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Acta Math. Hungar.
106 (1–2) (2005), 1–16.
SPLITTING CLOSURE OPERATORS
G. C. L. BRÜMMER (Cape Town)∗ , E. GIULI (l’Aquila)† and
D. HOLGATE (Stellenbosch)‡
Dedicated to George Strecker on his sixty-fifth birthday
Abstract. Following the notion of splitting space due to Arkhangel’skiı̆ we
introduce and study splitting objects and splitting closure operators in any category. The classes of splitting objects are characterized for complete categories
admitting factorization structures for sinks. In the case of topological constructs
with (epi, embedding) structure, such classes are exactly the quotient-reflective
subcategories. The relationship between splitting and regular closure operators is
established. Examples are given in topological spaces and in modules.
1. Introduction and preliminaries
A topological space X is splittable over a space Y if for every M j X
there is a continuous map f : X → Y for which M = f −1 f (M ) . This
notion of splittable or cleavable space was introduced by Arkhangel’skiı̆ in
[3]. Subsequent topological investigations into splittable spaces, particularly
those which split over powers of R, have shown amongst other things that
splittable spaces are strongly related to approximation of functions by continuous maps (cf. [4, 5, 2]).
The present paper is a categorical study of “splittability”. We introduce and study, in a complete (E, M)-category, the notions of splitting class,
Spl (P), and of splitting closure operator, splP , determined by a class of objects P.
∗ The first author acknowledges support from the National Research Foundation, from the University of Cape Town and from the University of L’Aquila.
† The second author acknowledges partial support from MURST and from the University of
Cape Town.
‡ The third author is the corresponding author. Support of the National Research Foundation
under grant number 2053738 and the hospitality of the University of Bremen during the final
drafting is gratefully acknowledged.
Key words and phrases: splitting space, splitting class, splitting closure operator, regular closure operator, morphism factorization structure.
2000 Mathematics Subject Classification: 18A32, 18A40, 54B30, 13C13.
c 2005 Akadémiai Kiadó, Budapest
0236–5294/5/$ 20.00 2
G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
A class of objects which all split over given P-objects is called a splitting class. We introduce the notion of (strong) M-separating sources and
see that closure under these sources characterizes splitting classes. In certain
instances this reduces to being closed under mono-sources or even monomorphisms. In particular in topological constructs productive splitting classes
are exactly the quotient reflective subcategories. In categories of modules
the splitting classes coincide with the hereditary classes.
It is natural to consider a subspace M j X for which M = f −1 f (M )
to be closed in a certain sense. (We call such an M a P-piece of X when
the codomain of f is in P.) With this intuition we define the splitting closure operator by an intersection of P-pieces. Apart from investigating the
closure and its construction, we characterize classes that induce the same
splitting closure. If X is E-cowellpowered, the splitting closure can be used
to characterize productive splitting classes.
The relationship of splP with the regular closure regP is established. It is
shown that the two closure operators coincide if and only if regP is discrete
on the objects of P, for productive P.
Throughout the paper we consider a category X with subobjects; that is:
an (E, M)-factorization structure for morphisms, which extends to a factorization structure for sinks, is given in X . For simplicity we assume that X
is a complete category.
We denote by sub (X) the class of all (M-)subobjects of an object X.
Thus sub (X) is the family of all the equivalence classes m of morphisms
m
M → X of M with respect to the equivalence relation
m∼
=n⇔m5n
and n 5 m
where m 5 n means that there exists a morphism p such that m = n · p.
We will not distinguish between m and m (e.g., m = n means m ∼
=n
when m and n are taken as subobjects).
m
The image by f : X → Y of a subobject M → X is the second factor
f (m) : f (M ) → Y of the (E, M)-factorization of f · m, and the inverse image by f : X → Y of a subobject n : N → Y , denoted by f −1 (n), is given
by pulling back n along f . These assignments give an adjunction between
sub (X) and sub (Y ). Both assignments are order preserving and image is
left adjoint to inverse image.
Terminology used in this paper may be found in [1] for category theory,
in [11] and [13] for closure operators on categories, and in [14] for general
topology.
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SPLITTING CLOSURE OPERATORS
3
2. Splitting classes
We begin with a brief study of sources which have cancellability properties relative to subobjects. Closure under these sources characterizes the
classes of objects we wish to study.
