EXTENSIONS AND MAPPINGS OF TOPOLOGICAL SPACES
23–68
Laurent¸iu I. Calmut¸chi, Mitrofan M. Choban
Department of Mathematics, Tiraspol State University,
str. Gh. Iablocichin 5, Chi¸sin˘au, MD-2069, Republic of Moldova
Abstract In the present paper the class of extensions of topological spaces and the meth- ods of constructing of special extensions are investigated. The notions of quasi- compactness, compactness and double compactness are considered. Various problems of the theory of extensions are stated.
1991 MSC primary 54C25, 54D35, 54D40, 54D60, 54E15, 54F15; sec- ondary 54A05, 54C05, 54C20, 54D80, 54E18.
Key words: compactness, extension, compactification, remainder, uniform space, singular mapping, superperfect mappings.
Introduction
Compactness is one of the most important notions. A quasi-compactness is a class of spaces which is multiplicative, hereditary with respect to closed subspaces and contains an infinite T0-space.
The concept of a compact space was introduced by L. Vietoris [117], P.S.
Alexandroff and P.S. Urysohn [2] and is due to the works of E. Borel, H.
Lebesgue, K. Kuratowski, W. Serpinski, S. Saks (see [34, 44, 90]).
The general notion of compactness is due to the works of P. S. Alexandroff and P. S. Urysohn [2], E. Hewitt [64], R. Arens and J. Dugundji [7], L. Nachbin [83], S. Mrowka and R. Engelking [43,81], H. Herrlich [62], H. Herrlich and J.
Vander Slot [63], M. Huˇsek and J. de Vries [67], Z. Frolik [51], R. N. Bhanmik and D. N. Misra [19], G. Viglino [118], A. P. Shostak [140] (see [44]).
For every spaceE there exists the minimal quasi-compactness P such that E ∈P (see [43, 44, 81]).
Theory of compactifications is a wide and vast branch of topology and its applications.
One-point compactification of the plane was studied by G. Riemann and compactifications of open subsets of the plane were studied by C. Caratheodory in connection with some problems of analytic functions. The notion of the ex- tension was used by R. Dedekind and G. Cantor in the theory of real numbers and by F. Hausdorff in the theory of metric spaces (see [30, 34, 44, 90, 121]).
Let P be a quasi-compactness. A generalizedP-extension of a space X is a pair (eX,f), whereeX ∈P,f :X→eX is a continuous mappings and the
23
setf(X) is dense ineX. Iff is an embedding, theneX is called aP-extension or a P-compactification of the spaceX.
The general problems of the theory ofP-extensions are the following.
First General Problem: To find the methods to construct and study the P-extensions and specialP-extensions of a given spaceX.
Second General Problem: To study the class GE(X) of all generalized P- extensions of a given space X.
Third General Problem. Under which conditions the classGE(X) is a com- plete lattice?
Fourth General Problem. Let GE(X) be a lattice and letβPX be the maxi- mal element inGE(X). To study the properties of spacesβPX and βPX\X.
Fifth General Problem. Let X and Y be spaces. Under which conditions there exists aP-extensioneX ofXsuch thatY andeX\Xare homeomorphic?
Various important problems of the theory of extensions were formulated in [3, 12, 17, 34, 49, 59, 90, 103, 119, 121, 129].
The purpose of the present paper is to investigate the class ofP-extensions of topological spaces and the methods of constructing of new P-extensions of topological spaces.
In Section 1 we discuss the general notions and problems. We introduce the notion of double compactness. In the final part of the section we give examples and concrete problems of the theory of extensions.
Section 2 is devoted to investigation of the methods of construction of ex- tensions.
The method of perfect mappings was used by M. C. Raybom [89] in the constructions of Hausdorff compactifications for locally compact spaces. We introduce the method of superperfect mappings for arbitrary spaces. These methods are used for investigation of the lattice of compactifications (see [1, 5, 27, 32, 55, 58, 68, 71,
72, 74, 76, 82, 95, 106, 114, 116, 119, 124]).
The method of singular mappings was introduced in [32] for construction of the Hausdorff compactifications of locally compact spaces.
The Wallman-Shanin method was introduced by W. H. Wallman [122] and N. A. Shanin [96, 97, 98, 99]. The notion of the base-ring was introduced by O. Frink [50], E. F. Steiner [106, 109], V. I. Zaitsev [128]. In [50] O. Frink formulated the problem: Is every Hausdorff compactification of a completely regular space of the Wallman-Shanin type? The problem of O. Frink was studied by many authors (see [49, 55, 79, 85, 90, 105, 106, 109, 113]) and it was negatively solved by V. M. Uljanov [115].
The spectrum of rings (see [15, 29, 52, 53, 58, 65, 66, 84, 90, 110, 111, 119, 124]) was used by L. I. Calmutskii [24, 28, 131, 132, 133] to introduce the notion of spectral compactifications.
In Section 3 we study the uniform extensions of completely regular spaces.
The construction of maximal uniform extensionµX of a spaceX is due to J.
Diendonn´e and to F. Hausdorff [44]. The concept of a uniform space and the notion of a complete uniform space were introduced by A. Weil (see [44]). The completions of separable metric spaces were studied by J. M. Aarts and P. V.
van Emde Boas [1]. The completions of arbitrary metric spaces were studied by V. K. Bel’nov [20,127]. An important part of the methods of construction of extensions of a space is to present the “new points” of the extension as a space with concrete properties. We simplify and extend the “Bel’nov’s gluing method” to theory of uniform completions of arbitrary completely regular spaces.
In this article we shall use the following notation:
We denote by clXAor clAthe closure of a setA in a spaceX.
We denote by |A|the cardinality of a set A.
We denote by w(X) the weight of a space X.
The interval [0, 1] is denoted by I.
On the setN ={1, 2, . . .}we consider only the discrete topology.
We use the terminology from [44,34,90].
General notions and problems
Let L be a partially ordered set. Fix a non-empty subset A of L. We consider that a=∨A ifa ≥x for everyx ∈ A and if b≥ x for each x ∈ A, thenb≥a. We consider thatc=∧Aifc≤x for everyx∈Aand if b≤xfor each x∈A, thenb≤c.
The setL is called:
- an upper semi-lattice if there exists the element ∨L and for every two elements x, y∈L there exists the elementx∨y=∨{x, y};
- a lower semi-lattice if there exists the element∧L and for every two ele- ments x, y∈Lthere exists the element x∧y=∧{x, y};
- a complete upper semi-lattice if for every non-empty subsetA ⊆Lthere exists the element ∨A;
- a lattice if Lis an upper semi-lattice and a lower semi-lattice;
- a complete lattice if Lis a lower semi-lattice and a complete upper semi- lattice.
We mention that in the complete latticeLfor every non-empty subsetA⊆L there exists the element ∧A.
LetLbe a complete upper semi-lattice andM be a non-empty subset ofL.
If for every two elements x, y ∈ M we have x∨y ∈M, then M is called an upper subsemi-lattice of L. In the similar way there are defined the notions of a lower subsemi-lattice and of a sublattice.
1.1. Extensions of spaces
1.1.1. Definition. A g-extension of a space X is called a pair (Y, f), where Y is a non-empty T0-space, f : X → Y is a continuous mapping and {clYf(A) :A⊆X} is a closed base of the spaceY.
1.1.2. Definition. Ag-extension (Y,f) of a spaceXis called an extension of X iff is an embedding of X inY.
1.1.3. Remark. If (Y,f) is ag-extension of a spaceX, then the setf(X) is dense inY.
Denote by E(X) the family of all extensions of a space X and by GE(X) the family of all g-extensions of the space X. The family GE(X) is partially ordered in the standard way: (Y1, f1) ≤(Y2, f2) if there exists a continuous mapping ϕ : Y2 → Y1 such that f1(x) = ϕ(f2(x)) for every x ∈ X, i. e.
f1 =ϕ◦f2.
