AT INFINITY
MARTIN MARIA KOV ´AR
Received 15 October 2004; Revised 18 November 2005; Accepted 28 November 2005
We study the behavior of certain spaces and their compactificability classes at infinity.
Among other results we show that every noncompact, locally compact, second countable HausdorffspaceX such that each neighborhood of infinity (in the Alexandroff com- pactification) is uncountable, hasᏯ(X)=Ꮿ(R). We also prove some criteria for (non-) comparability of the studied classes of mutual compactificability.
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1. The notation and terminology
Throughout the paper we mostly use the standard topological notions as in [1] or [2], however with a few exceptions. Space always refers to topological space, usually consid- ered without any additional separation axioms. Especially compactness is understood without the Hausdorffseparation axiom. For the terminology related to θ-regularity, we refer the reader to [3,4]. An ordinal is the set of smaller ordinals, and a cardinal is an initial ordinal. LetSbe a set. The cardinality ofS we denote by|S|. Let (X,τ) be a topological space. If we do not wish to specify its topology explicitly, we will sometimes, for our convenience, speak less precisely about the spaceX. Conversely, if we decide to specify the topology of a spaceXintroduced in some previous steps, we will usually de- note it byτorτX (in the case that we will work simultaneously with more topological spaces or more topologies on the same set). In a space X, a pointx∈X is in the θ- closure of a setA⊆X (x∈clθA) if every closed neighborhood ofxintersectsA. A filter baseΦinX has aθ-cluster pointx∈Xifx∈
{clθF|F∈Φ}. We say that a spaceX isθ-regular if every filter base inXwith aθ-cluster point has a cluster point. For more detailed characterization ofθ-regularity, the reader is referred to [3–5]. The pointsx,y in a spaceX areT0-separable if there is an open set containing only one of the points x,y. The pointsx,yareT2-separable if they have disjoint open neighborhoods. LetXbe a space. Two disjoint setsA,B⊆Xare said to be point-wise separated inXif everyx∈A, y∈BareT2-separable inX. Several modifications of local compactness have been de- fined by various authors in the literature. In this paper, we say that a space is (strongly) locally compact if its every point has a compact (closed) neighborhood. One can easily
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 24370, Pages1–12
DOI10.1155/IJMMS/2006/24370
check that a space is strongly locally compact if and only if it isθ-regular and locally compact. We will often use the following simple, but important property of strongly locally compact spaces. If X is strongly locally compact andγX is a compactification such thatX and the remainderγX\X are point-wise separated inγX, thenX is open in γX. A filter in a space X is said to be ultra-closed if it is maximal among all filters inX having a base consisting of closed sets [1]. By the Wallman compactification ofX, we mean the setωX=X∪ {y|yis a nonconvergent ultra-closed filter inX}. The sets
(U)=U∪ {y|y∈ωX\X,U∈y}, whereUis open inX, constitute an open base of ωX. For more detail, we refer the reader to [1]. Some properties of the Wallman com- pactification of aθ-regular space were studied in [5].
2. Preliminaries and introduction
Let us recall some notions and results from the previous papers [6–8]. Let (X,τX), (Y,τY) be spaces with X∩Y =∅. We say that the spaceX is compactificable by the space Y or, in other words, X,Y are said to be mutually compactificable if there is a compact topology τK on K=X∪Y such that the topologies on X,Y induced byτK coincide withτX,τY, respectively, and the setsX,Y are point-wise separated in (K,τK). Then we say that the topologyτKisᏯ-acceptable. Recall that mutually compactificable spaces are alwaysθ-regular, and any two disjoint strongly locally compact spaces are always mutually compactificable [6].
Let Top be the class of all topological spaces. For any pair of two spacesX,Z, we define X∼Zif for every nonempty spaceY disjoint fromX,Zthe spaceXis compactificable byY if and only ifZ is compactificable byY. It can be easily seen that∼is reflexive, symmetric, transitive, and hence it is an equivalence relation. Let us denote byᏯ(X) the equivalence subclass of Top with respect to∼containingXand call it the compactificabil- ity class ofX. For any spacesX,Z, we putᏯ(X)Ꮿ(Z) if for every nonempty spaceYit holds: if the spaceXis compactificable byYdisjoint fromX,Z, thenZis compactificable byY. The relationis reflexive, antisymmetric, transitive, and hence it is an order rela- tion between the compactificability classes. If for some spacesX,Zit holdsᏯ(X)Ꮿ(Z) butᏯ(X)=Ꮿ(Z), we writeᏯ(X)Ꮿ(Z). We proved in [7] that every compactifica- bility class contains aT1 representative, but there are compactificability classes with no Hausdorffrepresentatives.