Definition 2.1. A source (fi : X → Yi )i∈I of X -morphisms is said to
be:
(1) M-separating if for all n, m ∈ sub (X), fi (n) = fi (m)∀i ∈ I ⇒ n = m.
(2) Strong M-separating if for all m ∈ sub (X) there exists i ∈ I such that
−1
fi fi (m) = m.
Remarks 2.2. (1) Clearly any strong M-separating source is M-separating.
(2) (fi : X → Yi )i∈I is M-separating iff for every m ∈ sub (X), m =
V −1
fi (m) :
I fi
V
⇒: For all i ∈ I, fi (m) 5 fi ( I fj−1 fj (m) ) 5 fi (fi−1 fi (m) ) =
V
fi (m), thus m = I fi−1 fi (m) .
V
V
⇐: If fi (m) = fi (n) for all i ∈ I then m = I fi−1 fi (m) = I fi−1
fi (n) = n.
(3) Note that the above holds also if I = ∅ since an empty source is Mseparating iff sub (X) = {1X }. (In particular in Set the empty source with
domain ∅ is M-separating but not strong M-separating, where M is the
class of monomorphisms.)
(4) For morphisms as singleton sources, the notions of M-separating and
strong M-separating coincide. We will thus refer to such morphisms simply
as M-separating.
(5) Any m ∈ M is M-separating.
Examples 2.3. (1) In topological categories over Set with M the class
of embeddings, M-separating sources are those which separate points and
subsets, i.e. (fi : X → Yi )i∈I is M-separating iff (∀M j X)(∀x ∈ X \ M )
∃i ∈ I, fi (x) 6∈ fi (M ) .
(2) If Xcof is an infinite set endowed with the cofinite topology and S is
the 2-point space with one non-trivial open, then the source
(f : Xcof → S)f ∈Hom (Xcof ,S)
is M-separating in Top but not strongly M-separating. (Here M is the
class of embeddings.)
(3) The product (πi : X 2 → X)i∈{1,2} for X 6= ∅ is an extremal monosource in Set that is not an M-separating source.
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G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
Definition 2.4. In X subobjects are separating if for morphisms
f, g : X → Y , f (m) = g(m)∀m ∈ sub (X) ⇒ f = g.
In any topological construct, with M the embeddings, subobjects are
separating. In Grp this is not the case: Consider f, g : Z3 → Z3 with f = id,
g(x) = 2x (mod 3).
Proposition 2.5. (1) If in X subobjects are separating then every Mseparating source is a mono-source.
(2) If E is stable under pullback along monomorphisms then every monomorphism is M-separating.
Proof. (1) Let (fi : X → Yi )i∈I be an M-separating source with morphisms u, v : A → X such that fi u = fi v foreach i ∈ I. For each m ∈ sub (A)
we have: fi u = fi v ⇒ fi u(m) = fi v(m) ⇒ u(m) = v(m) ⇒ u = v.
(2) [16], Lemma 2.1.
We now turn to investigating object classes that are closed under the
above defined sinks and morphisms. Using properties of the image/inverse
image adjunction we make the following observation.
Lemma 2.6. If a source (fi : X → Yi )i∈I factors through a morphism
V
f : X → Y , then for each m ∈ sub (X), f −1 f (m) 5 I fi−1 fi (m) .
In particular if (fi )I is M-separating then so is f .
Proof. If fi factors fi = gi f for each i ∈ I we have for any m ∈ sub (X):
fi−1 fi (m) = (gi f )−1 (gi f )(m) = f −1 (gi−1 (gi f )(m) )
= f −1 gi−1 (gi f (m) ) = f −1 f (m) .
V
This for every i ∈ I, hence f −1 f (m) 5 I fi−1 fi (m) .
If (fi )I is M-separating then f −1 f (m) 5 m and equality follows.
Proposition 2.7. Let P be a productive class of X -objects, then:
(1) P is closed under small M-separating sources iff P is closed under
small strong M-separating sources iff P is closed under M-separating morphisms.
(2) If moreover E is stable under pullback along monomorphisms and in
X subobjects are separating, the above properties are further equivalent to P
being closed under monomorphisms.
Proof. (1) All that requires proof is that closed under M-separating
morphisms implies closed under small M-separating sources. This follows
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5
immediately from Lemma 2.6 by factoring a small source (fi : X → Pi )i∈I ,
with each Pi ∈ P, through the product
hfi i
π
i
X −→ ΠI Pj −→
Pi
and hfi i is M-separating.
