aa r X i v : . [ m a t h . G M ] M a r A Brief Survey on Fibrewise General Topology
Giorgio NORDO
Abstract
We present some recent results in Fibrewise General Topology withspecial regard to the theory of Tychonoff compactifications of mappings.Several open problems are also proposed.
1. Introduction
Mapping are more general object than of topological spaces. In fact, it isevident that any space can be trivially identified with the continuous mapping ofthat space to a single-point space.Since 50’s, this simple fact suggested the idea to consider properties for map-pings instead of the traditional ones for spaces in order to obtain more generalstatements.First steps in this direction were moved by Whyburn [W , W ] and Dickman[D], but only in 1975, Ul’janov [U] introduced the notion of Hausdorff mapping (formerly called separable ) to study the Hausdorff compactifications of countablecharacter.Later, Pasynkov [P ] generalized and studied in a systematic way to the con-tinuous mappings various other notions and properties concerning spaces like theseparation axioms T , T , T , T , the regularity , the complete regularity , the normality , the compactness and the local compactness . A considerable part ofthese new definitions and constructions is based on the notion of partial topo-logical product (briefly PTP) introduced and studied by Pasynkov in [P ]. Themain properties of PTP’s, included an analogous for mappings of the EmbeddingLemma, are proved in detail in [P ].Some weaker separation axioms for continuous mappings such as semiregularity and almost regularity were introduced and studied in [CN]. The problem of theirproductivity was investigated in [N ].Let us note that Pasynkov’s papers ([P ], in particular) have inspirated Jamesto give a slightly different approach to the same topic (see [J]).The generalization to mappings of notions originally defined for spaces belongsto the more general branch of the Fibrewise General Topology (sometimes called
General Topology of mappings ) and, from a categorial point of view, it means1o pass from the study of the property of the category
Top to those of thecategory
Top Y whose objects are the continuous mappings into some fixed space Y and whose morphisms are the continuous functions commutating the triangulardiagram of two objects.Because a property P Y of Top Y can be considered as a generalization of somecorresponding property P of Top it must coincide with P when Y is a single-point space, that is when every object f : X → Y of Top Y can be identified withthe space X , i.e. with an object of Top .In rare case, the analogous P Y in Top Y of some property P of Top is quiteevident. For example, to P = compactness corresponds P Y = perfectness. Inother cases (e.g. for the separation axioms T and T ) the property P Y can beobtained by requiring that the corresponding property P holds on every fibre ofthe mappings, but for many properties for mappings, it is necessary to give newdefinitions which are more complex than the corresponding ones for the spaces.The notion of compactification (i.e. perfect extension) of a continuous mappingwas given first by Whyburn in 1953 [W , W ].It is worth mentioning that in [N ] it is presented a filter based method whichallows us to build perfect extension of every function (not necessarily continuous)between two arbitrary topological spaces.However, the first general definition of compactification for mappings analogousto the well-known notion for spaces, was given by Pasynkov in [P ]. In fact, inthat paper, using techniques based on PTP’s, a method is described to obtainTychonoff (i.e. completely regular, T ) compactifications of Tychonoff mappingsbetween arbitrary spaces, and it is proved that the poset T K ( f ) of all the Ty-chonoff compactifications of a Tychonoff mapping f : X → Y admits a maximalcompactification βf : β f X → Y which is the exact analogous, in Top Y , of theStone– ˇCech compactification of a Tychonoff space.Let us note that K¨unzi and Pasynkov [KP] have completely described the set T K ( f ) of all the Tychonoff compactifications of a Tychonoff mapping f : X → Y by means of presheaves of the rings C ∗ ( f − ( U )) with U open set of Y .Recently, Bludova and Nordo [BN] have shown that if a mapping f : X → Y is Hausdorff compactifiable (i.e. it has some Hausdorff compactification) then thereexists the greatest (called ”maximal”) compactification χf : χ f X → Y in the set HK ( f ) of all the Hausdorff compactifications of f .An extension to mappings of the notion of H-closedness (see [PoW] or [CGNP]for a complete survey) was given in [CFP] by Cammaroto, Fedorchuk and Porter,while some other generalizations to the mappings of the concepts of realcompact-ness and Dieudonn´e completeness were introduced and studied in [IP], [BuP],[MuP] and [P ].The notion of perfect compactification given by Skljarenko in [S] was recentlygeneralized to the mappings in [NP] and, in [ ? ], it was proved that both the max-2mal realcompactification vf and the Dieudonn´e completion µf of a Tychonoffmapping f are perfect extensions of f .
