torsion
a
C
i
hypothesis
B is any essential extension, then by
If C
B.
C
hence
and
B
C
c__
is injectlve we may assume that C c__ A, so C torsion implies
is
injective. Therefore
B
that
means
whlch
extensions
essential
Thus B has no proper
of E, then TE c__B
subgroup
torsion
the
denotes
TE
If
A.
BeE for some subgroup E of
A
(/)
0. Thus E Is torsion free.
E
0, hence TE
Let P be a torsion group and H the inJectlve hull of P.
and so TEC_B
(+)
By hypothesis H
ABELIAN GROUPS IN A TOPOS OF SHEAVES
95
T e F where T is a torsion group and F is a torsion free group. If F # O, then since
H is an essential extension of P it follows that P
F
0, a contradiction, since P
0 which shows that H is a torsion group. Since every essential
Hence F
is torsion.
H,
extension of P has an embedding into
it follows all essential extensions of P are
torsion, hence the result.
THEOREM 4.3.
For a Boolean locale,
essential
preserve
in AbShL
extensions
torsion.
b
BU.
Let C
Consider an essential extension B of a torsion group A in AbShL.
PROOF.
For any U
denote the torsion subgroup of B(3.4).
g U be the largest element in
Let W
L consider an arbitrary element
+U such that
blW
We claim W is
CW.
+U, S 0 such that S A W 0. Now for
S A W
0.
0 implies V
In
W and so V + V A W
#
0. Since B D_ A is an essential extension therefore there exists a
particular
g AV c__ CV.
Now C is the torsion subgroup of B
S and m g Z such that 0
V
g CV.
CV implies
But then V
O, a contradiction, since
and 0 #
#
E
AV. Hence W is dense in +U. Since L is Boolean we have W
U, thus
0
BU c_ CU for all U g L and so B
C. Hnce B is torsion.
+U.
dense in
V
any
S,
If not, then there exists S
blV
CV gives V
E
his
mblV
mblB
blV
mblV
REMARK.
On the other hand, one can see that if essential extensions preserve
torsion in AbShL, then it does not necessarily follow that L is Boolean.
counterexampl e:
Here is a
B
Consider L
3.
+lh
B
If
B
are torsion in Ab.
given by
and B
2
By ([I], Proposition 2.3) the inJectlve hull of B is
E(B 2)
A
B
is torsion in AbSh3, then both
2
E(Ker h)
x
Hence
which is torsion in AbSh3.
E(B
2
Hence all essential extensions of B are torsion, although L
3 is not Boolean. Of
course the remark is a special case of the following more general result which shows
that there are non-Boolean L such that essential extensions in AbShL preserve torsion.
THEOREM 4.4. For any finite L essential extensions in AbShL preserve torsion.
PROOF.
Let B be any essential extension of the torsion group A. Then for an
arbitrary
U
U V
U e L.
AU,
V...V U and 0
k
a e
U2
then
malU
A
n
that
implies
torsion
nalU
e N such that
i
i
0 for all i and therefore ma
there
m
nln2...n k
i
for each U e L, AU is a torsion group in the category Ab.
0 and
cover
a
is
0 for all i
1,2,...,k.
0.
m #
If
This shows
_
Now, if there are V e L
such that
BV is not a torsion group then let S be minimal such that BS is not
torsion.
Then
U<S
U,
0 and for all U
<
S, BU is a torsic. ,..roup in Ab.
then since each BU is torsion it follows by
prop.> :r
AbSh+W.
3.1 that
If W
B[W
is
By the same argument as above it follows BW is torsion and hence
BS of infinite order.
Since B
Consider an arbitrary b
A is an essenltal
torsion in
W<S.
S
extension,
there exists V
otherwise
0 # mb
AV
has
finite
order
e AV.
Then
b
will
have
finite
Hence
V
W.
nblW--
0.
But 0
and
contradiction, since b has infinite order.
But BW is torsion and so for some 0
0#mb[V
so
Z such that
S and 0 # m
n
N,
infinite order and so by the same argument 0 #
nblW
This implies
#
nb
a contradiction.
V
#
S, for
order,
b[W
#
a
0.