(2) Under the given conditions it follows from Proposition 2.5 that
monomorphisms and M-separating morphisms coincide.
The preceding notions will now be applied to our main topic of this section, namely splitting classes.
Henceforth P always denotes a class of X -objects.
Definition 2.8. (1) m ∈ sub (X) is a P-piece of X if there is a morphism f : X → P with P ∈ P and m = f −1 f (m) .
(2) The splitting class of P, denoted Spl (P), is defined by: X ∈ Spl (P)
iff m ∈ sub (X) ⇒ m is a P-piece of X. P is termed a splitting class if P =
Spl (P).
Remark 2.9. For a subobject m : M → X the following are equivalent:
(1) m is a P-piece.
(2) There exist f : X → P , P ∈ P, and n ∈ sub (P ) such that m = f −1 (n).
0
(3) There
exist f : X → P , P ∈ P and m ∈ sub (X) such that m =
−1
0
f
f (m ) .
Proposition 2.10. Spl (P) is the strong M-separating hull of P (i.e.
X ∈ Spl (P) ⇔ there is a strong M-separating source (fi : X → Pi )i∈I with
Pi ∈ P for each i ∈ I).
Proof. Clearly if (fi : X → Pi )i∈I is a strong M-separating source with
each Pi ∈ P then X ∈ Spl (P). On the other hand if X ∈ Spl (P) then
for
−1 f (m) and
each m ∈ sub (X) there is an fm : X → Pm such that m = fm
m
the source (fm : X → Pm )m∈sub (X) is strong M-separating.
This characterization has the following easily verified consequences.
Corollary 2.11. P is a splitting class iff P is closed under strong Mseparating sources.
Corollary 2.12. (1) P j Spl (P).
(2) P j Q ⇒ Spl
(P) j Spl (Q).
(3) Spl Spl (P) = Spl (P).
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G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
If X has products and is E-cowellpowered, then any source
(fi : X → Yi )i∈I factors through a single morphism. For each i ∈ I take
the (E, M)-factorization fi = mi · ei . Choose a representative set J j I for
the source (ei : X → Mi )i∈I and form the product to give hej i : X → ΠJ Mj .
fi
/Y
x< i
33 FF e
x
x
33 FFFi
xx
xx mi
33 FF"
x
x
33
M
3
i
hej i 3
33 O
33 πji
3
X3 FF
ΠJ M j
Fig. 1
For each i ∈ I there is a ji ∈ J such that πji · hej i = ei and thus (fi )I
factors through hej i.
This observation leads to the following theorem.
Theorem 2.13. (1) If in X subobjects are separating, then P is closed
under monosources ⇒ P is a splitting class.
(2) If X has products and is E-cowellpowered with E stable under pullback
along monomorphisms, then P is a productive splitting class ⇔ P is closed
under monosources.
Proof. (1) Follows from Proposition 2.5(1), Remark 2.2(1) and Corollary 2.11.
(2) If in the diagram above (fi )I is a monosource with each Yi ∈ P,
then ΠJ Mj ∈ P (since each mj ∈ M and thus M-separating, and P is productive) and hej i is a monomorphism. Hence by Proposition 2.5(2) hej i is
M-separating and by Corollary 2.11 X ∈ P.
Conversely if (fi )I is a strong M-separating source with each Yi ∈ P,
then ΠJ Mj ∈ P (since each mj is a monomorphism, and any class closed
under monosources is closed under products) and by Lemma 2.6 hej i is Mseparating. Hence X ∈ P and by Corollary 2.11 P is a splitting class.
Corollary 2.14. If X is a topological construct with (Epi, Embedding)
factorization structure, then P is a productive splitting class iff P is quotient
reflective in X .
Examples 2.15. 2.A. In the category A-Mod of left modules over a unitary ring A the splitting classes are easily characterized: they are precisely
the hereditary classes (here (E, M) = (Epi, Mono)). Indeed, if we assume P
hereditary, then X ∈ Spl (P) implies, in particular, that there exist P ∈ P
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7
and f : X → P such that f −1 f (0) = f −1 (0) = 0, which means that X is
a submodule of P , so it belongs to P. The converse is given by Corollary
2.11 and Remark 2.2(4).
