Compact condensations of Hausdorff spaces
aa r X i v : . [ m a t h . GN ] J u l Compact condensations of Hausdorff spaces
Vitalii I. Belugin
Krasovskii Institute of Mathematics and Mechanics, 620219, Yekaterinburg, Russia
Alexander V. Osipov
Krasovskii Institute of Mathematics and Mechanics, Ural Federal University,Ural State University of Economics, 620219, Yekaterinburg, Russia
Evgenii G. Pytkeev
Krasovskii Institute of Mathematics and Mechanics, Ural Federal University, 620219,Yekaterinburg, Russia
Abstract
In this paper, we continue to study one of the classic problems in generaltopology raised by P.S. Alexandrov: when a Hausdorff space X has a con-tinuous bijection (a condensation) onto a compactum? We concentrate onthe situation when not only X but also X \ Y can be condensed onto acompactum whenever the cardinality of Y does not exceed certain τ . Keywords: a τ -space, subcompact space, continuous decomposition, weaklydyadic compact, condensation Introduction
The question of when each space from class A admits a continuous bi-jection (such map is called a condensation) onto some space from class B isone of the natural questions in general theory, the subject of which is thestudy of relations between classes of spaces, performed by various types ofmappings. Email addresses: [email protected] (Vitalii I. Belugin),
[email protected] (Alexander V.Osipov), [email protected] (Evgenii G. Pytkeev)
Preprint submitted to ... July 27, 2020 n 1937 S. Banach posed a problem which can be formulated equiva-lently: when can a metric space have a condensation onto a compact metricspace? Independently, the following more general question is attributed toP.S. Alexandrov: when a Hausdorff space X has a condensation onto a com-pactum?It is natural to call such spaces as subcompact spaces. One of the firstand strong results were obtained by M. Katetov [16]: an H -closed Urysohnspace is subcompact.In the future, an active study of Hausdorff spaces which admit a conden-sation onto a compactum was continued in the works of I.L. Raukhvarger[31], V.V. Proizvolov [25], A.S. Parhomenko [22, 23], Y.M. Smirnov [34],N. Hadzhiivanov [36], V.K. Bel’nov [9], A.V. Arhangel’skii [4, 5], O. Pavlov[5], V.I. Belugin [10, 11], E.G. Pytkeev [21, 27, 28, 29], W.Kulpa and M.Turza´ n ski [17], H. Reiter [32], W.W. Comfort, A.W. Hager and J. van Mill[13] and many other authors.The fact that X \ Y can be condensed onto a compactum for every count-able Y was established for metrizable compacta by Raukhvarger [31], forproducts of metrizable compacta by Proizvolov [25], for diadic compacta byBelugin [10], for weakly diadic compacta (including polyadic and centeredspaces) by Kulpa and Turzanski [17], for zero-dimensional first countablecompacta by Belugin [11]. On the other hand Ponomarev [6] proved that ifwe remove from the remainder ω ∗ = βω \ ω of the ˇ C ech-Stone compactifi-cation of ω , a countable subset D then ω ∗ \ D has no condensation onto acompactum.It is well known that any locally compact admits a condensation onto acompactum (Parhomenko’s Theorem) [23]. It turned out that condensationsonto a compactum are relatively rare. At the same time, the most promisingway of research appeared to be the one started in [31]. I.L. Raukhvargerproved that for any metric compact space X and C ∈ [ X ] ω , the space X \ C admits a condensation P onto some compact space Y C . The condensation P : X \ C → Y C is a quotient map (projection) where the decompositionspace Y C is obtained from X by identifying the points belonging to the samemember of the decomposition D = { ( c, a ) : c ∈ C, a ∈ A } ∪ {{ b } : b ∈ X \ ( C ∪ A ) } for a countable set A ⊂ X \ C . Clearly, the condensation P canbe extended to a continuous map e P : X → Y C . This method (continuousdecompositions) is quite effective in the study condensations onto compactspaces. For example, by the method of continuous decomposition, it was2roved that any weakly diadic is a strictly a -space [17].In this paper, we concentrate on the situation when not only X but also X \ Y can be condensed onto a compactum whenever the cardinality of Y does not exceed certain τ .