If (Y1, f1), (Y2, f2)∈GE(X), ϕ:Y2→Y1 andψ:Y1 →Y2 are continuous mappings, f1 = ϕ◦f2 and f2 = ψ ◦f1, then ψ = ϕ−1 and ϕ and ψ are homeomorphisms. Thus (Y1, f1) = (Y2, f2) provided (Y1, f1) ≤ (Y2, f2) and (Y2, f2)≤(Y1, f1).
If i∈ {0,1,2,3,312}, then GE(X) = {(Y, f) ∈GE(X) : Y is a Ti-space} and Ei(X) =E(X)∩GEi(X) ={(Y, f)∈E(X) :Y is aTi-space}.
1.1.4. Proposition. Let f : X → Y be a continuous mapping of a space X into a Ti-space Y, the set f(X) is dense in Y and i ≥ 3. Then (Y, f)∈GEi(X).
Proof. LetF be a closed non-empty subset ofY andy∈Y\F. There exist two open subsetsU and V of Y such thatF ⊆U,y∈V and U∩V =Ø. We put Φ =clY(f(X)∩U). Then F ⊆Φ and y /∈Φ. Hence{clYA :A⊆f(X)} is a closed base of the spaceY. The proof is complete.
1.1.6. Definition. A pair (Y, f) is called a weak g-extension (wg- extension) of a spaceX iff :X→Y is a continuous mapping,Y is aT0-space and the set f(X) is dense inY.
We denote byWGE(X) the family of all wg-extensions of a spaceX, W E(X) ={(Y, f)∈W GE(X) :f is an embedding},
W GEi(X) ={(Y, f)∈W GE(X) :Y is aTi-space} and W GEi(X) =W E(X)∩W GEi(X).
1.1.7. Proposition. Let X be a non-empty T0-space. Then:
1. WE(X) is not a set.
2. WGE(E) is not a set.
3. Ifi≥2, thenW GEi(X) is a set.
4. GE(X)is a set.
Proof. Let Z be a non-empty T0-space and Z ∩X =Ø. We put Y = X ∪Z, f(x) = x for every x ∈ X,- Im0 = {H ⊆ X : H is open in X} ∪ {X∪V :V ⊆Z and V is open in Z}. Then Im0 is a T0-topology on Y and
(Y, f) ∈W E(X). Thus W E(X) is not a set. Hence W GE(X) is not a set, too.
Let m=|X|and τ = 22τ. If (Y, f) ∈GE(X)∪W GE2(X), then |Y| ≤τ. ThereforeW GE2(X)∪GE(X) is a set. The proof is complete.
1.1.8. Proposition. LetX be an infiniteT1-space. ThenW E1(X) is not a set. In particular,W E1(X) is not a set.
Proof. Let Z be a non-empty T1-space and Z ∩X = Ø. We put Y = X∪Z, f(x) =x for everyx∈X and Γ1 ={H ⊆X:H is open inX} ∪ {V ⊆ Y :V ∩Z is open in Z and the setX\V is finite}. Then Γ1 is a T1-topology on Y, Γ0 ⊆Γ1 and ((Y, Γ1), f)∈W E(X). The proof is complete.
1.1.9. Remark. If in the proof of Proposition 1.1.7 or of Proposition 1.1.8 the space Z is compact, then the space (Y,Γ0) or (Y,Γ1) is compact, too.
If (Y1, f1), (Y2, f2) ∈ W GE(X), then (Y1, f1) ≤ (Y2, f2) if there exists a continuous mappingϕ:Y2 →Y1 such thatf1 =ϕ◦f2.
1.1.10. Proposition. The relation≤ is an ordering onW GE2(X).
Proof. Is obvious.
1.1.11. Example. LetXbe a non-empty space. Then≤is not an ordering on WE(X).
LetZ be a non-emptyT0-space, Z∩X= Ø and b∈Z. Consider the space Y1 =Z∪X with the topology Γ0 ={U ⊆X :U is open in X}∪{V ∪X :V is open inZ}and subspace Y2 ={b} ∪X ofY1. Letf(x) =x for eachx∈X.
Then (Y1, f), (Y2, f) ∈ W E(X). We put ϕ(y) = y for every y ∈ Y2, f = ψ|X and ψ(y) = b for every y ∈ Z. Then the mappings ϕ : Y2 → Y1 and ψ : Y1 → Y2 are continuous and ϕ(x) = ψ(x) = x for each x ∈ X . Thus (Y1, f)≤ (Y2, f), (Y2, f)≤(Y, f) and (Y, f)6= (Y2, f) provided|Z| ≥2.
1.1.12. Example. Let X be an infinite T1-space. Then ≤ is not an ordering onW E1(X).
Let Z be a T1-space, |Z| ≥ 2, b ∈ Z, Y1 = Z ∪X be a space with the topology Γ1 ={U ⊆X :U is open in X} ∪ {V ⊆Y1:V ∩Z is open in Z and the setX\V infinite},Y2={b} ∪X be a subspace ofY1 andf(x) =xfor each x ∈X. Then (Y1, f), (Y2, f) ∈W E1(X), (Y1, f)≤(Y2, f),(Y2, f) ≤(Y1, f) and (Y1, f)6= (Y2, f).
LetXbe a space. On the classWGE(X) we consider the relation∼:(Y1, f1)∼ (Y2, f2) iff (Y1, f1) ≤(Y2, f2) and (Y2, f2) ≤ (Y1, f1). Obviously, ∼ is a re- lation of equivalence. Denote by W GE0(X) the classes of equivalence on WGE(X) and by W E0(X) the classes of equivalence on WE(X).
Obviously ≤is an ordering on the a class W GE0(X).
1.1.13. Proposition. LetH ={(Yα, fα)∈ W GE(X) : α ∈A} be a set, f(x) = (fα(x) :α ∈A) for every x∈X and Y be the closure of the set f(X) in the space Π{Yα:α∈A}. Then:
1. (Y, f)∈W GE(X) and we put (Y, f) =∨H.
2. (Yα, fα)≤(Y, f) for each α∈A.
3. If (Z, g) ∈ W GE(X) and (Yα, fα) ≤ (Z, g) for each α ∈ A, then (Y, f)≤(Z, g).
4. Ifi∈ {0,1,2,3,312} andYα is aTi-space, then Y is a Ti-space.
5. IfH∩W E(X)6=Ø, then (Y, f)∈W E(X).
Proof. For every β ∈ A we consider the projection ϕβ :Y → Yβ, where ϕβ(yα : α ∈ A) = yβ for any (yα : α ∈ A) ∈ Y. Then fα = ϕα ◦f for each α ∈A. The assertions 1, 2, 4 and 5 are proved. Let (Z, g) ∈W GE(X) and (Yα, fα) ≤ (Z, g) for each α ∈ A. For any α ∈ A we fix a continuous mappingψα :Z →Yα such that fα=ψα◦g. Consider the mappingψ:Z → Π{Yα :α∈A}, whereψ(z) = (ψα(z) :α∈A). The mapping ψis continuous, ψ(g(X)) =f(X) andψ(Z)⊆Y. The assertion 3 and Proposition are proved.
1.1.14. Question. Is it true that W E0(X) is a set for each topological space X?
Obviously, W E0(X) is a set for every space X iff W GE0(X) is a set for every space X.
1.1.15. Remark. Let X be a non-empty space, D0 be a singleton space, fm : X → D0 be the unique mapping of X into D0, fM(x) = x for each x ∈X. Then (X, fM) is the maximal element in WGE(X) and (D0, fM) is the minimal element inWGE(X). Obviously, (X, fM), (D0, fM)∈GE(X).