LetA=[1,∞),I=[0, 1] be equipped with the Euclidean topology induced fromR, andD= {0, 1}equipped with the discrete topology. By 0, we denote the constant function equal to 0. In [8], we proved that for anyk,n∈Nthe spacesAk,Rn,Ak×Rn,Iℵ0\ {0}, Dℵ0\ {0}are of the same class of mutual compactificability. Also we proved that ifX is a noncompact locally connected metrizable generalized continuum, thenᏯ(X)=Ꮿ(R).
All these spaces are uncountable but second countable, locally compact and Hausdorff.
On the other hand, it can be proved that there exist spaces compactificable byRwhich are not compactificable byN, but not conversely. Hence,Ꮿ(N)Ꮿ(R). Note that we omit the proof now, because it will follow as a corollary from a theorem that will be presented in the next section. Further, it seems that connectedness does not affect the compactifi- cability classes much becauseDℵ0 is homeomorphic to a subspace ofR, also known as the Cantor Discontinuum. Hence, there is a natural question whether it is true that every
uncountable, second countable, or separable locally compact Hausdorffspace must be of the same class of compactificability asR. Next we will give a counterexample for this conjecture. After some further investigation we will find out that the essential property which plays the most important role in determining the compactificability classes is the behavior at infinity.
Before we start, let us recapitulate a couple of theorems from the previous papers that we will need in our proofs in the next section. For the proofs of these result, we refer the reader to [7, Theorem 2.12] forTheorem 2.1, and to [8, Theorems 3.1 and 3.2, Corollary 3.2] for Theorems2.2,2.3,Corollary 2.1, respectively.
Theorem 2.1. LetXbe aT3.5 space which is not locally compact and letZbe a strongly locally compact (not necessarily Hausdorff) space. ThenᏯ(X) andᏯ(Z) are not comparable in the order.
Theorem 2.2. Let (X,τX) be a closed subspace of a strongly locally compact space (Z,τZ).
Then,Ꮿ(X)Ꮿ(Z).
Recall that byw(X) we denote the weight of a space (X,τ), that is, the least infinite cardinalw(X), such that (X,τ) has an open baseτ0⊆τwith|τ0| ≤w(X).
Theorem 2.3. Let (X,τ) be a locally compact Hausdorffspace withw(X)=m, wherem ℵ0. ThenᏯ(X)Ꮿ(Dm\ {0}).
Corollary 2.4. For anyk,n∈N, the spacesAk,Rn,Ak×Rn,Iℵ0\ {0},Dℵ0\ {0}are of the same class of mutual compactificability.
3. Main results
Our first theorem studies what happens with the classes of mutual compactificability if a closed compact subspace is collapsed to a singleton.
Theorem 3.1. LetH⊆X be a compact closed subspace of a topological spaceX. Define an equivalence relation∼onXby (x∼y)⇔[(x=y)∨(x,y∈H)] for anyx,y∈X. Let Y=X/∼be the quotient space. ThenᏯ(X)Ꮿ(Y).
Proof. Let (Z,τZ) be a space which is mutually compactificable by (X,τ),X∩Z=∅. We putK=X∪Zand denote byτK theᏯ-acceptable topology onK. We will show thatH is closed in (K,τK). Let y∈K\H. Ify∈X, then y∈X\H∈τX, so there existsU∈τK such thatX\H=X∩U. ThenU∩H=∅, soy∈U⊆K\H. Ify∈Z, then from the fact thatX,Zare point-wise separated in (K,τK) it follows that there areU,V∈τKsuch that y∈U,H⊆V, andU∩V=∅. But thenU∩H=∅, which implies thaty∈U⊆K\H.
Hence,His closed in (K,τK).
We will extend the equivalence relation∼toKby setting (x∼y)⇐⇒
(x=y) or ({x,y} ⊆H) for everyx,y∈K. (3.1) Let f :K→K/∼be the corresponding quotient mapping. We putL=K/∼and considerL with its quotient topologyτL. For simplicity, we may identify the singleton equivalence classes with their elements, so the quotient mapping f :K\H→L\ {h} restricted to
the open setK\H is the identity on K\H and a homeomorphism between the open subspacesK\HandL\ {h}. SinceZ⊆K\H=L\ {h}, the topology onZinduced from (L,τL) coincides with its original topologyτZ.