2.B. In the category Top of topological spaces, with the (Surjection,
Embedding)-factorization structure, the subcategories Top0 , Top1 , Haus
and FHaus of T0 -spaces, T1 -spaces, Hausdorff spaces and functionally Hausdorff spaces respectively, are splitting classes (see Corollary 2.14).
Spl (P) contains the discrete spaces whenever P contains a space with
more than one point.
If P is closed under the formation of coproducts (= disjoint unions) then
the class Spl (P) has the same property (note that Spl (P) need not be closed
under the formation of coproducts in general).
2.C. In Top if P consists of the Sierpinski two-point space then Spl (P)
is the door spaces, i.e., those spaces whose subsets are either open or closed.
This example shows that splitting classes need not be closed under squares:
the square of the Sierpinski space is not a door space since the diagonal is
neither open or closed.
If P contains the two-point indiscrete space (or any space admitting a
two point indiscrete subspace), then Spl (P) = Top.
If P contains the two-point discrete space then Spl (P) contains all the
discrete spaces, and Spl (P) coincides with the class of all discrete spaces if
P consists of discrete spaces.
2.D. In the category Tych of Tychonoff spaces if P consists of the countable power Rω of the real line R then Spl (P) consists of the spaces with the
countable function approximation property ([4], Theorem 2.8). A Tychonoff
space is said to satisfy the countable function approximation property if for
each real valued function f : X → R there exists a sequence (fn : X → R)
of continuous maps such that for each finite subset F of X and ε > 0 there
exists a natural number m such that f (x) − fm (x) < ε for each x ∈ F .
Heredity, closure under refinements, and to contain all discrete spaces are
necessary conditions for a class of spaces to be a non trivial splitting class
but these properties do not characterize the splitting classes in Top.
Question. Give an internal characterization of the (non trivial) splitting classes of Top.
3. Splitting closure
As in the previous section X will be a category with subobject class M
and P will be a class of objects of X . Recall ([11, 13]) that a closure operator
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G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
on the category X with respect to M is a family of maps
c = {cX : sub (X) → sub (X) | X ∈ X }
such that, for each f : X → Y and m, n ∈ sub (X),
(1) m 5 cX (m);
(2) m 5 n ⇒cX (m) 5 cX (n);
(3) f cX (m) 5 cY f (m) .
It is a well known fact that every subclass of subobjects which is closed under pullback defines by intersection a closure operator which is in addition
idempotent.
Proposition 3.1. The class of all P-pieces is closed under pullback.
Thus, for each class P of objects, the formula
splP
X (m) =
^
n ∈ sub (X) | n is a P-piece, m 5 n
defines an idempotent closure operator on X .
Proof. If m is a P-piece of X by means of g : X → P then, for each
morphism f : T → X, f −1 (m) = (g · f )−1 g(m) since the composition of
two pullback squares is a pullback square, consequently f −1 (m) is a P-piece
of T .
Definition 3.2. A closure operator c is said to be a splitting (closure)
operator if there exists a class P such that c = splP . By adjointness we have:
splP
X (m) =
=
=
^
^
^
{f −1
=
f −1 (n) | f : X → P, P ∈ P, m 5 f −1 (n)
f −1 (n) | f : X → P, P ∈ P, f (m) 5 n
f (l) | f : X → P, P ∈ P, m 5 f −1 f (l) }
^
{f −1
f (m) | f : X → P, P ∈ P }.
The final characterization usually being the most convenient.
Definition 3.3. The M-separating hull of P is defined by: X ∈
MSH (P) ⇔ there is an M-separating source (fi : X → Pi )i∈I with each
Pi ∈ P.
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Noting that M-separating sources are closed under composition, we indeed have a hull operator with the following properties for any object class P:
(1) P j MSH (P).
(2) P j Q ⇒ MSH (P)
j MSH (Q).
(3) MSH MSH (P) = MSH (P).
Clearly since strong M-separating sources are M-separating we have
P j Spl (P) j MSH (P).
Proposition 3.4. For a class Q of X -objects, splP is discrete on Q iff
Q j MSH (P).
Proof. For an X -object X and any m ∈ sub (X), splP
X (m) = m iff m =
−1 f (m) | f : X → P, P ∈ P
iff
the
all-source
from
X to P is Mf
}
{
separating.
