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MAGILL-TYPE THEOREMS FOR MAPPINGS
GIORGIO NORDO and BORIS A. PASYNKOV
Received 9 December 2002
Magill’s and Rayburn’s theorems on the homeomorphism of Stone-ˇCech remainders and some of their generalizations to the remainders of arbitrary Hausdorff compactifications of Tychonoff spaces are extended to some class of mappings.
2000 Mathematics Subject Classification: 54C05, 54C20, 54D40, 54D50.
1. Introduction. In 1968, Magill [8] proved the following theorem.
Theorem1.1. For two locally compact Tychonoff spacesXandY, the Stone- ˇCech re-mainders (i.e., the rere-mainders of the Stone- ˇCech compactifications)βX\XandβY\Yare homeomorphic if and only if the posetsK(X)andK(Y )of all Hausdorff compactifications ofXandY are isomorphic.
In 1973, Rayburn [12] gave the following definition.
Definition1.2. A Tychonoff spaceXis calledk-absolute ifβX\Xis ak-space. It is proved in [12] that aTychonoff spaceXis k-absolute if and only if cX\Xis a
k-space for somecX∈K(X)and if and only if cX\Xis ak-space for anycX∈K(X).
All locally compact Tychonoff spaces arek-absolute because their Stone-ˇCech remain-ders are compact. Consequently, the following Rayburn’s theorem generalizes one half ofTheorem 1.1.
Theorem1.3. For any pair ofk-absolute spacesX,Y, if the posetsK(X)andK(Y ) are isomorphic, then the Stone- ˇCech remaindersβX\XandβY\Y are homeomorphic.
We note that the second half of Theorem 1.1cannot be generalized tok-absolute spaces (see [12, Example (B)]).
In [2], both Theorems1.1and1.3are generalized to arbitrary compactifications of two Tychonoff spaces.
We note that results concerning the extension ofTheorem 1.1to mappings are also contained in [5], but they are different from ours (see the remark at the end of this paper).
2. Preliminaries. Throughout this paper, space will mean a topological space and mapping will mean a continuous function. Terms and undefined concepts are used as in [4]. In this section, we recall some definitions and results from [2]. Some additional notions concerning fibrewise general topology (FGT) can be found in [9,10].
Definition2.1. LetX,Y,Z be spaces andλ:X→Y,µ:X→Zmappings. We say thatλis equivalent toµand we will writeλ≡µif there exists some homeomorphism
h:Y→Zsuch thatµ=h◦λ.
Evidently, the homeomorphismhis unique.
We will identify equivalent mappings, and so we can consider the setᏯ(X)of all the continuous maps from a fixed spaceXonto other spaces.
Definition2.2. Letλ, µ∈Ꮿ(X). We say thatλfollowsµand we will writeλ≥µif there exists some continuous mappingh:Y→Zsuch thatµ=h◦λ.
It is evident that(Ꮿ(X),≥)is a poset.
Definition2.3. Letλ∈Ꮿ(X). We say that
(i) λissimpleif there exists a unique pointtλ∈λ(X)such that|λ−1({tλ})|>1 and |λ−1({t})| =1 for everyt∈λ(X)\{t
λ};
(ii) λis finite simpleif there exists a nonempty finite setT ⊂λ(X)of points such that|λ−1({t})|>1 for everyt∈T and|λ−1({t})| =1 for everyt∈λ(X)\T.
We will suppose from this moment thatXis a Hausdorff space and thatᏼ(X)denotes the poset (as a subposet ofᏯ(X)) of all perfect onto mappings ofX. Clearly,λ(X)is Hausdorff for anyλ∈ᏼ(X).
Definition 2.4. A mapping λ ∈ᏼ(X)is called a dual point if it is simple and |λ−1({t
λ})| =2.
LetᏰ=Ᏸ(X)denote the set of all dual points of ᏼ(X)andᏲ(X)= {λ∈ᏼ(X):
λis finite simple}∪{idX}.
Definition2.5. A familyᏲ⊂Ᏸ is said to be a3-vertex family if for any distinct
α, β∈Ᏺthere exists someγ∈Ᏸ\Ᏺsuch thatγ >inf{α, β}.
