Disconnectednesses and closure operators

WSAA 13
F. Cagliari; Marcello Cicchese
Disconnectednesses and closure operators
In: Zdeněk Frolík and Vladimír Souček and Jiří Vinárek (eds.): Proceedings of the 13th Winter
School on Abstract Analysis, Section of Topology. Circolo Matematico di Palermo, Palermo, 1985.
Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 11. pp. [15]--23.
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DISCONNECTEDNESSES AND CLOSURE OPERATORS (*)
F. CAGLIARI AND M. CICCHESE
Abstract
Closure operators which characterize disconnectednesses and r e l a t i v e
disconnectednesses are i n t r o d u c e d . Such operators are used to f i n d c o n d i tions under which a r e l a t i v e disconnectednes is a disconnectedness.
AMS Subject C l a s s i f i c a t i o n : 54B30, 18B30, 18A40.
§1. Preliminaries
(**)
In t h i s paper we denote by T the c l a s s of a l l topological
spa-
ces, by T. ( i = 0,1,2) the c l a s s e s of T.-spaces, by Sing the c l a s s
of spaces which have at most one p o i n t . Moreover we denote by P an
a r b i t r a r y nonempty subclass of T and by P the category of spaces
of P and continuous f u n c t i o n s . Of course P i s a f u l l
subcategory
of T.
Let X be a space and xeX.
1.1 DEFINITION. We c a l l P-component of x in X the l a r g e s t subspace Y of X containing x such
that
for each' PeP and for each
f: Y —-p, f i s constant (see [11], p . 2 9 7 ) .
1.2 DEFINITION. We c a l l P-quasicomponent of x in X the l a r g e s t
subspace Y of X containing x such t h a t for each PeP and for each
(*)
This paper is in f i n a l form and no version of it w i l l be submitted
for p u b l i c a t i o n elsewhere.
( * * ) Notations and d e f i n i t i o n s not e x p l i c i t l y given are from [ 6 ] . Moreover, the functions we consider are always continuous functions between t o pological spaces.
16
F. Cagliari and M. Cicchese
f: X — P , f|Y is constant (see [11], p.297).
1.3. DEFINITION. A space X is called totally P-disconnected if
its P-components are singletons, totally P-separated if its P-quasicomponents are singletons (see [11], p.297).
We denote by UP the class of all totally P-disconnected spaces
and by QP the class of all totally P-separated spaces.
It follows immediately from the definitions that
1.4
PcQPcUP .
1.5 DEFINITION. A class P of spaces is called disconnectedness
if P = UP, and relative disconnectedness if P = QP.
§2. The closure operators E
and K .
Let f: A —*• B be a continuous function.
2.1 DEFINITION, f is said to be
PeP and for every g ,g : B —*• P
g
l
= g
P-cancellable
such
that
if for every
g f = g f,
we have
2*
Suppose now X is a space containing B as subspace.
2.2 DEFINITION, f is said to be P-cancellable rel X if for every PeP and for every g ,g : X -»• P such that (g |B)f = (g |B)f,
we have g j B = g2|B.
2.3. PROPOSITION. If P* is
a
class
of
spaces
such
that
P c P ' c Q p , we have that f: A —* B is P'-cancellable (or P-cancellable rel X) iff f is P-cancellable (or P-cancellable rel X ) .
PROOF. Since PcP» if f is P-cancellable it is obvious that f
is P-cancellable too.
Conversely, suppose f: A —*• B is P-cancellable. Let P'eP1 and
g ,g : B —•P 1 be functions such that g f = g f. Then
PeP and for every h: P' —• P we have
n
hg f = hg f
and
for
every
therefore
n
g 1 - g p - Since P'eQP, the class of all continuous functions from
P' whose range is in P distinguishis the points (see [10], 3.3).'It
follows that g
= g .
Similar arguments can be used to prove the proposition when
Disconnectednesses and closure operators
17
the function is cancellable rel X.
From now on we denote by X an arbitrary topological space and by
A an arbitrary subspace of X.
P
2.4 DEFINITION. By E (A) we denote the largest subspace Y of X
X
such that A cY and the inclusion of A into Y is P-cancellable.
P
2.5 DEFINITION. By K (A) we denote the largest subspace Y of X
A
such that A cY
and
the inclusion
of
A into Y is P-cancellable
rel X.