1.1.16. Question. LetX be a space andH be a non-empty subset of the set GE(X). Is it true that∨H∈GE(X)?
1.1.17. Corollary. Leti≥2. Then W GEi(X) is a complete lattice.
1.1.18. Corollary. Let i≥2. Then W Ei(X) is a complete upper semi- lattice.
1.1.19. Corollary. Leti≥3. Then GEi(X) is a complete lattice.
1.1.20. Corollary. Let i ≥ 3. Then Ei(X) is a complete upper semi- lattice.
1.2. The canonical functor m:W GE(X)→GE(X)
Consider a topological spaceX. Fix awg-extension (Y,f) of the spaceX.
Let ΓY be the topology of the space Y. OnY consider a new topology ΓY f generated by the closed base {clYH :H ⊆f(X)}. There exist a set Yf and a mappingPY :Y → Yf such that PY−1(PY(H)) =H for everyH ∈ΓY f and Γ0Y f ={PY(H) :H ∈ΓY f} is aT0-topology on a set Yf.
If y ∈ Y, thenPY−1(PY(y)) = (∩{Y ∈ ΓY f : y ∈ U})∩(∩{Y \U : U ∈ ΓY f, y /∈U}). Consider the mapping P f :X → Yf, where P f =PY ◦f. By construction, (Yf, P f) ∈ GE(X). We put (Yf, P f) = m(Y, f), Yf = m(Y) and Pf=m(f).
The canonical functorm:W GE(X)→GE(X) is constructed.
From the construction it follows.
1.2.1. Proposition. If (Y, f)∈GE(X), thenm(Y,f)=(Y,f).
1.2.2. Question. Is it true that the functorm is covariant?
1.3. The canonical functors mi :W GE(X)→W GEi(X)
Fix i ∈ {0,1,2,3,312}. For every space Y there exist a unique Ti-space Y /i and a unique projectioniY :Y →Y /iwith the properties:
1. iY is a continuous mapping ontoY /i;
2. for every continuous mapping ϕ:Y →Z in a Ti-space Z there exists a unique continuous mapping ¯ϕ:Y /i→Z such that ϕ= ¯ϕ◦iY.
3. if ψ : Y → Z is a continuous mapping, then there exists a unique continuous mapping ¯ψ:Y /i→Z/isuch that ¯ψ◦iY =iZ◦ψ.
The spaceY /i with the projectioniY is called thei-replic of the spaceY.
Fix a space X. If (Y, f) ∈ W GE(X) , then we put fi = iY ◦ f and mi(Y, f) = (Y /i, fi). From the construction it follows.
1.3.1. Proposition. mi :W GE(X) →W GEi(X) is a covariant functor.
If (Y, f) ≤ (Z, g), then mi(Y, f) ≤ mi(Z, g). If (Y, f) ∈ W GE(X), then mi(Y, f) = (Y, f).
1.4. Compactness
The notion of compactness is due to E. Mrowka [81,43], E. Hewit [64], R.
Arens and S. Dugundji [7].
A class P of topological T0-spaces is called a strongly compactness if the following conditions are fulfilled:
C1. the classP is non-empty;
C2. there exists a spaceX ∈P such that |X|>2;
C3. the classP is multiplicative, i. e. if{Xα ∈P :α∈A} is a non-empty set of spaces from P, then Π{Xα :α∈A} ∈P;
C4. the class P is closed hereditary, i. e. if Y is a closed subspace of a space X∈P, then Y ∈P;
C5. if Y is a dense subspace of a spaceX ∈ P, then {clXA :A⊆Y} is a closed base of the spaceX.
A class of spacesP with propertiesC1−C4 is called a quasi-compactness.
A quasi-compactness P of Hausdorff spaces is called a compactness.
Fix a quasi-compactness P. For every space X we put W P GE(X) = {(Y, f)∈W GE(X) :Y ∈P},W P E(X) =W P GE(X)∩W E(X),P GE(X) = W P GE(X)∩GE(X) and P E(X) =P GE(X)∩E(X).
If P is a compactness, then W P GE(X) = P GE(X) and W P E(X) = P E(X). From Proposition 1.1.7. it follows that PGE(X) and PE(X) are the sets for each space X.
1.4.1. Theorem. Let P be a compactness and X be a space. Then
∨H∈W P GE(X) for every non-empty set H⊆W P GE(X).
Proof. Follows immediately from the conditions C3, C4 and properties of Hausdorff spaces.
1.4.2. Corollary. LetP be a compactness. ThenWPGE(X) is a complete lattice for every spaceX.
Denote by (βPX, βP) the maximal element of the latticeWPGE(X), where P is a compactness.
1.4.3. Corollary. LetP be a compactness,X be a space andW P E(X)6=
∅. Then:
1. WPE(X) is a complete upper semi-lattice;
2. (βPX, βP)∈W P E(X).
1.4.4. Theorem. LetP be a compactness. For every continuous mapping f : X → Y of a space X into a space Y there exists a continuous mapping βPf :βPX → βPY such thatβP ◦f =βPf◦βP. If every spaceZ ∈ P is a T2-space, then the mappingβPf is unique.
Proof. We may consider that f(X) is dense inY. Then g=βP ◦f :X→ βPY is a continuous mapping, the set g(X) is dense in βPY and (βPY, g) ∈ W P GE(X). Thus there exists a continuous mappingβPf :βPX→βPY such that g=βP ◦f◦βP. The proof is complete.
1.4.5a. Corollary. LetP be a compactness andf :X →Y be a contin- uous mapping of a space X into a space Y ∈ P. Then Y =βPY and there exists a unique continuous mappingβPf :βPX→Y such thatf =βPf◦βP. 1.4.5b. Remark. Let P be a quasi-compactness. Then there exists βPX∈W P GE(X) such that βPX ∈ ∨P GE(X).
1.4.6. Proposition. Let P be a strongly compactness. Then every space x∈P is a Hausdorff space, i. e. P is a compactness.
Proof. Let d(X) = min{|H| : H ⊆ X, clXH = H} be the density of a spaceX. Consider the spaceF={0,1}with the topology Im ={∅,{1},{1,0}}. Suppose thatX∈P andXis not aT1-space. Then the spaceF is embeddable in X. Suppose that F ⊆ X. Denote by b, c the cardinality larger than 2c (see Proposition 1.1.7). For some cardinal m the space Y is embeddable in Fm ⊆Xm([44],Theorem 2.3.26). LetZ be the closure of Y inXm. Then the spaceZ is separable and|Z| ≥ |Y|>2c. IfS∈P, then |S| ≤exp(exp(d(S))).
Thus |S| ≤2c for every separable spaceS ∈P. Therefore every spaceS ∈P is aT1-space.
Suppose that X ∈ P and X is not a T2-space. There exist two distinct points a, b∈X such that V ∩W 6=∅ provided V and W are open subsets of X, a∈ V and b ∈ W. Fix a cardinal number τ > exp(exp(|X|)). We put Φ ={a, b}. InXτ we consider the diagonal ∆(X) (see [44], p.110). LetY be the closure of the set ∆(X) inXm. Then Φτ ⊆Y,|∆(X)|=|X|,d(Y)≤ |X|,
|Y| ≤exp(exp(|X))< τ|and |Φτ|= 2τ = exp(τ), a contradiction. The proof is complete.
1.5. Double compactness
A class P of topological T0-spaces is called a double compactness if the following conditions are fulfiled:
D1. the classP is non-empty;
D2. there exists a spaceX∈Dsuch that|X| ≥2;
D3. if Γ is a topology of the space X ∈ P then there is determined the completely regular topology dΓ on X such that (X, dΓ) ∈ P, Γ ⊆ dΓ and ddΓ =dΓ;
D4. if f :X → Y is a continuous mapping of a space (X, Γ) into a space (Y, Γ′) and X, Y ∈ P, then f is a continuous mapping of the space (X, dΓ) into a the space (Y, dΓ′);
D5. if {(Xα,Γα) : α ∈A} is a non-empty set of spaces, (Xα,Γα) ∈P for eachα∈A, X = Π{Xα:α∈A}, Γ is the product of topologies Γαon Xand Γ′ is the product of topologies dΓα onX, then Γ′⊆dΓ;
D6. if (X, Γ)∈ P, Y ⊆ X and Y is a closed subset of the space (X, dΓ), then (Y, Γ|Y)∈P and d(Γ|Y)⊇dΓ|Y, where Γ|Y ={U ∩Y :U ∈Γ} for the topology Γ on X.