2. Extension to mappings of notions for spaces
Throughout all the paper, the word ”space” will mean ”topological space” onwhich, no separation axiom is assumed and all the mappings will be supposedcontinuous unless otherwise specified. If X is a space, τ ( X ) will denote the familyof all open sets of X .For terms and undefined concepts we refer to [E].For any fixed space Y , we consider the category Top Y where Ob ( Top Y ) = { f ∈ C ( X, Y ) : X ∈ Ob ( Top ) } is the class of the objects and, for every pair f : X → Y , g : Z → Y of objects, M ( f, g ) = { λ ∈ C ( X, Z ) : g ◦ λ = f } is the class of the morphisms from f to g , whose generic representant is denotedfor short by λ : f → g .A morphism λ : f → g from f : X → Y to g : Z → Y will be called surjective (resp. closed , dense ) if λ ( X ) = Z (resp. λ ( X ) is closed in Z , λ ( X ) is dense in Z ).If λ : f → g is a surjective morphism, we will say that g is the image of f ( bythe morphism λ ) and we will write that g = λ ( f ).Moreover, we say that a morphism λ : f → g from f : X → Y to g : Z → Y isan embedding (resp. a homeomorphism ) if so is the function λ : X → Z .A mapping g : Z → Y is said an extension of f : X → Y if there exists somedense embedding λ : f → g (as usual, we shall identify X and f by λ ( X ) and g | λ ( X ) respectively).A morphism λ : g → h between two extensions g : Z → Y and h : W → Y ofa mapping f : X → Y will be called canonical if λ | X = id X .Let us introduce some notions and basic facts about partial products [P , P ].Given two spaces Y , Z and an open set O of Y , we consider the set P = ( Y \ O ) ∪ ( O × Z ) and the map p : P → Y defined by p | Y \ O = id Y \ O and p | O × Z = pr O where pr O : O × Z → O denotes the projection of O × Z onto O .We will call elementary partial topological product (briefly EPTP) of Y andfibre Z relatively to the open set O and we will denote it by P ( Y, Z, O ), the spacegenerated on P by the basis B ( Y, Z, O ) = p − ( τ ( Y )) ∪ τ ( O × Z ).The mapping p : P → Y above defined will be called the projection of the EPTP P = P ( Y, Z, O ) and it is routine to prove that it is a continuous, onto, openmapping. 3t is evident that the EPTP P ( Y, Z, ∅ ) is simply Y and that P ( Y, Z, Y ) coincideswith the usual product space Y × Z .Now, let Y be a topological space, { Z α } α ∈ Λ be a family of spaces and { O α } α ∈ Λ be a family of open sets of Y . For every α ∈ Λ, let P α = P ( Y, Z α , O α ) be theEPTP of base Y and fibre Z α relatively to O α and p α : P α → Y be its projection.We will call partial topological product (PTP for short) of base Y and fibres { Z α } α ∈ Λ relatively to the open sets { O α } α ∈ Λ the fan product of the spaces { P α } α ∈ Λ relatively to the mappings { p α } α ∈ Λ , i.e. the subspace P = ( t = h t α i α ∈ Λ ∈ Y α ∈ Λ P α : p α ( t α ) = p β ( t β ) ∀ α, β ∈ Λ ) and we will denote it by P ( Y, { Z α } , { O α } ; α ∈ Λ).For every α ∈ Λ, the restriction π α = pr α | P : P → P α of the α -th canon-ical projection pr α will be called the α -th short projection , while the fibrewiseproduct of the mappings { p α } α ∈ Λ , i.e. the continuous mapping p : P → Y de-fined by p α ◦ π α = p for any α ∈ Λ will be said the long projection of the PTP P ( Y, { Z α } , { O α } ; α ∈ Λ).In case O α = Y for every α ∈ Λ, the PTP P ( Y, { Z α } , { O α } ; α ∈ Λ) coincides(up to homeomorphisms) with the usual product Y × Q α ∈ Λ Z α , if | O α | = 1 forevery α ∈ Λ, the PTP P ( Y, { Z α } , { O α } ; α ∈ Λ) coincides with the usual Ty-chonoff product Q α ∈ Λ Z α of its fibres, while if O α = ∅ for any α ∈ Λ, the PTP P ( Y, { Z α } , { O α } ; α ∈ Λ) is simply (homeomorphic to) the space Y . Definitions.