Definition2.6. A 3-vertex familyᏲ⊂Ᏸis called apoint familyif it is maximal (i.e., if there is no 3-vertex family properly containingᏲ).
Lemma2.7. IfᏲis a 3-vertex family consisting of more than one element, then the setXᏲ={λ−1({t
λ}):λ∈Ᏺ}is a single point (which will be denoted byJᏵ(Ᏺ)).
Definition2.8. LetᏵ⊂ᏼ(X)such thatᏲ(X)⊂Ᏽandx∈X. We putKᏵ(x)= {δ∈
Ᏸ:x∈δ−1({t
δ})}.
Evidently, for|X|>2,
JᏵKᏵ(x)=x for anyx∈X, (2.1)
KᏵJᏵ(Ᏺ)=Ᏺ for any point familyᏲ⊂Ᏸ. (2.2)
In [2], dual points are characterized only by means of the order inᏵ. It follows from this that if, for Hausdorff spacesXj, we haveᏲ(Xj)⊂Ᏽj⊂ᏼ(Xj)withj=1,2 and
i:Ᏽ1→Ᏽ2is a poset isomorphism, theni(KᏵ1(x))is a point family inᏵ2.
We recall [2] that a one-to-one mappingf:X→Y between Hausdorff spaces is called ak-homeomorphism iffis continuous on compact subsets ofXand its inversef−1is
continuous on compact subsets ofY. Clearly, ak-homeomorphism betweenk-spaces is a homeomorphism.
Theorem2.10. LetXjbe a Hausdorff space andᏼ(Xj)the set of all perfect onto map-pings ofXj(withj=1,2). IfX1andX2arek-homeomorphic and they arek-spaces, then
ᏼ(X1)andᏼ(X2)(and soᏲ(X1)andᏲ(X2)) are isomorphic. LetᏲ(Xj)⊂Ᏽj⊂ᏼ(Xj) forj=1,2. IfᏵ1andᏵ2are poset isomorphic, thenX1andX2arek-homeomorphic, and
if, additionally,X1andX2arek-spaces, then they are homeomorphic. More precisely, if
i:Ᏽ1→Ᏽ2is a poset isomorphism and|X1|>2, then the functionhi:X1→X2, such that
hi(x)=JᏵ2(i(KᏵ1(x)))for anyx∈X1, is ak-homeomorphism.
Now, letX be a Tychonoff space and letK(X) denote the poset of all Hausdorff compactifications ofX(see, e.g., [3]).
For anycX, dX∈K(X)such thatcX < dX, let
πdc=πdcX:dX →cX (2.3)
be the canonical map (i.e., it is continuous andπdc|X=idX). Thenπdc−1(cX\X)=dX\Xand the mapping
r πdc=r πdcX
def
= πdc:dX\X →cXX (2.4)
is perfect and onto, that is,r πdc∈ᏼ(dX\X). FixeX∈K(X)and let
K(eX)=cX∈K(X):cX≤eX (2.5)
(in particular,K(βX)=K(X)). In [2], the function
σeX:K(eX) →ᏼ(eX\X) (2.6)
was defined byσeX(cX)=r πecand the following lemma was proved.
Lemma2.11. σeXis an isomorphism of the posetsK(eX)and
ᏸ(eX)def= σeX
The following two lemmas were also proved in [2].
Lemma2.12. Ᏺ(eX\X)⊂ᏸ(eX).
Lemma2.13. IfXis a locally compact space, thenᏸ(eX)=ᏼ(eX\X).
3. On the homeomorphisms of two pairs of spaces
Definition3.1. A spaceX and a closed subsetAare called apair of spaces and denoted by(X, A).
Definition3.2. Suppose(X, A)and(Y , B)are pairs of spaces, whereX,Y are Haus-dorff. Then a homeomorphism (ak-homeomorphism)h:X→Y is called a homeomor-phism (ak-homeomorphism) of the pair(X, A)onto the pair(Y , B)ifh(A)=B.