LetV∈τY. Then f−1(V)∈τX, so there existsU∈τKsuch that f−1(V)=U∩X. We putW=(U∩Z)∪V. Clearly,V=W∩Y. Further,
f−1(W)=f−1(U∩Z)∪f−1(V)=(U∩Z)∪f−1=(U∩Z)∪(U∩X)
=U∩(Z∪X)=U∩K=U∈τK. (3.2)
Then,W∈τLbecause the quotient mapping is continuous. It follows thatτYis weaker or equal to the topology onYinduced byτL. Conversely, letW∈τLand denoteV=W∩Y. ThenU=f−1(W)∈τK, soU∩X∈τX. We have
U∩X= f−1(W)∩X= f−1(W∩L)∩X=f−1W∩(Z∪Y)∩X
= f−1(W∩Z)∪(W∩Y)∩X=
(W∩Z)∪f−1(W∩Y)∩X
=[W∩Z∩X]∪
f−1(V)∩X=∅∪f−1(V)= f−1(V).
(3.3)
Letg:Y→Xbe the quotient mapping given by the original (not extended) equivalence
∼onX. Theng is a restriction of f, sog−1(V)= f−1(V)∈τX, and by the definition of the quotient topology we haveV∈τY. Hence, the topology onY induced from (L,τL) coincides withτY.
Lety∈Y,z∈Z. We will show thaty,zhave disjoint open neighborhoods in (L,τL).
Ify=h, then{y,z} ⊆K\H=L\ {h}. There existU,V∈τKsuch thaty∈U,z∈V, and U∩V=∅. We putP=U\H,Q=V\H. ThenP,Q⊆K\H=L\ {h}, soP,Q∈τLand we havey∈P,z∈Q, andP∩Q=∅. Lety=h. SinceX,Zare point-wise separated in (K,τK) andHis compact, there existU,V∈τKsuch thatH⊆U,z∈V, andU∩V=∅. We putP=(U\H)∪ {h}. Then
f−1(P)=f−1(U\H)∪f−1{h}
=(U\H)∪H=U∈τK. (3.4) Similarly, f−1(V)=V∈τK. ThenP,V∈τL,y=h∈P,z∈V, and
P∩V=
(U\H)∩V∪
{h} ∩V=∅∪
{h} ∩V⊆ {h} ∩(K\H)
= {h} ∩ L\ {h}
=∅. (3.5)
Hence,Y andZare point-wise separated in (L,τL).
Finally, the compactness of (L,τL) follows from the continuity of the quotient mapping f :K→L. Now we have aᏯ-acceptable topology onL=Y∩Z, soᏯ(X)Ꮿ(Y).
Corollary 3.2. LetX=N∪[−1, 0] with the Euclidean topology induced fromR. Then Ꮿ(X)=Ꮿ(N).
Proof. Clearly, [−1, 0] is the closed compact subspace ofX, so by the previous theorem we haveᏯ(X)Ꮿ(N). The converse inequality follows from the fact thatNis also a closed subspace ofX, which is strongly locally compact, and fromTheorem 2.2.
Now we will study the behavior of spaces at infinity and its influence on the compact- ificability classes. For that purpose, we will need some auxiliary assertions. The following lemma is a variation on Cantor-Bendixson theorem (cf., [2, Problem 1.7.10-11, page 59]).
Lemma 3.3. Let (X,τ) be a second countable space and letM⊆X be an uncountable set.
Then there is a closed setC⊆Xsuch thatM\Cis (at most) countable and for every open setO⊆X, eitherC∩O=∅orM∩Ois uncountable.
Proof. Letσ⊆τbe a countable base for the topologyτ. First, it is easy to see that if the condition stated in the theorem holds for every open basic setO∈σ, then it holds also for every open set fromτ. Therefore, we may restrict our considerations toσinstead ofτ. By transfinite induction, for some ordinalδ, we define a family{Oα|α < δ} ⊆σas follows:
(1) if for everyO∈σeitherM∩Ois empty or uncountable, thenC=clXMhas all the required properties and we are done. Otherwise, there existsO1∈σsuch that O1∩Mis nonempty and countable,
(2) now, suppose that we have already chosen {Oβ|β < α}for some ordinalα. If there exists a basic open setO∈σsuch thatO∩(M\
β<αOβ) is nonempty and countable, we letOαbe any such setO. Otherwise, we stop and putδ=α.