P
P
It can be easily proved that the operators E and K are Moore
X
X
closures and that if f: X —*Y is a continuous function we have:
2.6 E*{A)CZK*(A)
;
2.7 K*(A) = X 4=* E^(A) = X ;
A
A
2.8 f(E^(A)) CEy(f(A)) ; f (K*(A) ) CK*(f (A) ) ;
2.9 the followings are equivalent:
(i)
f is P-cancellable;
(ii) E^(f(X)) = Y ;
(iii) Ky(f(X)) = Y .
P
The operator KX was introduced in [12] and studied in [4], The
P
operator E coincides with the epiclosure defined in [2] when P is
X
productive, hereditary and XeP.
When there is no confusion about the class P, we indicate the
introduced operators only by E
and by K .
X
X
2.10 PROPOSITION. If P' is a class of spaces such that PcP'cQP,
we have
P
E
pi
x = Ex
P
:K
Pi
x =Kx •
PROOF. It follows immediately from 2.3.
2.11 PROPOSITION. Let xeX. We have:
P
(a) Ex({x}) is the P- component of x in X;
P
(b) K ({x}) is the P-quasicomponent of x in X.
X
PROOF,of(a)
It follows
immediately
from the
that
V is a
subspace
X such
that xeX,
the inclusion
j: fact
{x} —*V
is if
P-cancel-
-8
F. Cagliari and M. Cicchese
lable iff for each PeP the functions from V to P are all constant,
(b) It can be proved in a similar way as (a).
2.12 COROLLARY, (a) UP is the class of all spaces X whose points
P
are E -closed.
*
P
(b) QP is the class of all spaces X whose points are K -closed.
x
PROOF. It follows from 2.11.
2.13 PROPOSITION. The followings are equivalent:
(a) P C T 2 ;
(b) A C E v ( A ) ;
x
(c) ACK^(A) .
PROOF, (a) => (b) It follows from the fact that the inclusion
j: A ^ A is T -cancellable and therefore P-cancellable.
(b) => (c) It follows from 2.6.
(c) => (a) See [12] (p.555).
2.14 LEMMA. A space X belongs to QP iff the
diagonal A
is
x
p
K
-closed.
XxX
PROOF. If XeQP, the projections p ,p : XxX —• X coincide exactly
on A , and therefore, (see 2.3) K
X
(A ) = A
X
XXX
Conversely, suppose K
^
X
(A ) = A . Then there are two functions
XXX
X
X
f,g: XxX — P , with PeQP, such that f|A
= g|A
X
and f(x,y) ?- g(x,y)
X
whenever x -- y. If z is. an arbitrary point of X, we consider the
embedding j: X —•XxX defined by j(x) = (x,z). We have: fj(z) = gj(z)
and fj(t) ?- gj(t) for every teX-(z}. Hence K^({z}) = {z}, and from
2.12 XeQP.
2.15 PROPOSITION. The followings are equivalent:
(a) QPCQP* ;
(b) KJAOKJ'(A).
PROOF, (a) => (b) It follows easily from the definitions and
2.3.
(b) => (a) If (b) holds for each space X we have
K
XXX(AX)CKXXX(V'
Disconnectednesses and closure operators
P
If XeQP, by 2.14 we have
K
XxX(
A
x)
=
A
and so
x
A
is K
x
P'
xxx"ClOSed#
19
By
2.14 again we have XeQP1.
2.16 COROLLARY. The followings are equivalent:
(a)
P CTQ ;
(b)
b x (A)CE^(A) ;
(c)
b x (A)CK^(A) .
PROOF, (a) --=> (b) It follows from the fact that the inclusion
j: A —*-b (A) is T -cancellable (see [13]) and therefore P-cancelX
o
lable.
(b) =» (c) It follows from 2.6.
(c) =* (a) It follows from 2.15 and [12] (p.557).
Examples.
Let S be a singleton, C the two-points indiscrete space, D the
Sierpinski dyad, I the real interval [0,1].
If P = {S} then QP = UP = Sing and EV(A) = K V (A) = X.
X
X
If P = {C} then QP = UP = T and E V (A) = K V (A) = A.
X
X
If P = {D} then QP = UP = T
and EV(A) = K V (A) = b v (A), where
0
X
X
X
b x (A) is the b-closure of A in X (see [13] , 2.5; [12], p.557).