LetXbe a Hausdorff space andAa closed subset ofX. For anyλ∈ᏼ(X), let
resXA(λ)=λ:A →λ(A), (3.1)
that is, resXA(λ)is the corestriction ofλtoA.
Evidently, resXA(ᏼ(X))⊂ᏼ(A)and resXA:ᏼ(X)→ᏼ(A)is monotonous. It is not difficult to prove the following lemma.
Lemma3.3. resXA(ᏼ(X))=ᏼ(A),resXA(Ᏺ(X))=Ᏺ(A), andresXA(Ᏸ(X)∪{idX})=
Ᏸ(A)∪{idA}.
LetᏲ(X)⊂Ᏽ(X)⊂ᏼ(X)andᏲ(A)⊂Ᏽ(A)⊂ᏼ(A). Then, clearly,
resXA
KᏵ(X)(x)
\idA
=KᏵ(A)(x) ∀x∈A. (3.2)
Theorem3.4. LetXandY be Hausdorff spaces, letAbe a closed subset ofX, and letBbe a closed subset ofY,
min|X|,|Y|≥3, min|A|,|B|≥2, Ᏺ(X)⊂Ᏽ(X)⊂ᏼ(X), Ᏺ(A)⊂Ᏽ(A)⊂ᏼ(A),
Ᏺ(Y )⊂Ᏽ(Y )⊂ᏼ(Y ), Ᏺ(B)⊂Ᏽ(B)⊂ᏼ(B),
(3.3)
and let
iXY:Ᏽ(X)→Ᏽ(Y ), iAB:Ᏽ(A) →Ᏽ(B) (3.4)
be poset isomorphisms such that
iAB◦resXA=resY B◦iXY. (3.5)
Then
(i) in the casemin{|A|,|B|} ≥3fork-homeomorphismshiAB:A→BandhiXY :X→Y
(seeTheorem 2.10),
(ii) in the casemin{|A|,|B|} =2, there exists a homeomorphismhiAB(x):A→Bsuch
that (3.6) is also true. Thus
hiXY(A)⊂B, hiAB=hiXY :A →B, (3.7)
and sohiXY is ak-homeomorphism of(X, A)onto(Y , B).
If, additionally,XandY arek-spaces, thenhiXY andhiAB are homeomorphisms, and
sohiXY is a homeomorphism of(X, A)onto(Y , B).
Proof. First, let min{|A|,|B|} ≥3.
Letx∈A. Then, byTheorem 2.10,hiXY andhiAB arek-homeomorphisms and
hiAB(x)=JᏵ(B)
iAB
KᏵ(A)(x) by (3.2)
=JᏵ(B)
iAB
resXA
KᏵ(X)(x)
\idA
by (3.5) and sinceiABis a poset isomorphism =JᏵ(B)resY BiXYKᏵ(X)(x)\idB
by (2.2) and sinceiXY is a poset isomorphism
=JᏵ(B)
resY B
KᏵ(Y )
JᏵ(Y )
iXY
KᏵ(X)(x)
\idB
by the definition ofhiXY
=JᏵ(B)resY BKᏵ(Y )hiXY(x)
\idB by (3.2) =JᏵ(B)KᏵ(B)hiXY(x) by (2.1)
=hiXY(x).
(3.8)
Now, let|A| =2. SinceiABis a poset isomorphism,|Ᏽ(B)| = |Ᏽ(A)| =2 and so|B| =2. There is a unique dual pointλ∈Ᏽ(X)such that λ−1({t
λ})=A. Let A= {x1,x2}. Evidently,KᏵ(X)(x1)∩KᏵ(X)(x2)= {λ}. Then, by (2.2),
KᏵ(Y )
hiXY
xi
=KᏵ(Y )
JᏵ(Y )
iXY
KᏵ(X)
xi
=iXY
KᏵ(X)
xi
fori=1,2. (3.9)
Hence
iXY
{λ}=iXY
KᏵ(X)
x1∩KᏵ(X)
x2
=iXYKᏵ(X)x1∩iXYKᏵ(X)x2 =KᏵ(Y )
hiXY
x1∩KᏵ(Y )
hiXY
x2.
