General Topology and its Applications 10 (1979) 175-181
i3 North-Holland Publishing Company
Th. MARNY
Institiitftir Mathematik I, l-We UniversitiitBerlin, Hiittenweg9,lOOOBerlin 33, Federal Republic of
Germany
Received 1 March 1977
In ohis paper the lattice of all epireflective subcategories of a topological category is studied by
defining the To-objects of a topological category. A topological category is called universal iff it is
the bireflective hull of its To-objects. Topological spaces, uniform spaces, and nearness spaces form
universal categories. The lattice of all epireflective subcategories of a universai topological
category splits into two isomorphic sublattices. Some relations and consequences of this fact with
respect to Cartesian closedness and simplicity of epireflective subcategories are obtained.
AMS (1970) Subj. Class.: Primary 18A40, 54AOS; Secondary
lHD15,54DlO
topological category
epireflective subcategory
To-object
saturated object
universal kernel
Cartesian closed category
simple epireflective subcategory
Introduction
Davis [3~ pointed out that there exists a bijection between the collection of all
subcategories of the category To of topoiogical spaces defined by separation axioms
and the collection of all subcategories defined by regularity axioms. We show that
this correspondence gives rise to a bijection between two disjoint sublattices of the
lattice W of all epireflective subcategories of a topological! category. For some
topological categories (e.g. Top) W splits into two isomorphic sublattices one of
which consists of the bireflective subcategories. In each topological category there
exists an extrernal epi-reflection such that each epireflection is either a birefiection or
the composition of a bireflection followed by this extaernal epi-reflection. The reader
is expected to be familiar with the concept of topological categories by Herrlich [4,8,
aJ category as defined by Herrlich [9]. In particular, for any set
ists precisely one -structure on X. We also 25583118
X with cardinality one, ther
that there exists a ‘unique -structure on the
pty set. In the following, all
176
Th. Mamy / On epiaeflective subcategories
subcategories are ass ed to be both full and isomorphism-closed. We denote b
the subcategory of all -objects whose underlying set contains at most one ele
denotes the subcategory of all i
ch has (0, 1) as its underlying set,
powered (epi, extremal mono)-category (resp. a bi-co-well-powered (bi, initial)category, an extremal epi-co-well-powered (extremal epi, mono)-category) wi
products, we obtain the following (see [5, 6, 10, 171).
is epireflective (resp. bireflective, extrem
A subcategory
if and only if it is closed under the formation of
ubobjects, mono-subobjects)
is epirefiective in A and contains Ind or, equivalently,
This characterization theorem implies that for each class E of A-objects there
exists a smallest E containing epireflective subcategory (resp. bireflective subcategory, extremal epi-reflective subcategory) of A called the epireflective hull R
(resp. the bireflective hull I the extremal epi-reflective hull QE) of E in A. The
notation QE is due to Preul3 [20, 211 since QE consists of all A-objects whose
E-qua&components are trivial.
Theorem 2. The following statements are equivalent for any A-object A:
(resp. A E Z/E, A E QE).
(2) A is an extremal mono-subobject (resp. an initial-subobject, a mono-subobject) of a product of objects of E.
(3) There exists an extremal mono-source (resp. an initial-source, a mono-source)
1source fram A to is an extremal mono-source (resp. an initial -source,
a mono -source ).
-reflection of A is an extremal monomorphism (resp. an initial
morphism, a monomorphism ).
-reflection (resp. the QE-reflection) of any A-object A is the bimorphism in
the (bi, initial)-factorization (resp. the extremal epimorphism in the (extremal epi,
mono)-factorization) of the R -reflection of A.
be the lattice of all epireflectiv
nee not be 3 set.
are disjoint sublattices of
closed under the formation of infima an
177
T”n.hfarny / 0~ epireflective subcategories
(note that there exists an extremal monomorphism in each extremal m.ono-source
with domain Iz). Since
is the smallest element of
2 has a largest element
-spaces. We now
-morphism f : 12
+ A is
for each birejlectivesubcategory
roof.(1) If To is not extremal epi-reflective in , then, according to the
definition of ToA, QTOA is bireflective in A, Hence IToA c QTOA. Thus it follows
from Corollary 1 tha: T0A = R&A = I&A n QToA = ITOA. This contradicts the
fact that ToA is not bireflective in A.
(2) Since each non-constant morphism with domain 12 is an extremal monomorphism and 12eiToA, each morphism f : I2-, A is co. lstant. Conversely, this condition
implies that 12@R(A). Hence A E R(A) c ToA.
(3) Since each birefkctive subcategory of a topological category is a topological
category, ToBexists. The indiscrete objects of A aoldB coincide. Thus (2) implies (3).