1.5.1. Proposition. LetP be a class of spaces,Xbe a space,{Yα :α∈A} be a non-empty family of subspaces of the space X,Y =∩{Yα :α∈A} and Yα∈P for each α∈A. Then:
1. ifP is a double compactness, then Y ∈P; 2. ifP is a compactness, then Y ∈P.
Proof. We may consider that X=Yα for someα∈A. If X is aT2-space, then Y is a closed subspace of the space Π{Yα : α ∈ A}. The assertion 2 is proved. If P is a double compactness, then Y is a closed subspace of the space Π{Yα:α∈A}in the topologydΓ. The assertion 1 and Proposition are proved.
Fix a double compactness P. For every space X we put P GE(X) = {(Y, f)∈W GE(X) :Y ∈P and f(X) is a dense subset of the space (Y, dΓ) and P E(X) =W E(X)∩P GE(X).
From the condition D6 it follows that PGE(X) andPE(X) are sets.
1.5.2. Theorem. Let P be a double compactness. Then PGE(X) is a complete lattice for every spaceX.
Proof. Let {(Yα, fα) :α∈A}be a non-empty subset of the set PGE(X).
Denote by Γα the topology of the space Yα and by Γ the topology of the space Π{Yα :α ∈A}. Consider the mapping f :X →Π{Yα :α ∈A}, where f(x) = (fα(x) :α∈A) for eachx∈X. LetY be the closure of the setf(X) in the space (Π{Yα :α∈A}, dΓ). Then (Y, f) ≥(Y2, f2) for each α∈A. From the condition D4 it follows that if (Z, g)∈P GE(X) and (Z, g)≥(Yα, fα) for each α ∈A, then (Z, g) ≥(Y, f). Thus (Y, f) =∨{(Yα, fα) :α ∈A}. The proof is complete.
1.5.3. Corollary. LetP be a double compactness, let X be a space and P E(X)6=∅. Then PE(X) is a complete upper semi-lattice.
1.5.4. Theorem. LetP be a double compactness. Then:
1. for every continuous mapping f : X → Y of a space X into a space Y there exists a unique continuous mapping βPf : βPX → βPY, βP ◦f = βPf◦βP;
2. for every continuous mapping f : X → Y of a space X into a space Y ∈P there exists a unique continuous mapping βPf :βPX →Y such that f =βPf ◦βΓ.
Proof. Let Z be the closure of the set βP(f(X)) in the space (βPY, dΓ).
Then (Z, βP ◦f) ∈ P GE(X) and the assertion 1 is proved. If Y ∈ P, then βPY =Y. The proof is complete.
1.6. Examples
1.6.1. Example. Let C be the class of compact Haussdorff spaces. Then C is a strongly compactness. If (Y, f) ∈ CGE(X), then we say that (Y, f) is a g-compactification of X. If (Y, f) ∈ CE(X), then (Y, f) is called a compactification of X. For every spaceX theg-compactificationβX =βPX is the Stone- ˇCechg-compactification ofX. IfX is a completely regular space, thenβX is the Stone- ˇCech compactification of X.
1.6.2. Example. LetC0 be the class of zero-dimensional compact spaces.
ThenC0is a strongly compactness. IfindX >0, thenC0E(X)=Ø. IfindX=0, thenmf X =βC0X is the Morita-Freudenthal compactification of X. We put mf X =βC0X and (mf X, mf) = (βG0X, βG0). Theg-compactificationmfX is called the maximal zero-dimensional g-compactification of the space X.
1.6.3. Example. Let X be a completely regular space. A subset L of X is called bounded in X if the setf(L) is bounded in the space of realsR for every continuous function f :X →R. A space X is called µ-complete if the closureclL of every bounded subsetL is compact. LetCµbe the class of all µ-complete spaces. Then Cµ is a strongly compactness. The g-extension (βGµX, βCµ) = (µ∗X, µ∗) is called the maximalµ-completion of the spaceX.
If X is a completely regular space, then (µ∗X, µ∗) ∈ E(X) and µ∗X is the µ-completion of X.
1.6.4. Example. Let R be the space of reals. A space Z is called a realcompact space if it is homeomorphic to a closed subspace of some space RA. The class R of all realcompact spaces is a strongly compactness. Theg- extension (νX, ν) = (βRX, βR) is the maximal g-realcompactification of the space X. If X is a completely regular space, then νX is the realcompacti- fication of X and (νX, ν) ∈ E(X). Every realcompact space is µ-complete.
ThereforeνX ≤µ∗X and µ∗X⊆νX.
1.6.5. Example. LetU be the class of all complete uniform spaces. IfX is a completely regular space, then byUX we denote the universal uniformity on X (see [44]). Every uniform space is considered and a topological space too. Thus for every space X in UGE(X) the maximal element (µX, µ) is determined, where µX is a complete uniform space, µ:X→ µX is a contin-
uous mapping and the set µ(X) is dense in µX. The space µX is called the Diendonn´e completion of the space X. If X is completely regular, then µX is the completion of the uniform space (X, UX). IfX =µX, then the space X is called a Deudonn´e complete space. Every Deudonn´e complete space is µ-complete. For every spaceXwe may consider thatµ∗X ⊆µX ⊆νX⊆βX.
1.6.6. Example. LetPbe a compactness such that every space (Y,Γ)∈P be completely regular. For every space (X, Γ)∈P we put dΓ = Γ. Then P is a double compactness. Therefore every compactness of completely regular spaces may be considered as a double compactness.
1.6.7. Example. For every space (X, Γ) we put cΓ = {U ∈ Γ : U is a compact subset} and dΓ is the topology generated by the open base {U1∩U2∩ ... ∩Un:n∈N, U1, U2, ..., Un∈Γ} ∪ {X\U :U ∈cΓ}.
A space (X, Γ) is called a spectral space ifcΓ is an open base of the spaceX, U∩V ∈cΓ is an open base of the spaceX,U∩V ∈cΓ for all U, V ∈cΓ and (X, dΓ) is a compact Hausdorff space. LetS be the class of all spectral spaces.
Then S is a double compactness. For every T0-space X we haveSE(X)6=Ø, i. e. βS:X→βSX is an embedding.
1.6.8. Proposition. If (X, Γ) is a spectral space, then:
1. (X, Γ) is a compact T0-space;
2. (X, dΓ) is a zero-dimensional compact space;
3. dΓ = Γ if (X, Γ) is aT1-space.
Proof. Is obvious (see [132]).
1.6.9. Remark. The class of spectral compactifications of a space X was studied in [24,28,131,132,133].
1.6.10. Example. LetE=[0,1],F ={2−n:n∈N}and Im be the topology generated by the base {{t ∈ E :a < t < b}: a, b are real numbers}∪{Vn = {t ∈ E : t < 2−n} \F : n ∈ N}(see [2] or [44], Example 1.5.7). Then E is a T2-space and E is not regular. If X is the subspace of irrational numbers of E or X =E\F, then {clEH :H ⊆ X} is not a closed base of E. Thus E /∈ P for every compactness P. There exists a minimal quasi-compactness P of Hausdorff spaces such that E∈P. ThereforeP is a compactness andP is not a strongly compactness.
1.6.11. Example. Let P be the class of all compact T0-spaces. Then P is a quasi-compactness and P is not a compactness. ObviouslyOP E(X)6=∅ for everyT0-spaceX.