(3.10)
Thus, forη=iXY(λ),η−1({tη})= {hiXY(x1), hiXY(x1)} =hiXY(A). But, by (3.5),ξ=
resY B(η)=resY B(iXY(λ))=iAB(resXA(λ)). Since resXA(λ)is a dual point inᏵ(A)andiAB is a poset isomorphism,ξis also a dual point inᏵ(B). Thus,B=ξ−1({t
ξ})=η−1({tη})=
hiXY(A).
4. Extensions of Magill’s and Rayburn’s theorems to mappings
iff1=f2◦h. The mappingsfj:Xj→Y (forj=1,2) are (k-) homeomorphic if there exists a (k-) homeomorphism off1ontof2.
It is not difficult to prove the following lemma.
Lemma4.2. For mappingsfj:Xj→Y (withj=1,2) and for (k-) homeomorphism
h:X1→X2of spacesX1andX2, the following conditions are equivalent: (i) his a homeomorphism off1ontof2;
(ii) h(f−1
1 ({y}))⊂f2−1({y})for everyy∈Y;
(iii) there is a function hy :f1−1({y})→f2−1({y}) such that hy =h:f1−1({y})→ f2−1({y})for everyy∈Y.
Given a Tychonoff spaceXand a closed subsetAofX, we may define a function
kresXA:K(X) →K(A) (4.1)
such that
kresXA(cX)=clcX(A) for everycX∈K(X). (4.2)
If eX, cX ∈ K(X)and cX < eX, then πecX(cleX(A))=clcX(A), πecX(A)=A, and
(πecX:A→A)=idA.
Consequently, (eA = kresXA(eX)) > (cA =kresXA(cX)) and πecA =πecX : eA →
cA. Thus, kresXA is monotone andr πecA=r πecX :eA\A→cA\A, that is,r πecA= reseX\X,eA\A(r πecX). It follows from this that(σeA◦kresXA)(cX)=σeA(cA)=r πecA= reseX\X,eA\A(r πecX)=reseX\X,eA\A(σeX(cX)).
We have then proven the following lemma.
Lemma4.3. IfeX∈K(X)andeA=kresXA(eX), then
reseX\X,eA\A◦σeX=σeA◦kresXA|K(eX). (4.3)
Letf:X→Y be a mapping to a Tychonoff spaceY and let βf+:βX→βY be the (usual) continuous extension off over the Stone-Cˇech compactificationsβX, βfX=
(βf+)−1(Y ), andβf =βf+:βfX→Y. Evidently, the mappingβf is perfect. We note
thatX isC∗-embedded inβfXbecauseX⊂βfX⊂βX. Recall thatf is called a WZ-mapping [6] (resp., a Z-mapping) if (βf )−1({y})=cl
βfX(f−1({y})) for everyy ∈
Y (resp., if f (Z) is closed for any zero-setZ in X). It is clear that (βf )−1({y})= β(f−1({y})) if the space X is normal and f is a WZ-mapping. It is known [6] that
every Z-mapping is a WZ-mapping.
Theorem 4.4. LetXj be a Tychonoff space, let Y be a compact Hausdorff space, letfj:Xj→Y be a WZ-mapping, leteXj be a Hausdorff compactification ofXj, and letefj:eXj→Y be a continuous extension offj (thus,efjis a compactification of fj) for j=1,2. Let also Xjy =fj−1({y}),eXjy =cleXj(Xjy)(i.e.,eXjy =kresXjXjy(eXj))
forj=1,2, and suppose that there exist poset isomorphismsi:K(eX1)→K(eX2)and
iy:K(eX1y)→K(eX2y)such that
Then the remaindersefj\fj
def
=efj:eXj\Xj→Yofefjforj=1,2arek-homeomorphic (more exactly, ifmin{|eX1\X1|,|eX2\X2|} ≥3, then the functionhσ
eX2◦i◦σeX−11:eX1\X1→
eX2\X2is ak-homeomorphism ofef1\f1ontoef2\f2). If, additionally,X1andX2are
k-absolute spaces, then the remaindersef1\f1andef2\f2are homeomorphic.