A mapping f : X → Y is said to be T [P ] if for every x, x ′ ∈ X such that x = x ′ and f ( x ) = f ( x ′ ) there exists a neighborhood of x in X whichdoes not contain x ′ or a neighborhood of x ′ in X not containing x .A mapping f : X → Y is said to be Hausdorff (or T ) [U, P ] if for every x, x ′ ∈ X such that x = x ′ and f ( x ) = f ( x ′ ) there are two disjoint neighborhoodsof x and x ′ in X .We will say that f : X → Y is compact if it is perfect (i.e. it is closed and everyits fibre is compact).A mapping f : X → Y is said to be completely regular [P ] if for every closed set F of X and x ∈ X \ F there exists a neighborhood O of f ( x ) in Y and a continuousfunction ϕ : f − ( O ) → [0 ,
1] such that ϕ ( x ) = 0 and ϕ ( F ∩ f − ( O )) ⊆ { } .A completely regular, T mapping is called Tychonoff (or T ) [P ]. Remark.
It is easy to verify that all the previous properties in
Top Y coincidewith the corresponding ones in Top provided | Y | = 1 and that every continuousmapping f : X → Y has such a property if f both the spaces X and Y havethe corresponding properties (in particular, they are P− functions in the sense of[CN]). 4 efinition. A restriction f | X ′ : X ′ → Y to X ′ ⊆ X of a mapping f : X → Y issaid a closed restriction of f , if X ′ is a closed subset of X .Obviously (see for example [PoW]), every closed restriction of a compact map-ping is compact too.Most well-known statements which hold in the category Top have correspon-dent ones (and hence generalizations) in
Top Y . The following properties areessentially given in [P ] (detailed proofs can be found in [N ]). PROPOSITION 2.1.
Every image λ ( f ) of a compact mapping f : X → Y iscompact too. PROPOSITION 2.2.
Every closed restriction f | X ′ of a compact mapping f : X → Y is compact too. PROPOSITION 2.3.
Every compact restriction f | X ′ of a Hausdorff mapping f : X → Y is a closed restriction of f . PROPOSITION 2.4.
Let λ and µ be morphisms from a mapping f : X → Y to a Hausdorff mapping g : Z → Y and D be a dense subset of X . Then if λ | D = µ | D , the morphisms λ and µ coincide. PROPOSITION 2.5.
Every morphism λ : f → g from a compact mapping f : X → Y to a Hausdorff mapping g : Z → Y is perfect.
3. Compactification of mappings
Let f : X → Y be a mapping. We say that a mapping c : X c → Y is a compactification of f (in Top Y ) if it is a compact (= perfect) extension of f .This approach to the notion of the compactification of a mapping was proposedby Whyburn [W ], but in the most general situation, this notion was studied firstby Pasynkov in [P ]. Remark.
A different variant of compactifications of mappings was examined byUljanov [U]. But, it is a common opinion that Uljanov’s definition is not naturalfor non-surjective mappings because, in that case, a compact mapping is not itsown compactification.
Definitions.
Let c : X c → Y and d : X d → Y be two compactifications of amapping f : X → Y (in Top Y ). We say that: • c is projectively larger than d (relatively to f ) and we write that c ≥ f d (or c ≥ d , for short) if there exists some canonical morphism λ : c → d . • c is equivalent to d (relatively to f ) and we write that c ≡ f d (shortly, c ≡ d ) if there exists a canonical homeomorphism λ : c → d .5he following useful result is given in [BN]. PROPOSITION 3.1.
Let c : X c → Y and d : X d → Y be two Hausdorffcompactifications of a mapping f : X → Y . Then c ≡ f d if and only if c ≥ f d and d ≥ f c . Definition.