V
Thus MSH (P) is the largest class on which splP is discrete. As the following proposition shows, the connection between MSH (P) and splP is even
stronger. MSH (P) is (characterized as) the largest class Q for which splP
= splQ , while for any Q, P j Q j MSH (P) ⇒ splP = splQ .
Proposition 3.5. splP = splQ ⇔ MSH (P) = MSH (Q).
Proof. ⇒: Follows since by Proposition 3.4, MSH (P) is characterized
by splP .
⇐: Follows from the observation that splP = splMSH (P) . On the one
hand P j MSH (P) ⇒ splMSH (P) 5 splP . To verify the reverse inequality,
consider a morphism f : X → H in the construction of splMSH (P) (m) for
m ∈ sub (X) and H ∈ MSH (P). Since H ∈ MSH (P) there is an M-separating source (fi : H → Pi )i∈I with each Pi ∈ P.
f
m
fi
M −→ X −→ H −→ Pi .
Since f (m) ∈ sub (H), f (m) =
f
−1
V
I
fi−1 (fi f (m) ). Thus,
^
−1
−1
f (m) = f
fi (fi f (m) )
I
=
^
(fi f )−1 (fi f )(m) = splP (m).
I
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G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
Hence we observe,
splMSH (P) (m) =
^
{f −1
f (m) | f : X → H, H ∈ MSH (P)}
= splP (m).
Corollary 3.6. For given P, the following object classes induce the
same splitting closure operator as P.
(1) Her (P) = {X | ∃m : X → P, m ∈ M, P ∈ P};
(2) Spl (P);
(3) If E is stable under pullback along monomorphisms, Mono (P) = {X |
∃m : X → P , m a monomorphism, P ∈ P};
(4) If X is E-cowellpowered and P is productive, E(P), the E-reflective
hull of P in X .
Since for morphisms the notions of M-separating and strong M-separating coincide, when sources to P can be factored through morphisms, Mseparating and strong M-separating sources also coincide. This then leads
to a coincidence of the hulls Spl (P) and MSH (P) and P-pieces are exactly
the splP -closed subobjects.
Proposition 3.7. If X is E-cowellpowered then for a productive class
P, Spl (P) = MSH (P).
Proof. Let X ∈ MSH (P). As in the discussion of the preceding Theorem 2.13, consider an M-separating source (fi : X → Yi )i∈I with each
Yi ∈ P, factor each fi = mi · ei , take a representative set J j I for the source
(ei )I and form the product ΠJ Mj as in Fig. 1.
The morphism ΠJ mj : ΠJ Mj → ΠJ Yj ∈ M. Since P is productive,
P j Spl (P) and Spl (P) is closed under M-subobjects it follows that ΠJ Mj
∈ Spl (P). But hej i is M-separating (Lemma 2.6), so X ∈ Spl (P).
Proposition 3.8. If X is E-cowellpowered and P is productive, then for
any X -object X it holds that m is a P-piece of X iff splP
X (m) = m.
Proof. Necessity is clear. For sufficiency use the argument as above
to factor the source of all morphisms with domain X and codomain in P
through a single morphism f = hej i. Then from Lemma 2.6 we conclude
that
^
m = splP
(m)
=
{g−1 g(m) | g : X → P, P ∈ P } = f −1 f (m) . X
Another instance where any source to P factors through a single morphism is when P is (weakly) reflective in X . Recall that a subcategory R
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11
of X is called weakly reflective if for every object X there is an object RX
in R and a morphism rX : X → RX such that, for each object Y in R and
morphism f : X → Y there exists a morphism f 0 with f 0 · rX = f .
Proposition 3.9. If P is weakly reflective in X then for every subobject
m : M → X we have
−1
splP
X (m) = rX rX (m) .
Proof. The subobject rX −1 rX (m) is a P-piece which contains m and
for each morphism f : X → P , P ∈ P we have
−1
f −1 f (m) = (f 0 · rX )
(f 0 · rX )(m)
−1
= rX
f 0−1 (f 0 rX (m) ) = rX −1 rX (m) . Finally we compare splP to the widely studied regular closure operator.
Recall from [11] that the regular closure regP
X (m) induced by a class P of
objects is defined as the intersection of all equalizers of pairs of morphisms
from X to objects of P and containing m. The comparison between the
splitting closure and the regular closure is established by the following
Proposition 3.10. If P is closed under the formation of squares then
splP 5 regP .