Proof. Fixy∈Y. LetRej=eXj\XjandRejy=eXjy\Xjyforj=1,2. By Lemmas2.11and2.12, the mappingsσeXj:K(eXj)→(Ᏽj
def
= σeXj(K(eXj)))and
σeXjy :K(eXjy)→(Ᏽjy
def
= σeXjy(K(eXjy)))are poset isomorphisms andᏲ(Rej)⊂Ᏽj,
Ᏺ(Rejy)⊂Ᏽjy forj=1,2. Hence,i12=σeX2◦i◦σ−
1
eX1:Ᏽ1→Ᏽ2andi12y=σeX2y◦iy◦σ−
1
eX1y:Ᏽ1y→Ᏽ2yare poset
isomorphisms too.
We consider the following diagram:
KeX1 i
kresX1X1y
KeX2
σeX2
kresX2X2y
Ᏽ1
σeX−1
1
i12
resRe 1Re1y
Ᏽ2
resRe 2Re2y
Ᏽ1y i12y
σeX−1
1y
Ᏽ2y
KeX1y
iy
KeX2y. σeX2y
(4.5)
By (4.4), the external part of it is commutative. We prove that its internal part is commutative too, that is, that
resRe2Re2y◦i12=i12y◦resRe1Re1y. (4.6)
By (4.3), forj=1,2, we have (see the diagram)
resRejRejy◦σeXj=σeXjy◦kresXjXjy|K(eXj). (4.7)
Thus, sinceσeXj,σeXjy are isomorphisms,
kresXjXjy|K(eXj)◦σ−
1
eXj=σeX−1jy◦resRejRejy,
resRejRejy
Ᏽj
⊂Ᏽjy.
Hence,
resRe2Re2y◦i12=resRe2Re2y◦σeX2◦i◦σ− 1
eX1
=σeX2y◦kresX2X2y◦i◦σ−
1
eX1
=σeX2y◦iy◦kresX1X1y◦σ−
1
eX1
=σeX2y◦iy◦σ−
1
eX1y◦resRe1Re1y
=i12y◦resRe1Re1y.
(4.9)
ByTheorem 3.4, in the case min{|Re1|,|Re2|,|Re1y|,|Re2y|} ≥3, the homeomorphisms
hi12:Re1→Re2andhi12y:Re1y→Re2yare such that
hi12y=hi12:Re1y →Re2y, (4.10)
and in the case min{|Re1|,|Re2|} ≥3, min{|Re1y|,|Re2y|} ≥2, there exists a homeomor-phismhi12y:Re1y→Re2y such that (4.10) is also true.
Finally, in the case min{|Re1|,|Re2|} ≥3, min{|Re1y|,|Re2y|} ≤1, the existence of a homeomorphismhi12y:Re1y→Re2ysuch that (4.10) holds is evident.
Thus, in the case min{|Re1|,|Re2|} ≥3, (4.10) holds for everyy∈Y.
In the case min{|Re1|,|Re2|} ≤2, the existence of homeomorphismshi12:Re1→Re2
andhi12y:Re1y→Re2ysuch that (4.10) holds is evident.
Sincefjis a WZ-mapping,Rejy=(efj)−1({y})\fj−1({y})forj=1,2, and so
hi12y=hi12:
ef1−1{y}\f1−1{y} →ef2−1{y}\f2−1{y} for everyy∈Y .
(4.11)
ByLemma 4.2,hi12 is ak-homeomorphism ofef1\f1ontoef2\f2.
Corollary4.5[2]. LetX1,X2be Tychonoff spaces and leteX1,eX2be their Hausdorff compactifications. IfK(eX1)andK(eX2)are poset isomorphic, then the remainderseX1\
X1andeX2\X2arek-homeomorphic, and they are homeomorphic if, additionally,X1,
X2arek-absolute spaces.
Proof. It is sufficient to applyTheorem 4.4to (the simplest) mappingsfjofXjto the single point spaceY forj=1,2.
In particular, inCorollary 4.5, foreXj=βXj (j=1,2), we have Rayburn’sTheorem 1.3. Thus,Theorem 4.4is a generalization of this theorem to mappings.