BS is again of
Hence BS is a
96
K.R. BHUTANI
torsion group which contradicts the definition of S.
This shows B is a torsion group
_
in AbShL.
REMARK 3.4.
Recall from (2.1) that the finite locales L are exactly those L in
which both ACC and DCC hold.
It is therefore of interest to note that there exists an
L which satisfies DCC but for which essential extensions in AbShL do not preserve
Here is an example which is actually the same as that considered in (3.11)
If A and its injectlve hull B
A are as in 3.11, then B Is
not torsion because its torsion subgroup Is proper.
torsion.
for a different purpose:
THEOREM 4.5.
If essential extensions preserve torsion in AbShL,
then for all
L, the following are true:
U
(i)
Essential extensions preserve torsion in
AbSh+U.
(ll) Essential extensions preserve torsion in AbSh+U.
PROOF. Let B be any essential extension of the torsion group A in AbSh+U. Since
AbSh+U
AbShL preserves essential extensions [I] and also torsion
the functor
By
is an essential extension of the torsion group
(2.9), it follows
:
EuB
hypothesis
A.
EuB
is torsion in AbShL.
(EuB)
Therefore
B is again torsion since
preserves torsion (2.9), hence the result.
U given by (W)
L
(ll) Consider the local lattice homomorphlsm
AbSh/U
AbShL(2.6) where (,A)W A(U V W), W
Then
produces
the functor
:
W V U.
,:
,
L
Let B be an essential extension of the torsion group A in AbSh+U. We claim that B is
a
preserves torsion. Let 0
torsion. We first show that
(,A)W A(U V W).
U in /U, and 0
Since A is torsion, there is a cover (U V W)
ni N such
I
iEl
that
nialU i
find a cover W
,A
(nla) IUi ^
VI
I
(U
W)
_
0 all i
i^ W)
(U V W)
So we can for a cover W
I.
0 for all i
in L such that
V
in
E
Hence for
I.
L, such that
is torsion in AbShL.
0
nlal(U i ^
0
W
VIE
(,A)W,
a e
I
(U
i^ W)
we can always
W), and that proves
0
b
in
take
extesnlons
essential
that
preserves
show
AbSh/U.
there
exists
in
A
essential
is
B
V
L.
Since
W
B(W
U),
(,B)W
(V A
WVU implies V
AV. But U < V and V
Z such that 0
W V U and m
V
V U e A((VA W) V U). Thus for 0 b E
W) V U and therefore 0 mb
@,B)W, there
To
mblV
I(VVW)
is
(V AW).
<_ W
such that 0 #
an essential extension of
mblV A
,A
in
,A) (vAw)
W
AbShL.
Finally we show that
So, let ,P be a torsion group in AbShl for some P
P(W U)
a e (,P)W
0
U, then 0
a e PW, W
group implies, that there is a cover
nlalW I
W
Vig I
alWle
W
0 all i g I, where
(,P)WI
(W V U) in /I. then we get 0
i
that P is torsion in
Z.
for some m
AbSh/U.
PW, and
in L, and 0
VielWI
,
,B
This shows
is
reflects torsion.
If
so
,P
n.1
being
a
torsion
N such that
P(W V U). If we consider the cover
i
nlal(Wi
V U) for all i
I, which proves
Thus, in order to prove (ll), we consider an essential
Then by the above argument ,D is an
extension D of the torsion group C in AbSh+U.
But ,C is torsion since C is torsion, hence
essential extension of ,C in AbShL.
AbSh+U.
ABELIAN GROUPS IN A TOPOS OF SHEAVES
Now
by hypothesis #,D is torsion.
in AbShU. Hence the result.
REMARK 4.6.
y
X is a
U + U N Y,
0X
then /CY
BY,
subspace
for some topological space X and
given
being
isomorphism
by
the
Hence, by the last proposition, essential extensions preserve
#CY.
U
reflects torsion and that proves D is torsion
As a special case, if L
closed
c
,
97
torsion in AbShY, if they do in AbShX.