Let To: WI + W2 and I: W+ WI defined by To(B)= TOB and I(C)= IC.
Furthermore, let Tb : W3 + W2 be the domain restriction of To to the sublaatice
W3 = {B EW 1@d c B c ITOA} of WI and let I’ :W2.a W3 be the codomain restriction
of I to w3.
Theorem3. T&:
3+ Wa and I’ : W2 + W3 are imerse isomorphisms.
Proof. Obviously, To and I are order preserving aslsignments. Thus it remains to be
shown that (a) TJC = C for each C E Wa and (b) I’& = B for each BE W3.
(a) Clearly, Cc IC n ToA = TolC. Let A = (X, 6)~ To%‘C.Since A EIC, there
exists an initial-source (A, (h :A + Ci)r) fram A to C. In ordt,zrto prove that (A, (h)~)
is an extremal mono-source, suppose that it does not separate some elements nl,
x2 E X. Choose a map g : (0, 1)+X with g({O, 1)) = (xl, x2}*This, together with the
initiality of (.A, (f&) implies that g : 12 + A is an A-morphism. Then it follows from
A E ToA that g must be constant, hence x1 =x2. Thus, .4 E RC = C.
Let (Y, ~)EB and let ro:(Y, q)+(Yo, qo) be the ToA(b) Clearly, ITQBc
reAection of (Y, 7). It follows from (Y, Q)E ITOA and theorem 2 that ro is initial.
Since ro is surjective, there exists a map s : Yo+ Y such that rot0 s = 1 y. The fact that
ro is initial now implies taat s : (Yo, 70) + (Y, q) is an A-morphism. To verify that s is
pose that f :X+ Y. is a map such that s 0f: (&!C,
E)+ (Y, r)) is an
morphism, Then f : (A?,,()
+ ( ~0) is an A-morphism since the composit
roos$(X,~)+(Y0,qo)
is an -rlrr>rphism such that ro 0 s 0 f = f. Thus( Yo, 90) E
. Since ro is initial, we conclude (Y, q)~ IT0
Th. Many / On epitepective subcategories
178
Theorem 2 and proof (b) of the preceding theorem yield the following proposition.
are epircflective
subcategories of a topological ca&gory
, the following hold:
the bimorphism in the (bi, initiul)-factorization of the
(2) The C-reflection is the composition of the
reflection.
-reflection followed by the
TO
In any topologicalcategory each epireflectionis either a bireflectionor the
composition of a bireflectionfollowed by the ToA-reflection.
nition 1. A topological category A is called universal iff the ToA-reflection of
-object is initial.
The following are equivalent:
(3) W splitsinto the two isomorphic sublattices
= I(% n ToA)for each birejiectivesubca
(4)
roof. Clearly, (1) and
(& --r,(3). Recall that
WI = w3. Thus, (3)
ctive subcategory of A,then does not belong to Wz
SEW3. Thus, theorem (3) and proposition (1) yield B = Il’ii
A bireflectivesubcategory of
t is contained in ITOA. In
bireflectivesubcategory of
.
.
is universal if and only if it belongs to
ular; there exists a largest universal
is called the universal kernel of A. The objects of IToA are called
saturated.
16. LetA=(XJ)bean
-object and PA= {(x, y ) G
I (k VI9&,YI) is
rete} where &,,,) denotes the initial -structure on (x,
th respect to the
inclusion {n, y} + X and the -structure 6. Consider the following:
(1) A is saturated.
(2) PA is an equivalence relation such that the corresponding quotient map is initial.
alence relation such that the corresponding quotient map is the
en (1) and (2) are equivalent and imply (3).
Th. Mamy / On epireflectivesubrategories
17’9
-reflection of A. It suffices to
(1) =$ (2), (1) =+ (3). Let ro : A + A0 be the
prove that Spay iff to(x) = ro(y). Observe that XPA}’iff there exists an A-morphism
f : 12+ A such that f ((0, I}) = {x, y}. Therefore since A0 E To , it fc&jws from
that ye(x) = ro(y)
Proposition 1 that Spay imfllies r&) = ro(y). Conversely, supp
and choose a map f : (0, 1)+ X such that f ((0, 1}) = {x, y j. Then a00f is constant and
thusrocf:1z-*A
isan -morphism. Since A is saturated, ro is initial which implies
that f:I+A
is an -morphism. hence .~p~y.