Theorem 4.6. Let Xj be a locally compact Tychonoff space, let Y be a compact Hausdorff space, letfj:Xj→Y be a WZ-mapping, let eXj be a Hausdorff compactifi-cation ofXj, and letefj:eXj→Y be a continuous extension offj forj=1,2. Let also
if there exist poset isomorphismsi:K(eX1)→K(eX2)andiy:K(eX1y)→K(eX2y)such that (4.4) holds.
Proof. One half of the theorem follows fromTheorem 4.4.
Let Rej = eXj\Xj and Rejy =eXjy\Xjy for all y ∈ Y and j = 1,2. Now, sup-pose that the remaindersef1\f1and ef2\f2 are homeomorphic. Then there exists a homeomorphismh:Re2→Re1such thatef2\f2=(ef1\f1)◦h. Hence, the mappings hy =h:Re2y→Re1y are homeomorphisms for ally∈Y. Sincefj is a WZ-mapping,
(efj)−1({y}) is a compactification offj−1({y}), and so eXjy =(efj)−1({y}) for all
y∈Y andj=1,2. Evidently,i12:ᏼ(Re1)→ᏼ(Re2)andi12y:ᏼ(Re1y)→ᏼ(Re2y), such thati12(λ)=λ◦hforλ∈ᏼ(X1)andi12y(λ)=λ◦hy forλ∈ᏼ(X1y)and ally∈Y, are poset isomorphisms.
We prove that (4.4) holds for ally∈Y.
Indeed, for everyλ∈ᏼ(Re1)andy∈Y, we have
i12y◦resRe1Re1y(λ)=i12y
λ|Re1y
=λ|Re1y◦hy
=(λ◦h)|Re2y=
i12(λ)|Re2y
=resRe2Re2y◦i12(λ).
(4.12)
By Lemmas2.11and2.13,σeXj:K(eXj)→ᏼ(Rej)andσeXjy:K(eXjy)→ᏼ(Rejy), for
j=1,2 andy∈Y, are poset isomorphisms. Hence,i=σ−1
eX2◦i12◦σeX1andiy=σ− 1
eX2y◦
i12y◦σeX1y are poset isomorphisms fory∈Y andj=1,2.
We consider the diagram obtained from the previous one by replacingᏵj andᏵjy byᏼ(Rej)andᏼ(Rejy)forj =1,2 and σeX−11, σ
−1
eX1y,σeX2, σeX2y byσeX1,σeX1y, σ−
1
eX2, σeX−12y, respectively. By (4.12), its internal part is commutative. As above, in the proof of
Theorem 4.4, we can prove that its external part is commutative too, that is, that (4.4) holds.
Corollary4.7[2]. LetX1,X2be locally compact Tychonoff spaces and leteX1,eX2 be Hausdorff compactifications ofX1andX2, respectively. Then the remainderseX1\X1
andeX2\X2are homeomorphic if and only ifK(eX1)andK(eX2)are poset isomorphic.
Proof. It is sufficient to applyTheorem 4.6to the simplest mappingsfjofXjand
efjofeXjto the single point spaceY forj=1,2.
In particular, when inCorollary 4.7,eXj=βXjforj=1,2, we have Magill’s theorem from [8]. Thus,Theorem 4.6is a generalization of this theorem to mappings.
5. Reformulations of results obtained above and some examples. Some readers may find that Theorems4.4and4.6do not sound very natural. The reformulations, in the framework of FGT, sound better to us.
We will start with some definitions and results of FGT. A mapping is calledcompactif it is perfect.
Lemma5.1. For a compact Hausdorff spaceY, a mappingf:X→Yis compact if and only ifXis compact.
Definition5.2. A mappingf:X→Y is said to be T0[11] if for everyx, x∈Xsuch thatx=xandf (x)=f (x), there exists a neighbourhood ofxinXwhich does not containxor a neighbourhood ofxinXnot containingx.