Having the family{Oα|α < δ}, we putC=clX(M\α<δOα). Since{Oα|α < δ} ⊆σ, the ordinalδis countable. We have
M=
M\
β<α
Oβ
∪
M∩
β<α
Oβ
, (3.6)
and so
Oα∩M=
Oα∩
M\
β<α
Oβ
∪
Oα∩M∩
β<α
Oβ
⊆
Oα∩
M\
β<α
Oβ
∪
β<α
Oβ∩M.
(3.7)
The setO1∩Mis countable, from the previous steps (β) whereβ < α, we supposeOβ∩M countable and from the current step (α) we know thatOα∩(M\
β<αOβ) is countable.
Hence, by transfinite induction,Oα∩Mis countable as a countable union of countable sets. Then the set
M\C=M\clX
M\
α<δ
Oα
⊆M∩
α<δ
Oα
=
α<δ
Oα∩M (3.8)
is also countable. Suppose thatC∩O=∅for someO∈σ. Then, alsoO∩(M\
α<δOα)=
∅and from the last induction step (δ) it follows thatO∩(M\
α<δOα) is uncountable, because otherwise we could setOδ=O, which would contradict to the fact that (δ) is the last step of the induction. But then also the setM∩Ois uncountable and we can see that
Chas all the required properties.
The next lemma is a variation on Cantor’s well-known result thatDℵ0 embeds into every nonempty perfect set of reals (cf., [2, Problem 3.12.11, page 230; Problem 4.5.5, page 290]).
Lemma 3.4. Let (X,τ) be a locally compact, noncompact, second countable Hausdorff space,αX=X∪ {∞}the Alexandroffcompactification ofX. If every neighborhood of∞ is uncountable, then there exists an embeddinge:Dℵ0→αXwhich maps 0 to∞.
Proof. ByLemma 3.3, there is a closed setC⊆αXsuch that every open set that intersects Cis an uncountable set and such thatX\Cis countable. Clearly,∞ ∈C. Thus, without loss of generality we may assume thatC=αX. In other words, we may assume that every nonempty open subset ofαXis uncountable. In particular, it means that no point inαX is isolated.
SinceαXis second countable, it is metrizable. Letd:αX×αX→Rbe a metric onX, which induces the topologyτ. For a pointx∈X and a positive real number r >0, we denote
B(x,r)=
y|y∈X,d(x,y)< r. (3.9) LetD<ω=
n∈ωDnbe the set of all nonempty finite sequences whose members consist of 0’s and 1’s. Every sequences∈D<ωof the lengthnmay be extended by 0 or 1, respectively, to the sequence having the lengthn+ 1. We denote the extended sequence bys0 ors1, respectively. For everys∈D<ω, we define inductively an open setOs∈τas follows:
(1) we put x0= ∞, and as x1 we take any point from X. We also put r0=r1= (1/3)d(x0,x1),O0=B(x0,r0),O1=B(x1,r1).
(2) suppose thatOs=B(xs,rs) is defined for everys∈Dn. Lets∈Dn. The pointxsis not isolated, so there existsxs1∈B(xs, 2/3rs),xs1=xs. Further, we putxs0=xs, rs0=rs1=1/3d(xs0,xs1),Os0=B(xs0,rs0),Os1=B(xs1,rs1).
Letp∈Dℵ0. We denotepnthe restriction of the infinite sequencepto the firstnelements.
From the construction it follows that
Op1⊇clαXOp2⊇Op2⊇clαXOp3⊇Op3⊇ ···. (3.10) Since the space αX is compact, the intersection n∈NclαXOpn of the closed balls is nonempty and since limn→∞rpn=0, it contains exactly one element, saye(p). In particu- lar,e(0)= ∞. The mappinge:Dℵ0→Xis an injection. Indeed, letp,q∈Dℵ0,p=q, and letn∈Nbe the least number for whichpn=qn. We havee(p)∈clαXOpn,e(q)∈clαXOqn, but clαXOpn∩clαXOqn=∅. Thus e(p)=e(q). Finally, we will show that e:Dℵ0→X is continuous. Let O∈τ be an open set containing e(p). There is some m∈N such that e(p)∈Opm⊆O. The set U= {y|y∈Dℵ0, ym=pm} is open inDℵ0 since it is an intersection ofmsub-basic sets of the product topology onDℵ0. But if y∈U, then e(y)∈clαXOym+1⊆Oym=Opm⊆O. Hence,e:Dℵ0→Xis continuous. SinceDℵ0is com-
pact,eis a homeomorphism onto its image.
Finally, we can formulate and prove the main theorem.
Theorem 3.5. LetXbe a locally compact, noncompact, second countable Hausdorffspace.