1.6.12. Example. Let P be the class of all compact T1-spaces. Then P is a quasi-compactness andP is not a compactness. It is well–known that ωX ∈P E(X) for everyT1-space X.
1.7. Problems
1.7.1. Problem. LetP be a compactness or a double compactness.
1. Under which conditions the lattices PGE(X) and PGE(Y) are isomor- phic?
2. Under which conditions the upper semi-lattices PE(X) andPE(Y) are isomorphic?
3. Which topological properties of a spaceX are characterized in terms of the objectsPGE(X) and PE(X)?
4. Which properties of the lattice PGE(X) are characterized by the prop- erties of the spaceX?
5. Let X be a space and PE(X)6=Ø. Which properties of the upper semi- latticePE(X) are characterized be the properties of the space X?
The program of matching “interesting” topological properties of a com- pletely regular space X with “interesting” properties of the complete upper semi-lattice PE(X) is very important in the theory of extensions. N. Boboc and G. Siretchi [22] has proved that CE(X) is a lattice iff the space X is locally compact. In [76] K. D. Magil has proved that for two locally compact spaces X and Y the semi-lattices CE(X) and CE(Y) are isomorphic iff the spacesβX\X and βY \Y are homeomorphic.
Another program of investigation is to find the “interesting” compactness and double compactness.
1.7.2. Problem. LetP be a compactness or a double compactness, letX be a space and P E(X)6=Ø.
1. Find the methods of constructions the extension βPX, some extensions from PE(X) or all extensions PE(X).
2. Let Z be a space. Under which conditions there exists an extension (Y, f)∈P E(X) such thatY \f(X) andZ are homeomorphic?
3. Under which conditions there exists an extension (Y, f)∈P E(X) such that dim(Y \f(X))≥m, wherem∈N?
4. Let Z ∈ P. Under which conditions there exist an extension (Y, f) ∈ P E(X) and a closed subspace Z′ ⊆Y \f(X) such thatZ and Z′ are homeo- morphic?
2. Some methods of construction of extensions A mappingf :X →Y of a spaceX into a spaceY is called:
- a perfect mapping if f(X) = Y, f is continuous, closed and the fibers f−1(y), y∈Y, are compact;
- a superperfect mapping if f(X) = Y, f is continuous, perfect and there exists a compact set Φ⊆X such thatf−1(f(x)) ={x}for each x∈X\Φ;
- a singular mapping iff is continuous and the setf−1(V) is non-compact for any non-empty open subset V of Y;
- an almost perfect mapping iff(X) =Y, f is continuous, closed and there exists a closed compact set Φ ⊆X such that f−1(f(x)) = {x} for each x ∈ X\Φ.
2.1. Method of perfect mappings
LetX be an open dense subspace of a space eX,Y=eX\X and h:Y →Z be a perfect mapping onto a space Z. We put ehX =Z∪X and consider the mapping f :eX →ehX, where f(x) =x for every x ∈X and h =f|Y. On a space ehX we consider the quotient topology{W ⊆ehX :f−1(W) is open ineX}.
2.1.1. Property. The mapping f is perfect.
Proof. By construction, the mapping f is continuous and the fibers f−1(y), y ∈ ehX, are compact. Let F be a closed subset of the space eX.
ThenF1 =h−1(h(F∩Y)) is a closed subset ofY, Φ =F1∪F is closed subset ofeX and Φ =f−1(f(F)). Thusf(F) is closed inehX. The proof is complete.
2.1.2. Property. If i∈ {1,2,3,4} and eX is a Ti-space, then ehX is a Ti-space. Moreover, ifeX is a normal space, then ehX is a normal space.
Proof. The property to be aTi-space, i∈ {1,2,3,4}, is preserved by the perfect mappings.
2.1.3. Property. If eX andZ areT0-spaces, then ehX is aT0-space.
Proof. Obvious.
2.1.4. Property. X is an open dense subspace of the spaceehX.
Proof. Obvious.
We putLC(X) =∪{U :U is an open subset ofX andclXU is compact} – the set of locally compactness of a spaceX. LetRC(X)=X\LC(X). A space X is almost locally compact if the set LC(X) is dense in X. If RC(X)=Ø, then the spaceX is locally compact.
2.1.5. Theorem. Let eX be an extension of the almost locally compact spaceX, the setLC(X) is open in eX,Y=eX\LC(X),h:Y →Z is a perfect mapping onto a space Z and h−1(h(x)) = {x} for every x ∈ RC(X). Then there exist an extension ehX of a spaceX and a perfect mapping f :eX → ehX such that the setLC(X) is open in ehX.
Proof. Let X1 =LC(X). Then eX is an extension of the space X1 and the set X1 is open ineX. Properties 2.1.1 – 2.1.4 complete the proof.
2.2. Method of superperfect mappings
2.2.1. Theorem. If f : X → Y is an almost perfect mapping onto a T1-spaceY, thenf is superperfect.
Proof. There exists a closed compact subset Φ⊆Xsuch thatf−1(f(x)) = {x}for everyx∈X\Φ. LetF =f(Φ). Ify∈Y\F, thenf−1(y) is a singleton.
If y ∈F, then f−1(y) is a compact set as a closed subset of the subspace Φ.
Thus the fibers f−1(y) are compact. The proof is complete.
We say that a subset H of a space X is compact in X if the set clXH is compact.
A setN(f) ={x∈X :f−1(f(x))6={x}}is called the kernel of a mapping f :X →Y.
A mappingf :X→Y is almost perfect ifff(X) =Y,fis closed, continuous and the kernelN(f) is compact in X.
2.2.2. Theorem. Let X be a subspace of a space X1, Y = X1 \X, h:Y →Z be an almost perfect mapping onto a space Z and the set clYN(h) is closed inX. Then there exist a unique spaceSand an unique almost perfect mappingf :X1 →S such that N(f) =N(h) andh=f|Y.
Proof. We put S =X∪Z, Y1 = clYN(h), X2 =X1\clYN(h), f(x) = h(x) for every x∈Y and f(x) =xfor every x∈X. The space X2 is open in X1andg=h|Y1 :Y1 →Z1 =h(Y1) is a continuous closed mapping. OnSwe consider the quotient topology. Obviously, N(f) =N(h). By construction, f is a closed continuous mapping. The proof is complete.
2.2.3. Corollary. Let eX be an extension of a space X, Y=eX\X, h : Y →Z be an almost perfect mapping onto a spaceZ and the setclYN(f) be closed ineX. Then there exist an extensionehX of the spaceXand an almost perfect mapping f :eX →ehX such that:
1. Z is a subspace of the spaceehX and Z =ehX\X;
2. h=f|Y; 3. N(f) =N(h).
2.3. Method of singular mappings LetP be a quasi-compactness.
A spaceX is called locallyP-compact if for every pointx∈X there exists an open subset U ⊆X such thatx∈U and clXU ∈P.
We say that a mappingf :X→Y is aP-singular mapping if f is continuous and clXf−1(V)∈/ P for every non-empty open subset V ⊆X.
Consider that the compactness P fulfills the following conditions:
S1. If Y and Z are closed subspaces of a space X and Y, Z ∈ P, then Y ∪Z ∈P.
S2. If Y is a closed subspace of the space X, Y ∈ P, Z ∈ P provided Z ⊆ X \Y is a closed subset of X and X\Y=∪{V : V is open in X and clXV ⊆X\V}, thenX∈P.
In the class of regular spaces Condition S1 follows from Condition S2.
2.3.1. Construction. Let f : X → Y be a P-singular mapping of a locally P-spaceX into a compact space Y ∈P. Obviously that the setf(X) is dense in Y. We puteX =X∪Y, with the topology generated by the open base {U ⊆X :U is open in X} ∪ {V ∪(f−1(V)\U) :V is open in Y, U is open in X and clX ∈U P}.
Property 1. eX ∈P.