Definition5.3. A mappingf:X→Y is said to becompletely regular [11] if for every closed setFofXandx∈X\F, there exist a neighbourhoodOoff (x)inYand a continuous functionϕ:f−1(O)→[0,1]such thatϕ(x)=0 andϕ(F∩f−1(O))⊆ {1}.
A completely regularT0-mapping is calledTychonoff (or T3(1/2)).
It is not difficult to prove the following lemma.
Lemma5.4[11]. For a Tychonoff spaceY, a mappingf:X→Y is Tychonoff if and only ifXis Tychonoff.
Definition5.5. A compact Tychonoff mappingef:efX→Y is called aTychonoff compactificationof a Tychonoff mappingf:X→Y ifX⊂efX,Xis dense inefX, and
efX|X=f(more precisely, if some embeddinge:X→efXis fixed so thate(X)is dense inefXandf=ef◦e, but usually,Xande(X)are identified by means ofe).
Throughout the rest of the paper, we fix a spaceYand we will consider only Tychonoff mappings toY and their Tychonoff compactifications.
Definition 5.6. A mapping λ: dfX→ cfX between two compactifications cf :
cfX→Y and df :dfX→Y is called canonical if df =cf◦λ andλ|X =idX. In this case, one says that we have a canonical morphism λ:df →cf (and we write that
df > cf).
It is not difficult to prove thatdf andcf are homeomorphic if and only ifdf > cf
andcf > df (see, e.g., [1]).
It is proved in [11] (see also [1]) that all compactifications of a mapping toY form a set up to canonical homeomorphisms. This set will be denoted byT K(f ). Evidently, with respect to the just defined relation>,T K(f )is a poset.
In [11], it is also proved that there exists the maximal elementβf:βfX→YinT K(f ) and that, ifY is Tychonoff,βf may be obtained in the following way.
ByLemma 5.4,Xis Tychonoff. Hence, there exists the unique continuous extension
βf+:βX→βY off. ThenβfX=(βf+)−1(Y )andβf =βf+:βfX→Y.
For a compactificationef:efX→Y of a mappingf :X→Y, the mappingef\f=
ef:efX\X→Y is called theremainderofef.
A mappingf:X→Y is calledlocally compact[7] if for any pointx∈X, there exists a neighbourhoodOofxinXsuch thatf|clX(O):clX(O)→Y is compact.
It is not difficult to prove that
(ii) a mapping f :X →Y is locally compact if and only if X is open inβfX or, equivalently, ifXis open inefXfor any compactificationef:efX→Y off. Now,Theorem 4.4can be reformulated in the following way.
Theorem5.7. LetY be a compact Hausdorff space andefj:ef
jXj→Y a Tychonoff
compactification of a locally compact Tychonoff WZ-mappingfj:Xj→Y (forj=1,2). Let alsoXjy=fj−1({y}),eXjy=cleXj(Xjy)forj=1,2. Then the remaindersef1\f1and
ef2\f2are homeomorphic if and only if there exist poset isomorphismsi:K(ef1X1)→ K(ef2X2)andiy:K(eX1y)→K(eX2y)such that (4.4) holds.
Theorem 3.4may be reformulated analogously. But, even in the style ofTheorem 5.7, the formulation of Theorem 4.4 (andTheorem 3.4) seems too complicated, but this complexity may not be avoided. In order to explain why, for any fixed spaceY, consider the categoryTopY, where
ObTopY
=f∈C(X, Y ): X∈ObTop (5.1)
is the class of theobjectsand, for every pairf:X→Y,g:Z→Y of objects,
M(f , g)=λ∈C(X, Z): g◦λ=f (5.2)
is the class of themorphismsfromf tog, whose generic representation is denoted in short byλ:f→g.
So, the question is: may the passage from the categoryTopto the category TopY allow us to give simpler variants of Theorems4.4and4.6which can generalize Magill’s and Rayburn’s theorems?
In this connection, we will give two examples which demonstrate that in the frame-work ofTopY, such generalizations are impossible.