If every neighborhood of∞ ∈αXcontains uncountably many elements ofX, thenᏯ(X)= Ꮿ(R).
Proof. ByLemma 3.4,X contains a closed subspace homeomorphic toDℵ0\ {0}. Then from Theorems2.2,2.3, andCorollary 2.4, we get
ᏯDℵ0\ {0}
Ꮿ(X)ᏯDℵ0\ {0}
=Ꮿ(R), (3.11)
which consequently givesᏯ(X)=Ꮿ(R). The proof is complete.
So far, our effort was concentrated especially on the problem how to prove that the compactificability classes of two or more different spaces coincide. Perhaps it is just the right time to find some general conditions under which the compactificability classes of different spaces must differ.
LetTbe some class of directed sets with∅∈/ T. Define the following properties of a topological spaceX:
(i) propertyα(T): there exists an injective netϕ(A,≥) withA∈T,ϕ(A)⊆X, con- verging inX,
(ii) propertyα(T): no injective netϕ(A,≥) withA∈T,ϕ(A)⊆X, has a cluster point inX.
Lemma 3.6. Let (X,τX) be aT1space,I=[0, 1],K=ωX×I,Y=K\(X× {0}). We con- siderIwith its natural, Euclidean topology. LetτK be the product topology of the Wallman compactificationωXof (X,τX) with the Euclidean topology ofIand letτY be the subspace topology onY induced from (K,τK). Leta1,a2,. . .,b1,b2,. . .be two disjoint sequences in X having no cluster point inX. Let (Z,τZ) be a topological space disjoint fromY and let L=Y∪Zbe equipped with a topologyτLsuch that (Y,τY), (Z,τZ) are subspaces of (L,τL).
Letc1,c2,. . .be a sequence inZsuch that eachciis a cluster point of the netsαi(t)=(ai,t), βi(t)=(bi,t) with their values inYfort→0. Then the topologyτLis notᏯ-acceptable.
Proof. Suppose that all the conditions stated in the lemma are satisfied and the topology τLisᏯ-acceptable. We putF= {a1,a2,. . .},F= {b1,b2,. . .}. ThenF∩G=∅andF,Gare closed in (X,τX). DenoteV=X\G∈τX,(V)=V∪ {h|h∈ωX\X,V∈h}, where we identify the elements ofωX\Xwith the nonconvergent ultra-closed filters in (X,τX).
The set(V) is open inωX. Leth∈ωX\Xbe a cluster point of the sequencea1,a2,. . ..
ThenF∈h, because otherwise,(X\F) would be a neighborhood ofh, which does not contain any element froma1,a2,. . .. Then,V∈hbecauseF⊆X\G=V. Thenh∈(V).
Consequently, all cluster points of the sequencea1,a2,. . .are in(V). Analogously, ifg is a cluster point of the sequenceb1,b2,. . ., then necessarilyG∈g. ThenV=X\G /∈g, which givesg /∈(V). Therefore, all cluster points of the sequenceb1,b2,. . .are outside of(V), that is, inωX\(V). Then both sequencesa1,a2,. . .,b1,b2,. . .have no common cluster point inωX.
DenoteM=N×(0, 1]. For every (k,t), (l,s)∈M, we define
(k,t)≤(l,s)⇐⇒k≥l, t≤s. (3.12)
Further, for every (k,t)∈M, we put ϕ(k,t)=αk(t)=
ak,t, χ(k,t)=βk(t)=
bk,t. (3.13) The netsϕ(M,≥),χ(M,≥) have their values inY and they have no cluster point inX× {0}. Suppose that, for instance,ϕ(M,≥) has a cluster pointz∈Z. Then there exists a net ϕ(M,) finer thanϕ(M,≥), which converges toz. SinceK is compact,ϕ(M,) has a cluster point, sayy∈K. Butyis also a cluster point ofϕ(M,), which has no cluster point inX× {0}, so y∈K\(X× {0})=Y. However, sinceτLisᏯ-acceptable,Y andZ are point-wise separated in (L,τL). This is not possible sinceϕ(M,) converges toz∈Z and has a cluster pointy∈Y, both in the topologyτL, which onY coincides with the topology induced from (K,τK). Hence, the netϕ(M,≥), and similarly alsoχ(M,≥), has no cluster point inZ.
Letw∈Lbe the cluster point of the sequencec1,c2,. . .. TakeW∈τLsuch thatw∈W.