. Since r is initial, we
(2)+(1).Letr:A+Bbe
uotient map with respect
need only show that B is a
object. In fact, given an -morphism f : 12 4 B, it
morphism s: B 4 A
follows as in the proof (b) of Theorem 3 that there exists
suchthatros=l~.Thensof:&+Aisan
-morphism. Finally, by definition of PA
and r, r 0 s 0 f :I* + B which is f is constant.
pological category 0p of topological spaces is universal. If
= HO-spaces of K. CsBszar, B3 = regular spaces,
= zero-dimensional spaces, then
4 = Tychonoff spaces
To-spaces and in eat
The topological categories Unif of uniform spaces, Prox of proximity spaces, and
Neu of nearness spaces are universal. Concerning Near notice that ToNear. is the
category of N&paces (see [$I) and that the ToNear-reflection r. : (X, 6) + (X0, To)of
a nearness space (X, 6) is determined by r&)= ro(y) if! (~c)s{y) and &,=
(BE P*Xo 1ro’ 23E [}. Since cl4 @ corefines r&#t, ro is initial.
A degenerated example is presented by the topological category Born of
bornological spaces (see [13, 191). Since each two-element bornological space is
indiscrete, it follows that T,o olen= Sgi and ?IToBorn= Iad.
The topological category ‘UtiL of uniform limit spaces is not universal because the
universal kernel ITJJnifL is the category of satu;Gced (in the sense of [23]) uniform
limit spaces which is different from UnifL by an example of Wyler [23]. This fact has
motivated Definition 2.
The topological category PrT of pretopological spaces and hence all topological
categories which contain PIT as a bireflective subcategory such as the category PST
of pseudotopological spaces, Lii of limit spaces, and Con of convergence spaces are
not universal. For, there are three-element pretopological spaces A for
not an equivalence relation. Thsae A-objects A of a topological category
PA is an equivalence relation form the objects’ of a bircflective subcategory of
which is in general c’lifferent not only from A but also from IToA. For instance, an
infinite limit space A = (X9 q) such that 4 is generated by the set {(i, y) 1x, y E X}
(where R denotes the ultrafilter based on {x)) is not saturated though PA= X XX is an
equivalence relatioim.This example also shows that in the above proposition (3) does
not imply (1) or (2).
indiscrete if and only if
An A-object A of a universal top
pA=XxX This characterization of
Th. Marny / On epir$ective subcategories
180
Propodtion 6. Let C be an extremal ep i’-,eflective subcategory of a Cartesianclosed
mpological category A,. ‘Then C and IC are Cartesianclosed such that powers in them
are fbrmed in A.
Roof. The Cartesian closedness of C has been shown by Nel[l9]. It follows from the
fact that for A-objects A = (X9 6) and C(e, : A[A, C]+ C)XE~(where e,(f) = f(x)> is
a mono-source which implies that A[A, C] is a C-object whenever C is. Given
B E IC the C-reflection r: B + C is initial. Since the internal horn-functor
preserves initial sources, ALA, r] : A[A, B] 3 A[A, C] is initial,
[A, -1: A-,
hence A[& B] E IC.
Corollary 2. The universal kernel of a Cartesianclosed topologicalcategory is Cartesian
clxed. Especially, the universal kem2: IT’con, IT~Xh, und IToPsT of the Cartesian
closed topologicalcategories Con, Unr, and PST are Cartesianclosed.
Corollary 3. If A is a Cartesian closed topological hull of a universal topological
ctmgory, then A is universal. In particular, the ?opolo&calcategory Ant of Antoine
spaces is universal.
An epireflective subcategory of A is called A-simple iff it is the epirefiective hull of
a one-element class (see [5]>.*
Proposition 7. If B and C cc_ -:;+eflective subcategories of a topological category AI.
such that = IC and C = Xx _. !km B is A-simple if aqd only if C is A-simple.
Proof. Let B = R {B} and i‘F. . r + C be the TOA-reflection of B. According to
Proposition 2, C is a C-objee.. To prove R(C) = C, by Theorem 3 it is sufficient to
show that IR(c’} = B. Since r. is initial, we have B E I(C) which implies that
Y?%=
R(B)- I{B}c I(C) = IR{C). Trivially, I’(C&B.
X C = R(C), then we may assbmc that C is a non-empty A-object. Then
the product 12x C is non-empty and it folilows that I?(12 x C} = R{&, C} =
R(Indu{C})=
I(C)=
IR(C)=
IC= B.
Herrrich [5,6] has shown that TOTop is the only op-simple q&fle&re
subcategory of Top which is included by ToTop and which contains all ‘&spaces.
Combining this with Theorem 3 asad Proposition 7, we get the result:
o&on 8. Top and
drJp are the only Top-simple epirefiective subcategories of
which contain all T3-spies.
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Th. Marny / 0~1 egir&ectite subcategories
181
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