Example5.8. LetY=[0,2],X1=X2=N, andfj:Xj→Y be such thatfj(Xj)= {j} forj=1,2. ThenβfjXj=βN, βfj(βfjXj)= {j},βfj\fj=βfj:βfjXj\Xj→Y, and
so βfj\fj=βfj: βN\N→Y, with (βfj\fj)(βN\N)= {j}for j=1,2. Thus, the remaindersβf1\f1 andβf2\f2 are not homeomorphic, butT K(f1)andT K(f2)are poset isomorphic because they, in fact, coincide withK(N).
This example shows that an extension of Magill’s Theorem to the categoryTopYmust take into consideration fibres of objects ofTopY.
Example5.9. LetI= {(x,0)∈R2: 0≤x≤1},L= {(0, y)∈R2:−1≤y≤1},J= {(x,1)∈R2: 0≤x≤1},Y =I,S1=I∪L, andS2=J∪L. Letj=1,2. Putpr
j(x, y)=x for any(x, y)∈Sj. Letω1be the space of all finite and countable ordinal numbers. Thenβω1=ω1+1. PutXj=Sj×ω1and letπjbe the projection of the productXjonto its factorSj. Takefj=prj◦πj. Evidently,fjis either closed or open and all its fibres are countable compact. ThenβXj=Sj×(ω1+1)andRj=βXj\Xj is homeomorphic toSj. Letpjbe the projection of the productβXjonto its factorSj. Thenβfj=prj◦pj andβfj\fj=βfj:Rj→Y is homeomorphic toprj.
For anyt=(x,0)∈Y,(βfj)−1({t})=β(fj−1({t})), and, for anyt=(x,0)∈Y with
x >0, the remaindersRjt=(βfj)−1({t})\fj−1({t})=β(fj−1({t}))\fj−1({t})are sin-gle points. Hence,K(fj−1({t}))consists of only one element fort=(x,0)∈Y with
x >0. Fort=(0,0), the remainderRjt is homeomorphic toL. Evidently,T K(fj)and
K(f−1
j ({(0,0)}))are poset isomorphic toᏼ(L).
Now, it is evident that there exist poset isomorphisms i: T K(f1)→ T K(f2), it :
K(f1−1({t})) → K(f2−1({t})) for t ∈ Y and monotone functions m
tj : T K(fj) →
K(fj−1({t}))fort∈Y andj=1,2such that
(i) for any cfj: cfjXj →Y with cfj ∈T K(fj), mtj(cfj)=(cfj)−
1({t}), that is,
mtj(cfj)is the closure of fj−1({t})incfjXj;
(ii) mt2◦i=it◦mt1for everyt∈Y.
Example 5.9shows that the use of the posetsK(eX1)andK(eX2)(in particularK(X1)
andK(X2)) instead ofT K(f1)andT K(f2)is justified.
Remark 5.10. Let f : X →Y be a mapping between Tychonoff spaces M(f ) = {y ∈ Y : fis not closed atyorf−1({y})is not compact} and M1(f ) = {y ∈ Y :
|(βf )−1({y})\f−1({y})| =1}. Evidently,M1(f )⊂M(f ) and βf (β
fX\X)=M(f ). The following proposition is proved in [5].
Proposition5.11. Letf :X→Y andg:Z→T be locally compact mappings be-tween Tychonoff spaces and|M1(f )| = |M1(g)| =1. Then there exists a one-to-one cor-respondenceΨ:M(f )→M(g)such that(βf\f )−1({y})and(βg\g)−1({Ψ(y)})are
homeomorphic for anyy∈Y. If, additionally,M(f )andM(g)are discrete, thenβf\f
andβg\gare homeomorphic in the sense of [5], that is, there exist homeomorphisms
ϕ:βfX\X→βgZ\Zandψ:M(f )→M(g)such thatψ◦(βf\f )=(βg\g)◦ϕif and only ifT K(f )andT K(g)are poset isomorphic.
Acknowledgments. The authors would like to thank the anonymous referees for their very accurate reading and the valuable suggestions. This research was supported by MURST and CNR (GNSAGA). The second author was supported by KCFE and RFFI.
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E-mail address:[email protected]
Boris A. Pasynkov: Chair of General Topology and Geometry, Faculty of Mechanics and Math-ematics, Moscow State University, Moscow 119899, Russia