Let (k0,t0)∈M. There existsk≥k0such thatck∈W. However,ck is a cluster point of the netsαk(t),βk(t) fort→0. So, there existt,s∈(0, 1],t≤t0,s≤t0such thatαk(t)= (ak,t)=ϕ(k,t)∈Wandβk(s)=(bk,s)=χ(k,s)∈W. We have (k,t)≥(k0,t0) and (k,s)≥ (k0,t0), which mean thatw∈Lis a common cluster point of the netsϕ(M,≥),χ(M,≥).
Because of the previous paragraphw /∈Z, sow∈Y=K\(X× {0})=(ωX×(0, 1])∪ [(ωX\X)× {0}]. However, it is not possible thatw∈ωX×(0, 1], because in this case w∈ωX×(ε, 1] for sufficiently smallε >0. The setωX×(ε, 1] is open inτY, but it does not containϕ(k,t) orχ(k,t) if (k,t)≥(1,ε). Hence, it remainsw∈(ωX\X)× {0}. Then w=(v, 0), wherev∈ωX\X. But this implies that vis a common cluster point of the sequencesa1,a2,. . .,b1,b2,. . ., which is a contradiction.
Thus our assumption thatτLisᏯ-acceptable is incorrect.
Theorem 3.7. Let (X,τX) be aθ-regularT1 space containing a discrete infinite sequence of subspacesP1,P2,. . .with the propertyα(T). Then for any space (Z,τZ) with the property α(T) it followsᏯ(X)Ꮿ(Z).
Proof. SupposeᏯ(X)Ꮿ(Z). We putI=[0, 1] with the Euclidean topology,K=ωX× I,Y =K\(X× {0}). We consider the product topologyτK onK and the topologyτY
induced from (K,τK) onY. Clearly, the spaces (X,τX) andX× {0} with the topology induced from (K,τK) are homeomorphic. Since (X,τX) isθ-regular,ωX\X andXare pairwise separated inωX. Then, alsoX× {0}andY are pairwise separated in (K,τK).
ThenᏯ(X× {0})=Ꮿ(X)Ꮿ(Z). Hence, there exists aᏯ-acceptable topologyτLonL= Y∩Z, whereZis supposed to be disjoint fromY (ifY∩Z=∅, we will replaceZby a homeomorphic copy, disjoint fromY).
Letξi(Ai,≥) be an injective net inPisuch thatAi∈T, having a limitli∈Pi. For any α∈Aiandt∈(0, 1], we define
ζi,α(t)=
ξi(α),t∈Pi×(0, 1]⊆X×(0, 1]⊆Y. (3.14) If we consider (0, 1] as a directed set, then ζi,α is a net with values in Y and a limit (ξi(α), 0)∈X× {0}fort→0. Since the netζi,αhas a limit inX× {0}and the setsX× {0},
Y are pointwise separated in (K,τK),ζi,αhas no cluster point inY. Since (L,τL) is com- pact,ζi,α has a cluster point inZ, sayψi(α)∈Z. Then ψi(Ai,≥) is a net in Z, having Ai∈T.
Suppose thatψiis injective. Since (Z,τZ) has the propertyα(T), the netψi(Ai,≥) has no cluster point inZ, so it must have a cluster point inY, say y∈Y. We denoteMi= Ai×(0, 1] and for every (α,t), (β,s)∈Mi, we define
(α,t)(β,s)⇐⇒α≥β,t≤s, ϕi(α,t)=ζi,α(t)=
ξi(α),t. (3.15)
For everyi∈N, we have a netϕi(Mi,) with values inY. Let us show that ϕi(Mi,) has the limit (li, 0)∈X× {0}in (K,τK). ChooseV∈τXsuch thatli∈V andε >0. Then there existsα0∈Aisuch thatξi(α)∈V forαα0. Then, for every (α,t)∈Misuch that (α,ti)(α0,ε/2), we haveϕi(α,t)=(ξi(α),t)∈V×(0,ε). Then,ϕi(Mi,) converges to (li, 0)∈X× {0}. LetW∈τY be an open neighborhood of y∈Y. There existsU∈τL
such thatW=Y∩U. Let (α,t)∈Mi. There existsβ∈Ai,β≥αsuch that ψi(β)∈U.
Butψi(β) is a cluster point of the netζi,β. Hence, there existss∈(0, 1],s≤tsuch that ϕi(β,s)=ζi,β(s)=(ξi(β),s)∈U. Since the net ϕi(Mi,) has its values in Y, we have ϕi(β,s)∈W. Then y∈Y is a cluster point ofϕi(Mi,), which is a contradiction, be- cause in the previous step we proved thatϕi(Mi,) has the limit (li, 0)∈X× {0}and the setsX× {0},Y are point-wise separated in (K,τK). Hence, the only possible conclusion is thatψiis not injective.