A Hausdorff mapping f : X → Y will be called Hausdorff compact-ifiable if it has some Hausdorff compactification (in
Top Y ).In [BN], it is noted that the class of all Hausdorff compactifications of anyHausdorff compactifiable mapping f : X → Y forms a set modulo the equivalence ≡ f . Definition. If f : X → Y is a Hausdorff compactifiable mapping, HK ( f ) willdenote the set of all equivalence classes of Hausdorff compactifications of f .So, by 3.1, it follows that ( HK ( f ) , ≥ ) is a poset and, for any pair of Hausdorffcompactifications c, d ∈ HK ( f ) we can write c = d instead of c ≡ f d , that is wedo not distinguish between equivalent Hausdorff compactifications.In [P ], Pasynkov erroneously indicated that it is proved in [U] that everyHausdorff compactifiable mapping f : X → Y has a maximal one.This fact was also used in several following papers like [BuP], [CFP], [IP], [KP],[MuP], [M ], [P ], etc. but it is not correct because the Ul’janov’s definition isdifferent from the currently used one (given by Pasynkov in [P ]) that does notinclude the surjectivity.Anyway the existence of the maximal Hausdorff compactification was actuallyproved by Bludova and the author as direct consequence of the following moregeneral result. THEOREM 3.2. [BN]
For any Hausdorff compactifiable mapping f : X → Y , ( HK ( f ) , ≥ ) is a complete upper semilattice The projective maximum of ( HK ( f ) , ≥ ), i.e. the maximal Hausdorff compact-ification of f , will be denoted by χf : χ f X → Y .From this and by 2.4 and 2.5, it follows – in particular – that for any Hausdorffcompactification bf : X b → Y of a Hausdorff compactifiable mapping f : X → Y there exists a unique perfect canonical morphism λ b : χf → bf .In [P ], Pasynkov proved that every Tychonoff mapping has a Tychonoff com-pactification.Since it is easy to show that a Tychonoff mapping is Hausdorff, Proposition 3.1allow us to give the following: Definition.
For any Tychonoff mapping f : X → Y , we will denote by T K ( f )the set of all Tychonoff compactifications of f up to the equivalence ≡ f .For a mapping f : X → Y , let us denote by C ∗ ( f ) the family of all the6 artial mappings on f , i.e. of all the continuous bounded real-valued mappings ϕ : f − ( O ϕ ) → I ϕ defined from the inverse image by f of an open set O ϕ of Y toa compact subset I ϕ of the real line IR. Definitions.
A subfamily C = { ϕ : f − ( O ϕ ) → I ϕ } of C ∗ ( f ) is said to be: • separating the points of f if for every x, x ′ ∈ X such that f ( x ) = f ( x ′ ) thereexists some ϕ ∈ C such that x, x ′ ∈ f − ( O ϕ ) and ϕ ( x ) = ϕ ( x ′ ). • separating the points from the closed sets of f if for any closed set F of X and every x ∈ X \ F there exists some ϕ ∈ C such that x ∈ f − ( O ϕ ) and ϕ ( x ) / ∈ cl I ϕ ( F ∩ f − ( O ϕ )).It is shown in [P ] (see [N ] for a more detailed proof) that every Tychonoffcompactification bf : X b → Y of a Tychonoff mapping f : X → Y is uniquelydeterminated by a subfamily C = { ϕ : f − ( O ϕ ) → I ϕ : O ϕ ∈ τ ( Y ) } of C ∗ ( f )separating the points and the points from the closed sets of f and that bf coincideswith a particular restriction of the long projection p C : P C → Y of the PTP P C = P ( Y, { O ϕ } , { I ϕ } ; ϕ ∈ C ).Thus, the notion of PTP plays in the category Top Y the same role that thenotion of product space has in the category Top and, as matter of fact, theycoincide when | Y | = 1.In [P ], it is also proved that for any Tychonoff mapping f : X → Y thereexists, in ( T K ( f ) , ≥ ), a maximal Tychonoff compactification βf : β f X → Y that is determinated by all the whole family C ∗ ( f ) and characterized by someextension properties very similar to that of the Stone- ˇCech compactification.We have, in fact, the following: THEOREM 3.3.