By construction, Y ∈ P and X =eX \Y = ∪{U ⊆ X :U is open in eX and clXU ∈P}. If U is open in X and clXU, then cleXU =clXU. LetZ be a closed subspace of eX and Z∩Y =Ø. For every pointy∈Y there exist an open subsetVy of Y and an open subsetUy ofX such thatclXUy ∈P, y∈Vy
and Z ∩(f−1V \clXUy) =Ø. Since Y is compact, there exists a finite set F such that Y = ∪{Vy : y ∈ F}. Then Z ⊆ ∪{clXUy : y ∈ F}. By virtue of Condition S1,∪{clXUy :y∈F} ∈P and Z ∈P. Condition S2 completes the proof.
Property 2. X is an open dense subspace of the spaceeX.
Obviously,X is open ineX. Lety∈Y,V be an open subset ofY,U be an open subset ofX,y ∈V andclXU ∈P. Then the setW =V∪(f−1(V)\clXU) is open in eX and y ∈ W. Since clXf−1(V) ∈/ P, then W ∩X = f−1(V)\ clXU 6=Ø. Thus the setX is dense in eX.
Property 3. Let i ∈ {0,1,2} and X, Y be Ti-spaces. Then eX is a Ti-space.
Letx, y ∈eX and x6=y.
Case 1. x, y∈Y and i≤1.
If V is open in Y,x∈V and y /∈V, then W =V ∪f−1(V) is open in eX, x∈W andy /∈W.
Case 2. x, y∈Yandi= 2.
There exist two open subsetsV1 and V2 of Y such thatx∈V1, y∈V2 and V1∩V2=Ø. The sets Wi =Vi∪f−1(Vi) are open ineX, x∈W1, y ∈W2 and W1∩W2 =∅.
Case 3. x∈X and y∈Y.
There exists an open subset U of X such that x ∈ U and clXU ∈P. We put W=eX\clXU =Y ∪(f−1(Y)\clXU). The set W is open in eX,y ∈W and U∩W =∅.
Case 4. x, y∈X.
SinceX is an open subspace of the spaceeX andX is aTi-space, the proof is complete.
Property 4. If every closed subset Z of X is compact provided Z ∈ P, theneX is a compact space.
Proof. Obvious.
Property 5. Let ϕ :eX → Y be the mapping for which f = ϕ|X and ϕ(y) =y for all y∈Y. Thenϕis a continuous mapping.
Proof. If V is open inY, then ϕ(V ∪(f−1(V)\clU)) =V. The proof is complete.
Property 6. LetX be aT2-space and for every open subsetU of X with clXU ∈P there exists an open subsetW ofXsuch thatclXU ⊆W, clXW ∈P and clXW is a normal subspace of X. TheneX is a normal space.
Proof. Let F and Φ be two closed subsets of eX and F ∩Φ =∅. Case 1. F ⊆Y and Φ⊆Y.
There exists a continuous functionh:Y →[0,1] such thatF ⊆h−1(0) and Φ⊆h−1(1). We put g(x) =h(ϕ(x)) for everyx∈eX.
The function g:eX →[0,1] is continuous F ⊆g−1(0) and Φ⊆g−1(1).
Case 2. Φ∩Y =∅.
There exist the open subsets U and W of X such that Φ ⊆ clXU ⊆ W and clXW ∈ P. Then clXW is a normal subspace of X and the set clXW is closed in eX. There exists a continuous function h : X → [0,1] such that Φ⊆h−1(1) and (F∩X)∪(X\W)⊆h−1(0). We putg(y)=0 for everyy∈Y and g(x) =h(x) for everyx∈X. The function g:eX →[0,1] is continuous, F ⊆Y ⊆g−1(0) and Φ⊆g−1(1).
Case 3. F ⊆Y.
Let Φ1 =Y ∩Φ6=∅. There exists a continuous function g1 :eX →[−1,1]
such that F ⊆g1−1(1) and Φ1 ⊆g1−1(−1). The set U ={x ∈eX :g1(x) <0} is open in eX. We put g2(x) = sup{g1(x), 0}. The function g2 :eX →[0,1]
is continuous, F ⊆g2−1(1) and Φ1 ⊆U ⊆g−12 (0). The set Φ2= Φ\U is closed in eX and Φ2∩Y = ∅. There exists a continuous function g3 : eX → [0,1]
such that F ⊆g3−1(1) and Φ2 ⊆g3−1(0). Now we putg(x) =g3(x)·g2(x) for every x ∈ eX. The function g :eX → [0,1] is continuous, F ⊆ g−1(1) and Φ⊆g−1(0).
Case 4. F1=F∩Y 6=Ø and Φ1= Φ∩Y 6=∅.
There exists a continuous function g1 : eX → [0,2] such that Φ⊆ g1−1(0) and F1 ⊆ g1−1(2). The set U = {x ∈ eX : g1(x) > 1} is open in eX. Let F2 = F \U. The set F2 is closed in eX and F2 ∩Y = ∅. There exists a continuous function g2 :eX → [0,1] such that Φ⊆g2−1(0) and F2 ⊆g−1(1).
Now we put g(x) = min{1, g1(x) +g2(x)} for every x ∈ eX. The function g : eX → [0,1] is continuous, Φ ⊆ g−1(0) and F ⊆ g−1(1). The proof is complete.
2.3.2. Remark. In [31,32] the method of singular mappings was applied for the construction of Hausdorff compactifications of locally compact spaces.
2.4. Wallman-Shanin method
A family L of subsets of a space X is called an l-base on a spaceX ifL is a closed base andF ∪H, F ∩H∈Lfor all F, H ∈L.
LetLbe anl-base on the spaceX. AnL-filter in the spaceXis a non-empty familyξ of subsets of X which satisfies the following conditions:
F1. ξ⊆L and ∅∈/ ξ.
F2. IfF, H∈L,F ⊆H and F ∈ξ, thenH∈ξ.
F3. IfF, H∈ξ, thenF∩H ∈ξ.
A maximalL-filter is called anL-ultrafilter. A filterξis called a freeL-filter if∩ξ=∅.
A family L of subsets of the space X is called a net in the space X at a point x ∈X if for every neighbourhoodU of x there exists H ∈ Lsuch that x ∈H ⊆U. A family L of subsets ofX is a net in the spaceX ifL is a net of X at each point x∈X (see [9,10,11]).
For every pointx∈X we putξL(x) ={F ∈L:x∈F}.
2.4.1. Lemma. Let L be an l-base and x∈ X. The following assertions are equivalent:
1. L is a net of the spaceX at the point x;
2. ξL(x) is an L-ultrafilter.
Proof. Suppose that ξL(x) is an L-ultrafilter. If H ∈ L and x /∈H, then H /∈ ξL(x). Then H∩F = ∅ for some F ∈ ξL(x). Thus L is a net at the point x ∈ X. Consider that L is a net at the point x ∈ X, H ∈ L and H /∈ξL(x). Then there exists F ∈Lsuch thatx∈F ⊆X\H. ThusξL(x) is an L-ultrafilter. The proof is complete.
DenoteωLX={ξL(x) :x∈X} ∪ {ξ :ξ is a free L-ultrafilter}. We identify the pointx∈X with the filter ξL(x) and obtain X⊆ωLX. For every F ∈L we puthFi={ξ∈ωLX :F ∈ξ}. Let< L >={< F >:F ∈L}.
2.4.2. Lemma. For everyH, F ∈Lwe have< H∪F >=< H >∪< F >
and < H∩F >=< H >∩< F >
Proof. If H∪F ∈ ξ ∈ωLX, thenξ∩ {H, F} 6=∅. Thus < H∪F >=<
H > ∪ < F >. If H ∩F = ∅, then < H > ∩ < F >=< ∅ >= ∅. Let Φ =H∩F 6=∅. If Φ∈ξ ∈ωLX, then H, F ∈ξ and ξ ∈< H >∩< F >. If ξ ∈< H >∩< F >, then H, F ∈ξ and H∩ F ∈ξ. The proof is complete.