If P = {D }, where D
is the two-points discrete space, then QP
is the class of all totally separated spaces and UP is the class of
all totally disconnected spaces. Moreover
P
K (A) = n IB | A C B C X , B i s c l o p e n i n X } .
A
If P = {1} then QP is the class of all functionally Hausdorff
spaces. Moreover
P
K (A) = 0{B| ACBcCX, B is a zeroset in X} .
X
We observe that when P = {1} and in many other cases it is not easy
p
to know how the operator E v works and how the class UP is.
20
F. Cagliari and M. Cicchese
§3. Disconnectednesses and relative disconnectednesses.
UP and QP are subcategories of T closed under
products
and
injective functions. Therefore they are extremal epireflective in
T (see [8]).
»We indicate by
R: T —"UP, S: T —•QP
the corresponding epire-
flectors and by r : X —• RX and s : X —• SX the epireflection maps
X
X
associated to R and S respectively. We remind that r is the quoX
tient map which identifies the points of each P-component (see [1],
Th.3.7) and s is the quotient map which identifies the points of
X
each P-quasicomponent (see [10], p.304).
3.1. PROPOSITION. A function f: X — Y is P-cancellable
iff
Sf: SX —•SY is an epimorphism in QP.
PROOF. Let f: X — Y be P-cancellable, PeQP and f-.fp? SY — P
such that f,(Sf) = f_(Sf). Then f (Sf)sv = f_(Sf)sv .
1
(Sf)s
1
<_
A
= s f we have f s f = f s f. By 2.3
A
i Y
Y
hence f s v = fps . Since s
Conversely, let Sf be
f
Since
A
__
is QP-cancellable,
__ i
is an epimorphism
an
epimorphism
in
in QP.
f ,f : Y -*P are functions such that f f = f f,
functions g^g.: SY — P such that g ^
T,
we obtain
If PeP
there
and
exist two
= f. , g-S__ = f_.
Thus
g syf = g-syf , and therefore g1(Sf)s}_ = g_(Sf)sx- Since s x is an
epimorphism in T and Sf is an epimorphism in QP, we obtain g
= g .
3.2 PROPOSITION. K^(A) = s^1(K^ (sx(A))).
PROOF. By 2.8 we have K (A) c s " (K (s (A))). Suppose there
X
X X X
exists a point yes" (K (sv(A))) - K (A). Then we can find two funX
ctions
f ,f :X — P ,
X
X
X
with PeP,
such
that
f |A = f |A
and
f (y) ± f (y). If we consider the functions g , g : SX -»-P, such that
g.s^. = f1
, g_s x = f_ , we
obtain
g ^ t y ) 4 g-Sx(y) .
g |sx(A) = g_|sx(A) we deduce that sx(y) ^ K S X ( S X ( A ))>
Since
and this
is
absurd.
REMARK. We do not know whether an analogous proposition for the
P
operator E and the epireflection map r holds. By 2.13 it could
X
X
only be proved that such equality holds when P c T .
Disconnectednesses and closure operators
21
We remind that if P is productive and hereditary and XeP, for
each AtTX the inclusion j: A —"X is an extremal monomorphism iff
P
P
E (A) = A , and j is a regular monomorphism iff K (A) = A (see [2]).
X
X
As a consequence of this fact and of corollaries 3.5, 3.6 in
[3], we obtain the following
3.3 PROPOSITION. If P if a disconnectedness contained in T
and
1
different from Sing we have
E X (A) = K x (A) = A .
3.4 PROPOSITION. The following conditions, are equivalent:
(a)
UP = QP ;
(b)
E^ = K^ for each XeT ;
X
X
E^({x}) =K^({x}) for each XeT and xeX ;
(c)
X
X
(d)
K^(A)
X
K x (B)
PROOF, (a)
P
QP = UP and E
X
= K^(A) f o r e a c h X,A,B such t h a t A C B C X and
B
= B .
=> (b) If QP coincides with T, T or QIS},
P
= K (see examples in §2).
then
X
Moreover the only disconnectednesses which are not contained in
T
are T and T
(see [l], Prop. 2.10). Thus we have only to consi-
der the case QP = U P c T , with QP / Q S . If X is a space and A C X ,
by 3.3 we have
K
S
S X ( X (A))
K X (A) =
= sx(A) and by 3.2
s 1
x (K s x (s x (A)))
= s x 1 (s x (A)) .