Proof. If m : M → X is the equalizer of f, g : X →P , P ∈ P, then
P × P ∈ P and h = hf, gi : X → P × P satisfies h−1 h(m) = m so that m
is a P-piece.
We cannot avoid the assumption that P is closed under the formation
of squares in the above proposition. For instance splSS×S (∆S ) = S × S (see
Example 3.D below) and regSS×S (∆S ) = ∆S .
Observe that splP is the largest closure operator that is discrete on P,
m
f
since if c is discrete on P consider M −→ X −→ P where m ∈ M and P ∈ P:
c(m) 5 c(f −1 f (m) )
(m 5 f −1 f (m) )
5 f −1 (c f (m) )
= f −1 f (m)
(c-continuity of f )
(c is discrete on P).
V
Thus cX (m) 5 {f −1 f (m) | f : X → P , P ∈ P } = splP
X (m). This
allows the conclusion that:
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G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
Proposition 3.11. If P is closed under squares then splP = regP iff
is discrete on P.
regP
There are other well studied closure operator constructions that sometimes coincide with the splitting closure. Holgate introduced in [18] a new
class of closure operators (the so called pullback closure operators) and observed that these closure operators and the splitting closure operators coincide whenever P is E-reflective in X .
Castellini and Hajek [7] (see also [6], [9]) have pointed out that, in the
realm of a construct, the splitting closure operators are useful to build a
factorization of the connectedness-disconnectedness Galois connection in the
sense of Herrlich [17] and Preuss [19].
It should be noted that, in contrast with the usefulness of the splitting
closure operators for categorical connectedness, the beautiful theory of categorical compactness given in [8] becomes trivial when the closure operator
is a splitting operator and the theory is restricted to Hausdorff objects (due
to the fact that the splitting closure induced by a class P is discrete on P).
Examples 3.12. In what follows we will use the general result (cf. [11]):
If c is an idempotent and weakly hereditary closure operator of a category X
then (c-dense morphism, c-closed M-morphism) forms a factorization structure for morphisms of X .
3.A. In A-Mod there is an easy description of the splitting closure operator defined by a class R of modules:
^
splR
(M
)
=
M
+
ker (f ) | f : X → R, R ∈ R .
X
In particular, if R is a (weakly) reflective subcategory of X (with reflections
rX : X → RX) then the splitting closure is given by
splR
X (M ) = M + ker (rX ).
Every class R defines also a preradical rR by setting
^
rR (X) =
ker (f ) | f : X → R, R ∈ R
and the closure operator splR is weakly hereditary if and only if the preradical rR is idempotent (cf. [12]).
Since splR is always an idempotent closure operator, (splR -dense homomorphism, splR -closed monomorphism) is a factorization structure for
morphisms of A-Mod if and only if the preradical rR is idempotent.
The closure operator splR is discrete on X if and only if rR (X) = 0 and
it is indiscrete on X if and only if rR (X) = X.
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SPLITTING CLOSURE OPERATORS
13
In A-Mod splR = regR if and only if R is closed under the formation of
quotients.
3.B. Let spl0 denote the splitting closure operator induced in Top by
the class Top0 of all T0 -spaces. As pointed out in [7] spl0 has the following
description:
spl0 X (M ) = x ∈ X | clX {x} = clX {a} for some a ∈ M .
It is an hereditary operator.
Since spl0 is idempotent and (hereditary, hence) weakly hereditary (spl0 dense map, spl0 -closed embedding) is a factorization structure for morphisms
in Top. The spl0 -closure is strictly finer than the regular closure induced by
the T0 -spaces (which coincides with the Skula closure or b-closure, cf. [11]).
The closure operator spl0 is discrete on X if and only if X is a T0 -topological
space and it is grounded-indiscrete (i.e. spl0 X (M ) = X for non-empty M and
spl0 X (∅) = ∅) on X if and only if X is an indiscrete topological space.
One also sees at once that spl0 is additive and therefore a Kuratowski
closure operator on every topological space. Thus spl0 corresponds to a topology finer than the Skula topology, strictly finer on some spaces, but distinct
from the discrete topology on some spaces.