By the previous paragraph, there existαi,βi∈Aisuch thatαi=βiandψi(αi)=ψi(βi).
We put ai=ξi(αi), bi=ξi(βi), ci=ψi(αi)=ψi(βi). Since the net ξi(Ai,≥) is injective, ai=bi. We haveai,bi∈Pi, so the sequencesa1,a2,. . .andb1,b2,. . .are disjoint because of the discreteness of the collection{P1,P2,. . .}. For the same reason, they have no clus- ter point inX. Moreover, for everyi∈N,ci=ψi(αi)=ψi(βi)∈Z is a cluster point of the nets ζi,αi(t)=(ξi(αi),t)=(ai,t), ζi,βi(t)=(ξi(βi),t)=(bi,t) for t→0 in (L,τL). By Lemma 3.6, the topologyτLis notᏯ-acceptable, which is a contradiction. Therefore, it
must beᏯ(X)Ꮿ(Z).
Modifying the technique of the proof of the previous theorem analogously, we can obtain the following result.
Theorem 3.8. Let (X,τX) be aθ-regularT1space containing a discrete infinite sequence of subspacesP1,P2,. . .with the property|Pi|> κ(κis a cardinal number). Then for any space (Z,τZ) with the property|Z| ≤κit followsᏯ(X)Ꮿ(Z).
Proof. SupposeᏯ(X)Ꮿ(Z). We putI=[0, 1] with the Euclidean topology,K=ωX×I, Y=K\(X× {0}). We consider the product topologyτK onK and the topologyτY in- duced from (K,τK) onY. The spaceX× {0}, with the topology induced from (K,τK), is homeomorphic to (X,τX) and pairwise separated from Y in (K,τK). ThenᏯ(X× {0})Ꮿ(Z). Therefore, there exists aᏯ-acceptable topologyτLonL=Y∩Z(whereZis supposed to be disjoint fromY).
Leti∈N. For everyp∈Pi, we putζi,p(t)=(p,t). The netζi,p((0, 1],≤) has its values inYand has the limit (p, 0)∈X× {0}fort→0. We put
Qi=
q|q∈Z,qis a cluster point of the netζi,p
(0, 1],≤
. (3.16)
Then |Qi| ≤κ <|Pi|, so there existai,bi∈Pi⊆X,ai=bi such that the netsζi,ai(t)= (ai,t), ζi,bi(t)=(bi,t) have the same cluster point ci∈Z fort→0 in the topologyτL. Since the collection{P1,P2,. . .}is discrete in (X,τX), the sequencesa1,a2,. . .andb1,b2,. . . are disjoint and have no cluster point inX. ByLemma 3.6, the topology τL is notᏯ-
acceptable, which is a contradiction. Hence,Ꮿ(X)Ꮿ(Z).
The next corollary summarizes some relationships between the compactificability classes of some frequently used and well-known spaces. These properties follow from the theory presented in this paper and in the three previous papers [6–8].
Corollary 3.9. The following statements are satisfied:
(i) letX,Zbe two infinite discrete spaces with different cardinalities|X|>|Z|. Then Ꮿ(Z)Ꮿ(X),
(ii)Ꮿ(R\Q)Ꮿ(Q), (iii) CompᏯ(N)Ꮿ(R),
(iv)Ꮿ(Q),Ꮿ(R\Q), andᏯ((0, 1)ℵ0) are incomparable with Comp,Ꮿ(N),Ꮿ(R), (v) letX= {n−1/m|n,m∈N}be a subspace ofR. Then
Ꮿ(N)Ꮿ(X)Ꮿ(R), (3.17)
(vi) letω= ℵ0=N, and letω1= ℵ1be the first uncountable cardinal, both considered as ordinal number, with the interval topology. Then
Ꮿω×
ω1+ 1Ꮿω1
. (3.18)
Proof. (i) Let|X|>|Z| =κ≥ ℵ0. Then there exists a decomposition ofXinto countably many pairwise disjoint subsetsPi⊆X,i∈N, such that, for eachi∈N, we have|Pi| =
|X|> κ. FromTheorem 3.8 it followsᏯ(X)Ꮿ(Z). Further, by Theorem 2.2it holds Ꮿ(Z)Ꮿ(X). ThenᏯ(Z)Ꮿ(X).