It follows
K X (A) = s x 1 (s x (A)) = - ^ ( r ^ A ) ) = U(E X ( {x} ) | xeA} C E ^ A ) ,
hence, by 2.6, K X (A) = E X (A).
(b) => (c) Obvious.
(c) => (a) It follows immediately from 2.12.
(c) 4=> (d) It can be proved in a similar
way
as
in Prop.1.8
in [2], even though the present assertion is more general.
REMARKS, (a) If P is a class of Hausdorff spaces, the operators
P
P
P
P
E and K coincide if and only if QP = Sing. For if E v = K
X
X
X X -
22
F. Cagliari and M. Cicchese
and QP ^ Sing, for every X E P and AcX we have by 2.13 and 3.3:
AcE^(A) = K £ ( A ) = A .
This would imply that every XeP is discrete and this is not
possible. As a consequence we get again that in T
there are no
disconnectednesses different from Sing (see [1]).
(b) The notions given in this paper can be introduced in a topological category. In particular PreuB introduced and studied the
relative disconnectednesses in this more general setting ([10]).
The situation seems to be a little more complicated for the disconnectednesses. A reason is that in T the quotient space obtained by
identifying the points of each P-component is P-totally disconnected and this fact is not always true in a topological category.
For instance this is not true in the bireflective hull in T of the
Hausdorff spaces.
References
[1 ]
ARHANGEĽSKI I a n d R.
A.V.
nectednesses
[2]
F.
F.
D.
, Riv.
CAGLIARI
r i e s of
[4]
and
CAGLIARI
epiclosure"
[ 3]
in t o p o l o g y " ,
a n d S.
[5]
H.
HERRLICH,
[6]
H.HERRLICH
tes
Inc,
R.E.
Boston
category
delberg,
P.J.
(4)
115-122.
"On disconnectednesses
topics",
"Closure operators
and
to
in
subcatego-
appear.
induced
by
topological
appear.
Springer,
and G.E.
reflexionen
Berlin,
STRECKER,
und coref lexionen",
Heidelberg,
"Category
New Y o r k
theory",
Lecture
No-
1968.
Allyn
and
Bacon
1973.
spaces",
New Y o r k
NYIKOS,
subcategories
8 (1982),
and related
HOFFMANN, " F a c t o r i z a t i o n of cones
Hausdorff
[8]
Parma
MANTOVANI,
"Topologische
in M a t h . 7 8 ,
" E p i r e f l e c t i ve
CICCHESE,
DIKRANJAN a n d E . G I U L I ,
to
discon-
G e n . T o p . Appl. _5 (1975) 9 - 3 3 .
Mat. Unïv.
a topological
epireflections",
[7]
M.
WIEGANDT, "Connectednesses a n d
II,
L e c t u r e Notes in M a t h .
1982,
with
915,
applications
Springer,
weak
Berlin,
Hei-
148-170.
" E p i r e f l e c t i v e c a t e g o r i e s of H a u s d o r f f
Notes in M a t h .
to
340, S p r i n g e r ,
Berlin,
Heidelberg,
spaces",
N.Y.
1976,
Lectures
452-481.
Disconnectednesses and closure operators
23
[9]
G. PREUB, "Allgemeine Topologie", Springer V e r l a g , B e r l i n 1972.
[10]
G. PREUB, "Relative connectednesses and disconnectednesses in topolo-
[11]
G. PREUB, "Connection properties in topological categories and related
gical categories", Quaestiones Math. 2_, 297-306.
t o p i c s " , Lecture Notes in Math. 719, Springer, B e r l i n , Heidelberg, New
York (1979) 293-305.
[12]
S. SALBANY, "Reflective subcategories and closure o p e r a t o r s " ,
Notes in Math. 540, Springer,
Lecture
B e r l i n , Heidelberg, New York 1976,
548-565.
[13]
L. SKULA, "On a r e f l e c t i v e subcategory of the category of a l l
topolo-
gical spaces", T r a n s . Amer. Math. Soc. 142, (1969) 37-41.
F. CAGLIARI and M.CICCHESE
Dipartimento di Matematica
Uni versi ta
43100 PARMA ( I t a l y )
This work has been supported by research funds of the Ministero
Pubblica I s t r u z i o n e , I t a l y .

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