OnωLX we consider the topology generated by a closed base< L >.
We say that the extension Y of a space X is an end – T1-extension if the set {y} is closed in Y for every pointy ∈Y\X.
2.4.3. Theorem. IfLis an L-base of a space X, then:
1. ωLX is a compactification of the spaceX.
2. ωLX∈E(X).
3. ωLX is an end – T1-extension of X.
Proof. For everyF ∈Lwe have< F >∩X=F and< F > is the closure of F in ωLX. By construction, ωLX is a compact space. If ξ ∈ ωLX is an L-ultrafilter, then{ξ} is a closed subset ofωLX. The proof is complete.
2.4.4. Corollary. ωLX is aT1-space iff X is a T1-space andL is a net of the space X.
2.4.5. Definition. If L is the family of all closed subsets of a space X, thenωX =ωLX is called the Wallman compactification of the spaceX.
The compactification ωX is a T1-space iff X is a T1-space. The compact- ification ωX for a T1-space X was constructed by H. Wallman (see [122]).
The T1-compactifications of the typeωLX were constructed by N. A. Shanin [96,98].The general case was examined in [29,133].
A compactification bX of a space X is called the compactification of the Wallman-Shanin type if there exists anl-base ofX such that bX=ωLX.
In [18,100,115] it was proved that there exists a Hausdorff compactification bX of some discrete spaceXwhich is not of the Wallman-Shanin type.The pa- pers [15,18,24,29,49,50,70,79,85,100,109,113,132,133] contained sufficient con- ditions provided the compactification to be of the Wallman-Shanin type.
2.5. ωα–compactification Fix a space X and an l-base Lof X.
2.5.1. Definition. A compactificationbX of a spaceX is called an ωαL- compactification if there exists a continuous closed mapping f :ωLX → bX such thatf(x) =x for each x∈X.
If L is the family of all closed subsets X, then an ωαL-compactification is called an ωα-compactification. The ωα-compactifications of T1-space were introduced and examined by P. C. Osmatescu [87].
IfbX is anωαL-compactification of a spaceX, then the mappingf :ωLX→ bX is a natural projection if f is continuous closed and f(x) = x for every x∈X.
2.5.2. Proposition. LetbX be anωαL-compactification of a spaceXand f :ωLX→bX be the natural projection. Then:
1. f(ωLX) =bX;
2. f(ωLX\X) =bX\ X;
3. bX is an end-T1-extension of the spaceX;
4. f−1(x) ={x} for each x∈X;
5. bX∈E(X);
6. the natural projectionf :ωLX →bX is unique.
Proof. Let (Y1, f1) ∈ GE(X), (Y2, f2) ∈ W GE(X), ϕ : Y1 → Y2 be a closed mapping andf2 =ϕ◦f1. Then (Y2, f2)∈GE(X). Thus the assertion 5 is proved.
Since f is a closed mapping and the setf(ωLX) is dense inbX, thenbX= f(ωLX). The assertion 1 is proved.
Obviously,bX\X⊆f(ωLX\X).
If x ∈ ωLX \X, then the set {x} is closed in ωLX and the set {f(x)} is closed inbX. Therefore the assertion 3 is proved.
Letx ∈X,y ∈ωLX\X and f(y) =x. There exists an L-ultrafilter ξ such that y = ξ and y ∈ clωLXF for every F = ξ. Since f is continuous, then x ∈ clbXF for every F ∈ ξ. There exists H ∈ ξ such that x /∈ H. Then f(< H >) =clbXH and clbXH∩bX=H, a contradiction. The assertion 4 is proved.
Let f, g : ωLX → bX be two continuous mappings and f(x) = g(x) for all x ∈ X. Then f(< H >) = clbXH = g(< H >) for each H ∈ L, f(y) =
∩{clbXH : H ∈ L, y ∈< H >} and g(y) = ∩{clbXH : H ∈ L, y ∈< H >} for every y ∈ ωLX \X. Thus f(y)=g(y) for every y ∈ ωLX. The proof is complete.
2.5.3. Theorem. The set ΩL(X) of all ωαL-compactifications of the space X is a complete upper semi-lattice andωLX is the maximal element in ΩL(X).
Proof. Let {Yα : α ∈ A} be a non-empty subset of the set ΩL(X) and fα : ωLX → Yα be the natural projection of ωLX onto Yα. Consider the
mapping f : ωLX → Π{Yα : α ∈ A}, where f(y) = (fα(y) : α ∈ A) for every y ∈ ωLX. We put Y = f(ωLX). Then f is a continuous mapping, f|X is an embedding ofX intoY,Y is a compactification of X,f(x)=xfor each x ∈ X. For every α ∈ A there exists a projection gα :Y → Yα, where fα = gα ◦f. Since gα(A) = fα(f−1(A)) for each A ⊆Y, the mapping gα is closed. If F ⊆ ωLX, then f(F) = Y ∩Π{fα(F) : α ∈ A} and the mapping f is closed. Therefore Y = ∨{Yα : α ∈ A} and ΩL(X) is a complete upper semi-lattice. The proof is complete.
We put aLX = X∪ {a}, where a /∈ X and {U ⊆ X :U is open in X} ∪ {aLX\F :F ⊆ X, F is closed in ωLX} is the open base of the spaceaLX.
The mappingp:ωLX→aLX, wherep−1(a) =ωLX\Xandf(x) =xfor each x∈X, is continuous. ThusaLX ∈W E(X) and aLX is a compactification of X.
2.5.4. Theorem. The following assertions are equivalent:
1. ΩL(X) is a complete lattice;
2. aLX is a minimal element of the lattice ΩL(X);
3. the set X is open inωLX;
4. aLX is an end-T1-extension of X.
Proof. Let Y ∈ ΩL(X), y1, y2 ∈ Y\ X and y1 6= y2. We put Z = Y \ {y2}, ϕ(y) =y for everyy∈Z, ϕ(y2) =ϕ(y1) =y1 and onZ consider the quotient topology. Then ϕ:Y →Z is a closed mapping,Z is a compactifica- tion of X,Z ∈ΩL(X) and Z ≤Y. Thus the compactificationY ∈ΩL(X) is not a minimal element in ΩL(X) provided|Y \ X| ≥2.
Let Y be the minimal element in ΩL(X) and f :ωLX→ Y be the projec- tion. ThenY \X is a singleton,X is open inY,X =f−1(X) is open inωLX and Y =aLX.
If X is open in ωLX, then the mapping p : ωLX → aLX is closed. The proof is complete.
2.5.5. Theorem. LetX be a locally compact space and the l-base L be a net in the space X. Then:
1. X is an open subset of ωLX.
2. aLX is an ωαL-compactification of X.
3. aLX is the minimal element of the complete lattice ΩL(X).
Proof. For every point x ∈ X there exists an open subset Ux such that x ∈ Ux and the set Φx = clXUx is compact. Every filter ξ ∈ ωLX is an L- ultrafilter. IfF is a closed subset ofX, thenclωLXF =∩{< H >:H∈L, F ⊆ H}. Fix x ∈ X. There exists Hx ∈ L such that x /∈ Hx and X \Ux ⊆ Hx. Since Lis a net of X, there exists Fx ∈L such thatx∈Fx ⊆Ux∩(X\Hx).
Thus ξ(x) ∈/< Hx >. Therefore x ∈ ωLX\ < Hx >. If ξ ∈ ωLX\X, then there exists H ∈ ξ such that H∩Φx = ∅. Then H ⊆ X \Ux ⊆ Hx and ξ ∈< Hx >. Therefore Vx =ωLX\< Hx >is open in ωLX and x∈Vx ⊆X.
The assertion 1 is proved. The Theorem 2.5.4. completes the proof.