{I/nln
1,2,...}C_R there is a torsion group
{0} U
LEMMA 3.7. On the space X
C with a non-torslon essential extension.
for all n e 0. Then the
-(p
e Ab X by {0}
0, (n)
PROOFo Cnsider
/ AbShX [11 produces B
FA, (F)U
t."unctor F b
Ixl
lI
uA{x}
{ U
x
of B.
0 if 0 e U} in AbShX.
Z(p ), (0)
/
Let C be the torsion subgroup
BX and so the function
BX given by (0)=0,
Assume C B, then CX
This means there exists a
#(I/n) --a n where an has order p n n
1,2... is in CX.
kl#Ui=O,
N such that
Since 0 e U. for
for all i e I.
Uie I U i and 0 #
0 a
I and hence U. contains infinitely many {I/n, n e N} thus k.
Hence # # CX, which shows B is not a torsion group in AbShX. We now
contradiction.
cover
X
ki
#IUj,
some j
show that B is an essential extension of C.
Let 0
some i/n e
IW
=(I/n)
e
U.
{I/n}, then
W
If
Z(p),
elW
hence 0
#
e BU, then (I/n) # 0 for
0 is
of
order
finite
since
Thus B is an essential extesnlon of C which is
CW.
torsion, although B itself is not torsion.
THEOREM 4.8.
If X is a first countable Hausdorff space, then essential extensions
preserve torsion in AbShX iff X is discrete.
PROOF. (/) Suppose that X is not discrete.
Then there is a point x e X for
o
{x
which
is not open. Let the countable basic nelghbourhoods of x be arranged in
0
D
Denote by
N pick an element
the form
and for each n
X the subspace of X consisting of the points
Since the sequence
0
0}
U1D U2
xne Un
Un+l"
{xo,Xl,X2,...}.
{Xk}ke N converges
Hausdorff and so
being compact
to
is olosed in X.
0
is also closed in X.
For any
Hence {x
0 the subset
discrete then AbShX
A{x}, xeX
extension
Bl{x}
in
If X
HeelXe,
X
AbSh
and so if A
B{x}
g
D_AI{x}
Xe
is a first coutable, Hausdorff
HeelXa,
then each
Xe
is non-trlvial for infinitely many
Also 2
hence only finitely many X
is discrete.
closed subspace of X, hence by Remark 4.6,
But
X
is given to be first
countable and Hausdorff, therefore by Proposition 4.8, each
hence closed in X.
also
Thus B is torsion in AbShX.
where each
essential extensions preserve torsion in AbShX
X
_
{Xn
AbShX is a torsion
So, if B
space, and essential extensions in AbShX preserve torsion, :.
If X
0
A is an
A{x} is essential in Ab,
is a torsion group in Ab.
AbShX, then
hence each B{x} is a torsion group in Ab.
PROOF.
X
AbShX0,
group then clearly each
COROLLAY 4.9.
0
xn, n
{X
{X
{x })} N X is open in.
0
0
n
n
then easy to see that the subspace X consisting of {x0,xl,... is
0
R. By the above lemma essential extensions of
space {0} U. {i/nlneN
a contradiction to Remark 4.6 hence X is
not be torsion in
the space X
O. It is
homeomorphic to the
torsion groups need
discrete.
(/) If X is
essential
But X is
X is compact in X.
it follows that the space
x0,
X
a, then 2
X
is discrete.
subspace of X.
is first countable, Hausdorff.
But 2
Suppose
is compact,
But it is not discrete,
are non-trivial which implies that X is disrete.
K.R. BHUTANI
98
All finite L are spatial, and for all finite L, essential extensions in
RELRK.
AbShL
preserve
Hence
torsion.
there
many non-discrete
are
spaces
X such that
essential extensions preserve torsion in AbShX.
ACKNOWLEDGMENTS.
This is a part of my doctoral dissertation submitted to the
Graduate School of blcMaster University for partial fulfillment of the requirements for
the
degree of Doctor of
Professor
Bernhard
The author is grateful
Philosophy.
Banaschewskl,
for
his
valuable
to
guidance,
her supervisor,
encouragement,
and
I would also like to thank the Laboratory of
Statistical and Mathematical Methododlogy at the National Institutes of Health,
Bethesda, MD, for providing the facilities for the preparation and revision of this
criticism throught the research work.
paper.
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