(ii) For everyn∈N, we putPn=[2n, 2n+ 1]∩(R\Q). Then|Pn|>|Q| = ℵ0, and the family{Pn|n∈N}is discrete. ByTheorem 3.8, we haveᏯ(R\Q)Ꮿ(Q).
(iii) It is clear that CompᏯ(N). We putPn=[2n, 2n+ 1] for everyn∈N. We have
|Pn|>|N| = ℵ0, so byTheorem 3.8we getᏯ(R)Ꮿ(N). ButNis a closed subspace of R, so byTheorem 2.2we haveᏯ(N)Ꮿ(R). ThenᏯ(N)Ꮿ(R).
(iv) The spacesQ,R\Q, and (0, 1)ℵ0areT3.5, but not locally compact. The compact spaces as well asNorRare strongly locally compact. The assertion now follows directly from [7, Theorem 2.1].
(v) We takeT= {(N,≥)}, where we consider the natural, linear order≥onN. Then any injective netϕ(A,≥) withA∈Tis an injective sequence, which clearly has no clus- ter point in N. However, if we put Pn= {n−1/m|m∈N,m≥2} ∪ {n}, then Pn⊆ X contains an injective sequencen−1/2,n−1/3,n−1/4,. . .converging ton∈Pn. The
family{P1,P2,. . .}is discrete. ByTheorem 3.7it holdsᏯ(X)Ꮿ(N). On the other hand, Nis a closed subspace ofX, so byTheorem 2.2,Ꮿ(N)Ꮿ(X). Hence,Ꮿ(N)Ꮿ(X). For provingᏯ(X)Ꮿ(R), we may adjust (iii) and use the fact thatXis a closed countable subspace ofR.
(vi) We take T= {(ω1,≥)}, where ≥ is the order of ordinals. An injective net ϕ (ω1,≥) has no cluster point inω1, since any uncountable subset ofω1is not bounded. On the other hand,Pn= {n} ×(ω1+ 1) clearly contains values of an injective netϕn(ω1,≥) given byϕn(α)=(n,α), converging to the point (n,ω1)∈Pn. ApplyingTheorem 3.7, we
getᏯ(ω×(ω1+ 1))Ꮿ(ω1).
However, we will note that there are still much more open questions than answers and solutions. Among numerous very natural questions that one certainly may ask, we can only list some of them.
Question 3.10. Is Comp the only class aboveᏯ(N)? Is it true that a spaceXis compact if and only ifᏯ(X)Ꮿ(N)?
Question 3.11. Which compactificability classes have the representatives among the subspaces of the real line and which are their relationships?
Remark to the previous question that we know almost nothing about the compacti- ficability classes represented by the non-locally compact subspaces of the real line, like Ꮿ(Q) orᏯ(R\Q). Perhaps some set-theoretic axioms like CH may affect the next open question.
Question 3.12. Is it true that between Ꮿ(N) and Ꮿ(R) there are infinitely or even uncountably many compactificability classes?
Question 3.13. Is it true thatᏯ(ω1)Ꮿ(ω×(ω1+ 1))?
Question 3.14. Which relationships are there between the compactificability classes rep- resented by the spaces of ordinals?
Question 3.15. Which properties have the compactificability classes represented by spaces constructed from the classic examples that were given over the years (Sorgenfrey line, Niemytzki plane,. . .)?
It is known that each compactificability class has aT1 representative, however, there exist classes of mutual compactificability with no Hausdorffrepresentatives. Hence, we will close the paper by the following natural question.
Question 3.16. Is it true that every compactificability class contains a sober or soberT1
representative?
Acknowledgment
The author acknowledges support from Grant no. 201/03/0933 of the Grant Agency of the Czech Republic and from the Research Intention MSM 0021630503 of the Ministry of Education of the Czech Republic.
References
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[2] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann, Berlin, 1989.
[3] D. S. Jankovi´c,θ-regular spaces, International Journal of Mathematics and Mathematical Sci- ences 8 (1985), no. 3, 615–619.
[4] M. M. Kov´ar, Onθ-regular spaces, International Journal of Mathematics and Mathematical Sci- ences 17 (1994), no. 4, 687–692.
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[7] , The classes of mutual compactificability, to appear in International Journal of Mathe- matics and Mathematical Sciences.
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Martin Maria Kov´ar: Department of Mathematics, Faculty of Electrical Engineering and Communication, University of Technology, Technick´a 8, Brno 616 69, Czech Republic E-mail address:[email protected]