3.C. Let spl1 denote the splitting closure operator induced in Top by
the class Top1 of all T1 -spaces. It coincides with reg1 , the regular closure
operator induced by the class of all T1 -spaces. In fact reg1 is discrete on T1 spaces (cf. [10]) so that Proposition 3.11 applies. There is a natural closure
operator which is properly finer than spl1 but admitting the same separated
objects (= objects with closed diagonals). It is defined by
^
openX (M ) = {A ⊂ X | A is open and M ⊂ A}.
It is called the open closure operator induced by the Kuratowski closure. A
systematic study of open closure operators is made in [16].
It follows directly from the above description that open is (hereditary,
hence) weakly hereditary, consequently (open-dense map, open-closed embedding) is a factorization structure for morphisms of Top.
The closure operator open is discrete on X if and only if X is a T1 -space
and it is grounded-indiscrete on X if and only if X is an indiscrete topological
space.
3.D. Denote by S = {0, 1} the two-point Sierpinski space. The splitting
closure operator given by S has the following description:
splSX (M ) = open (M ) ∩ M .
Acta Mathematica Hungarica 106, 2005
14
G. C. L. BRÜMMER, E. GIULI and D. HOLGATE
This closure fails to be additive (e.g., splSS×S (∆S ) = S × S while both points
(0, 0) and (1, 1) are splS -closed). The heredity of splS is ensured by the heredity of both the open closure and the Kuratowski closure. A topological space
X is splS -separated if and only if it is a T1 -space. In fact splS is clearly discrete on T1 -spaces; S is not splS -separated; the class of separated objects
forms a quotient-reflective subcategory (cf. [11]); there is no intermediate reflective subcategory between Top and Top1 (cf. [15]). The closure operator
splS is discrete on the Sierpinski space but it is not so on the real line endowed with the upper topology. X is splS -grounded-indiscrete if and only if
it is an indiscrete topological space.
3.E. The splitting closure operator induced by a two-point discrete space
D is the well known “clopen closure”
^
splD
{M ⊂ A | A is clopen in X}.
X (M ) =
X is splD -discrete if and only if it is a discrete topological space and it is
splD -grounded-indiscrete if and only if it is a connected topological space.
The closure splD fails to be weakly hereditary (cf. [13]).
3.F. Any splitting closure operator splP on Top is finer than the Kuratowski closure whenever P contains a non T1 -space. In fact in such a case
it is not restrictive to assume that P contains either an indiscrete two-point
space (consequently splP is discrete) or the Sierpinski space (then splP is
finer than spl0 which is clearly finer than the ordinary closure). On the other
hand splP is properly coarser than the Kuratowski closure whenever P is
a class of discrete spaces, being the clopen operator if the class P contains
a non trivial space, the grounded-indiscrete closure if P = {point} and the
indiscrete closure if P = {∅}.
Finally splP is not comparable with the Kuratowski closure whenever
P ⊂ Top1 contains a non discrete space X. In fact splP is discrete on X
and it is indiscrete on the Sierpinski space in such a case. In particular the
Kuratowski closure is not a splitting closure operator. Also the idempotent
hull of the θ-closure is not a splitting closure operator since it is coarser than
the Kuratowski closure and is different from the clopen closure (see Example
3.E).
The diagram in Fig. 2, with → denoting properly finer, shows the comparison between some splitting closure operators. For each infinite cardinal
α
α, splα and splS will denote the splitting closure operator induced by the
cofinite space of cardinality α and the splitting closure operator induced by
the product of α copies of the Sierpinski two-point space S, respectively; I2
will denote the two-point indiscrete space.
Acta Mathematica Hungarica 106, 2005
SPLITTING CLOSURE OPERATORS
splφ
15
= indiscrete closure
6
splpoint= grounded – indiscrete
closure
6
splD = clopen closure
*
6
K
splS
splℵ0
@
I
@p
ppp
ppp
= Kuratowski closure
6
6
pp
pp
pp
p
6
p
α
splS
@
I
@
splα
6
pp
pp
pp
p
6
@
I
@p
ppp
ppp
p
@
I
@
spl1
spl0
6
@
I
@
splI2
= discrete closure
Fig. 2
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G. C. L. BRÜMMER, E. GIULI and D. HOLGATE: SPLITTING CLOSURE OPERATORS
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(Received May 22, 2003)
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
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Acta Mathematica Hungarica 106, 2005