2.5.6. Example. Let X be a non compact T1-space. Denote by L1 the family of all closed subsets of X. Fixξ∈ωX\X. We put L={F ∪H:F ∈ ξ, H∈L1}. Then Lis anl-base ofX and|ωLX\X |= 1. Thus ΩL(X) is a complete lattice and a singleton set. In this case ωLX =aLX.
2.5.7. Corollary. The set Ω(X) of allωα-compactifications of the space X is a complete upper semi-lattice with the maximal elementωX.
2.5.8. Corollary. If the space X is locally compact, then Ω(X) is a complete lattice with the maximal elementωXand minimal elementaX, where aX =aLX for thel-base L of all closed subsets of X.
2.5.9. Corollary. For aT3-spaceXthe following assertions are equivalent:
1. X is locally compact;
2. Ω(X) is a complete lattice.
IfXis a complete regular space X, then we denote bySC(X) the family of all Hausdorff compactifications. In this case the Stone- ˇCech compactification βXis the maximal element inSC(X) andSC(X) is a complete subsemi-lattice of the upper semi-lattice Ω(X).
2.5.10. Corollary (N. Boboc and G. Siretchi [22]). For a complete regular space X the following assertions are equivalent:
1. X is locally compact;
2. SC(X) is a complete lattice and sublattice of Ω(X).
If is well–know that the Stone- ˇCech compactification βX of a completely regular space X is aωα-compactification [3] of the Wallman-Shanin type (see [3, 50, 86, 96, 109]).
From Theorem 2.1.5 it follows.
2.5.11. Corollary. Let X be an almost locally compact space, L be an l-base ofXandf :ωLX\LC(X)→Y be a continuous perfect mapping onto a T1-spaceX such that f−1(x) =xfor every x∈X\LC(X). Then there exists a unique ωαL-compactification bX of the space X such that bX \ LC(X) is homeomorphic toY.
2.5.12. Corollary. Let X be a locally compact space, L be an l-base of X and Y be a T1-space.
1. If there exists a closed mapping f : ωLX\X → Y onto Y, then there exists a uniqueωαL-compactificationbX ofX such that the remainderbX\X is homeomorphic to Y.
2. If there exists a closed mapping f : ωX \X → Y onto Y, then there exists an ωα-compactification bX of X such that the remainder bX \X is homeomorphic toY.
From Theorem 2.2.2 it follows.
2.5.13. Corollary. LetL be anl-base of a spaceX and Y be a T1-space.
1. Iff :ωLX\X→Y is an almost perfect mapping ontoYandclωLXN(f)⊆ ωLX\X, then there exists a uniqueωαL-compactificationbX ofX such that the remainder bX\X is homeomorphic to Y.
2. Iff :ωX\X→Y is an almost perfect mapping ontoYandclωLXN(f)⊆ ωLX\X, then there exists an ωα-compactification bX of X such that the remainderbX\X is homeomorphic to Y.
2.6. Spectral compactifications
LetSE(X) be the set of all spectral compactifications of a space X.
A mappingg:X →Y of a spaceX into a spaceY is a spectral mapping if g is continuous and a set g−1(U) is compact provided the set U is open and compact in Y.
If Y, Z ∈ SE(X), then we consider that Z ≤ Y if there exists a spectral mapping g :Y → Z such that g(x) =x for every x ∈X. In this conditions SE(X) is a complete upper semi-lattice with the maximal elementβSX (see Example 1.6.7).
Let L be an l-base of a space X. The filter ξ ⊆ L is a simple L-filter if ξ∩ {F, H} 6=Ø providedF ∪H∈ξ and F, H∈L. Every maximal L-filter is simple. The filterξ(x) is simple for everyx∈X.
Denote by sLX the set of all simple L-filters. For every H ∈ L we put
<< H >>= {ξ ∈ sLX : H ∈ ξ}. Then << L >>= {<< H >>:H ∈L} is a closed base of the space sLX. We identify x ∈ X with ξ(x). ThenX is a subspace ofsLX, X is dense insLX and the setsLX\<< H >>is open and compact insLX for everyH∈L. ThussLX is a spectral compactification of X. We mention that ωLX ⊆sLX.
If bX is a spectral compactification of X, then L={X\U :U is an open and compact subset ofbX } is anl-base ofX and bX =ωLX (see [24,25]).
In the papers [24,131,132] the class of all spectral compactifications was constructed and studied using the functional rings.
We mention that the spectrum of the simple ideals of a ring in the Zariski topology is a spectral space (see [132]).
3. Uniform extensions of topology spaces
In the present chapter every space is assumed to be a completely regular T1-space.
A uniform space (X, U) is a set X and a family U of entourages of the diagonal ∆(X) ={(x, x) :x∈X}ofX inX×X which satisfies the following conditions:
U1. If V ∈U and V ⊆W, thenW−1={(x, y) : (y, x)∈W} ∈U. U2. If V, W ∈U, thenV ∩W ∈U.
U3. For every V ∈ U there exists W ∈ U such that 2W ⊆ V, where 2W ={(x, y) : there exists z∈X such that (x, z), (z, y)∈W}.
U4. ∩U = ∆(X).
Denote byu−w(X, U) the weight of a uniform space (X,U). On a uniform space (X,U) we consider the topologyT(U), generated by the uniformityU.
LetX be a space with the topologyT. We putu−w(X) = min{u−w(X, U) : T(U) =T}+ℵ0.
IfX is discrete or metrizable, thenu−w(X) =ℵ0.
A pseudometric on a space X is a function ρ : X×X → R into the reals such thatρ(x, x) = 0, ρ(x, y) =ρ(y, x) andρ(x, z)≤ρ(x, y) +ρ(y, z) for all x, y, z ∈ X. The pseudometric ρ is continuous if the sets B(ρ, x, r) ={y ∈ X:ρ(x, y)< r}, x∈X and r >0, are open in X.
Every uniformity is generated by a family of pseudometrics [44].
3.1. Lattice U E(X)
A uniform extension of a spaceX is a complete uniform space (eX,U) that containsX as a dense subspace.
Denote by U E(X) the family of all uniform extensions of a space X.
If (eX, U),(bX, V)∈U E(X), then we consider that (e(X), U)≥(bX, V) if there exists a uniformly continuous mappingg:eX →bXsuch thatg(x) =x for each x∈X.
3.1.1. Proposition. The set U E(X) is a complete upper semi-lattice for every non-empty space X.
Proof. See Example 1.6.5
In the present chapter we consider the following two problems.
Problem 1. LetP be a property,X be a space and (Y, V) be a complete uniform space with the property P. Under which conditions there exists a uniform extension (Z, U) ofX such that:
1. (Z, U) is a uniform space with the propertyP;
2. the uniform space (Y, V) is uniformly isomorphic to the subspace Z\X of (Z, U)?
Problem 2. Let X be a space and (Y, V) be a complete uniform space.
Under which conditions there exists a uniform extension (Z, U) of X such that (Y, V) is uniformly isomorphic to some subspaceH ⊆Z\X of the space (Z, U)?
Concrete results related to the solution of the problems of this type play an important role in the study of classes of spaces and complete uniform spaces.
3.2. Discrete subspaces and uniform extensions
A subsetLof a spaceX is strongly discrete if there exists a discrete family {Hx :x ∈L} of open subsets of X such that L∩Hx ={x} for every x ∈L.
For every space X we put DS(X) = {|L| : L is a strongly discrete infinite subset of X}and d(X) = min{|H|:H is a dense subset of the spaceX}.
If Y is a subspace of a space X, then we denote DS(X, Y) ={|H|:H ⊆ X\Y and H is a strongly discrete infinite subset of X}.
3.2.1. Proposition. LetY be a subset of a spaceX,ρanddbe continuous pseudometrics on the space X, r > 0, X1 ={x ∈X :d(x, y) <2r for some