For any Tychonoff compactification bf : X b → Y of a Ty-chonoff mapping f : X → Y , the following conditions are equivalent:(1) bf = βf ;(2) for every U ∈ τ ( Y ) and ϕ ∈ C ∗ ( f − ( U )) there exists a unique extension e ϕ ∈ C ∗ (( bf ) − ( U )) ;(3) for every compact Tychonoff mapping k : Z → Y and every morphism λ : f → k there exists a morphism e λ : bf → k which extends λ . Moreover, Theorem 3.2. allow us to obtain as immediate consequence thefollowing:
THEOREM 3.4. [BN]
The poset ( T K ( f ) , ≥ ) of all Tychonoff compactificationsof a Tychonoff mapping f : X → Y is a complete upper semilattice whoseprojective maximum is βf . PROPOSITION 3.5. [P ] For any Tychonoff compactification bf : X b → Y of a Tychonoff mapping f : X → Y there exists a unique (perfect) canonicalmorphism µ b : βf → bf such that µ b ( β f X \ X ) = X b \ X . | Y | = 1, X is a Tychonoff space, the domain β f X of βf coincides with the Stone ˇCech compactification βX of X , the domain X b of bf isa generic compactification of X and λ b : β f X → X b becomes the usual quotientmap (see for example [Ch]).In general, for a Tychonoff mapping f : X → Y , we have T K ( f ) ⊂6 = HK ( f )that is, unlike the corresponding case for spaces, there exist Hausdorff compact-ification which are not Tychonoff or, equivalently, there are compact Hausdorffmapping which are not Tychonoff. In fact, it was proved in [Cb] (see also [HI])that it is possible to build a perfect ( ≡ compact) mapping defined on a regular T but non Tychonoff space onto a Tychonoff space (that is the property T isnot an inverse invariant by perfect mappings) and since it is proved in [P ] that ifa mapping and its range are both completely regular, its domain is too, it followsdirectly that such a mapping can not be completely regular and hence Tychonoff.This is the reason why it is necessary to study the classes of Tychonoff andHausdorff compactifiable mappings separately.
4. Open problems.
It seems that the following questions might be interesting. Some of these prob-lems are published for the first time.
Problem 1.
It is well-known that the poset K ( X ) of all Hausdorff compactifi-cation of a Tychonoff space X can be completely characterized in terms of thefamilies of C ∗ ( X ) that separete points and points from closed sets of X (see, forexample, [Ch]). Is it possible to obtain such a similar characterization for the set HK ( f ) (the set T K ( f ) ) of all Hausdorff (Tychonoff ) compactification of a Hausdorff compacti-fiable (Tychonoff ) mapping f in terms of the separating families of C ∗ ( f ) ? Problem 2.
Magill has proved in [M] (see also [Ch]) that the posets K ( X ) and K ( Y ) of all Hausdorff compactifications of two locally compact spaces X and Y are isomorphic if and only if their Stone- ˇCech remainders βX \ X and βY \ Y are homeomorphic. Is it possible to find a definition of locally compact mapping that allow us to obtaina Magill-type theorem for mappings ?
Problem 3.
Is it possible to obtain analogous in
Top Y of properties like thecountably compactness, the paracompactness and the pseudocompactness ? roblem 4. Is there a consistent definition of metrizable mapping which extendsthe corresponding notion for spaces and allow us to obtain general metrizationtheorems ?
Problem 5.
Is it possible to extend to mappings other kind of Tychonoff exten-sion properties like the m -boundedness (see [PoW]) ? References [BN] BLUDOVA I.V., NORDO G.,
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Key words and phrases: partial topological product, T mapping, Hausdorff map-ping, completely regular mapping, Tychonoff mapping, compact mapping, com-pactification of a mapping, realcompact mapping, Dieudonn´e complete mapping. AMS Subject Classification:
Primary 54C05, 54C10, 54C20, 54C25; Secondary:54D15, 54D35, 54D60.
Giorgio NORDO
MIFT - Dipartimento di Scienze Matematiche e Informatiche, scienze Fisiche escienze della Terra, Messina University, Messina, ItalyE-mail: [email protected]@unime.it