1. Main definitions and notation
In this paper, all considered spaces are assumed to be Hausdorff topolog-ical spaces. We use a quotient space and related concepts.Let Y be a set. For a space X and a surjection f : X → Y , let τ ( f ) = { U ⊂ Y : f − ( U ) is open in X } . Then τ ( f ) is called the quotient topology for Y determined by f . Let X and Y be topological spaces. Let f : X → Y bea surjection. Then f is called a quotient map if the topology in Y is exactly τ ( f ); that is, U is open in Y if and only if f − ( U ) is open in X . The space Y is called the quotient space of X by f [37].Let X be a set. Let D be a decomposition of X ; that is, D is a coverof X such that any two distinct members are disjoint. Let P : X → D be the projection (i.e., P maps each point x ∈ X to the unique member of D containing x ). Let X ( D ) be the space D having the quotient topologydetermined by P (i.e., D ′ ⊂ D is open in X ( D ) if and only if P − ( D ′ ) isopen in X ). The space X ( D ) is called the decomposition space of X by D .Namely, the decomposition space X ( D ) is obtained from X by identifyingthe points belonging to the same member of D , and a subset of X ( D ) is openif and only if its inverse image by the projection P is open in X . The set A ⊂ X is called saturated with respect to the decomposition D , if A is theunion of some set of elements of D (i.e. for any element T ∈ D if T T A = ∅ ,then T ⊆ A ) [37].A decomposition D of a space X is called continuous if for any T ∈ D and for any open set U ⊇ T there is an open saturated set V such that T ⊆ V ⊆ U [1, 19].P.S. Alexandrov and H. Hopf proved that a decomposition D of a space X is continuous if and only if the projection P : X → X ( D ) is a closed map[3]. Using this result, it is easy to prove that the quotient space X ( D ) of acompact Hausdorff space X is a compact Hausdorff space if and only if thedecomposition D is continuous and consists of closed subsets of X .In this paper, we use the following notations: ω - the first infinite ordinal, ω - the first uncountable ordinal, ℵ - the first infinite cardinal number, Q , N and R are, as usual, the set of rational, natural and real numbers,3espectively. For an arbitrary set A and a cardinal number τ , [ A ] ≤ τ ( [ A ] <τ )will denote the set of all subsets of the set A of the cardinality ≤ τ ( < τ ). Aspace X is non-trivial provided that | X | > Theorem 1.1. (S. Iliadis) Let γX be a extremally disconnected compacti-fication of a space X such that the remainder X ∗ = γX \ X of cardinality < c . Then X is not subcompact space. We use the particular case of this theorem: Let X be an extremallydisconnected compact space without isolated points and C ∈ [ X ] ω . Then X \ C is not subcompact space.Let τ be an infinite cardinal number. • A compact space X is called an a τ -space provided that for each C ∈ [ X ] ≤ τ there is a condensation from X \ C onto a compactum [12].In particular, for τ = ℵ , a compact space X is called an a -space [11]. • A compact space X is called an strictly a τ -space provided that for each C ∈ [ X ] ≤ τ there is a condensation f from X \ C onto a compact space Y that f can be extended to a continuous map e f : X → Y [12].Note that any strictly a τ -space is an a τ -space and is a strictly a η -spacefor any η ≤ τ . For τ = ℵ , a compact space X is called a strictly a -space [11].A natural extension of the classes of a τ - and strictly a τ -spaces are classesof (strictly) a τ -subcompact and almost (strictly) a τ -subcompact spaces. Definition 1.2.
Let τ be an infinite cardinal number. • A space X is called a τ -subcompact provided that for each C ∈ [ X ] ≤ τ there is a condensation from X \ C onto an a τ -space. • A space X is called strictly a τ -subcompact provided that for each C ∈ [ X ] ≤ τ there is a condensation f from X \ C onto a strictly a τ -space Y that f can be extended to a continuous map e f : X → Y . • A space X is called almost a τ -subcompact provided that for each C ∈ [ X ] ≤ τ there is a condensation from X \ C onto a compact space. • A space X is called almost strictly a τ -subcompact provided that for each C ∈ [ X ] ≤ τ there is a condensation f from X \ C onto a compact space Y that f can be extended to a continuous map e f : X → Y .4ote that for an arbitrary τ ≥ ℵ , the following implications are true:strictly a τ -space −→ a τ -space ↓ ↓ strictly a τ -subcompact space −→ a τ -subcompact space ↓ ↓ almost strictly a τ -subcompact space −→ almost a τ -subcompact spaceDiagram 1.Further, we prove the strictness of all the implications in the Diagram 1. Proposition 1.3.
Let X be an (strictly) a τ -subcompact space and C ∈ [ X ] ≤ τ .Then X \ C is an (strictly) a τ -subcompact space. In particular, a space X \ C is an (strictly) a τ -subcompact space for any(strictly) a τ -space X and C ∈ [ X ] ≤ τ . Proposition 1.4.
There exists an (strictly) a -subcompact space X such thatit is not homeomorphic to Y \ C where Y is an (strictly) a -space and C ∈ [ Y ] ≤ ω .Proof. In [26], it was proved that any non- σ -compact Borel subset of a Polishspace admits a condensation onto a metric compact space. Consider anyBorel subset X of a Polish space that has a Borel order higher than the first.Since, for any S ∈ [ X ] ≤ ω , the space X \ S is Borel not σ -compact subset ofa Polish space , by Theorem 1 in [26], X \ S admits a condensation onto I ℵ .Hence, X is an (strictly) a -subcompact space.Assume that X is homeomorphic to Y \ C where Y is an (strictly) a -spaceand C ∈ [ Y ] ≤ ω . Then Y \ C is a G δ -set in Y and, hence, Y \ C is a Polishspace. It follows that X is Polish and it has a Borel order equal to 1. Theorem 1.5.
There is an almost strictly a τ -subcompact space which is not a τ -subcompact.Proof. Consider X = D L K , where D is a discrete set of cardinality τ + and K is an extremally disconnected compact space without isolated pointsof cardinality (2 τ + ) + . We prove that X is almost strictly a τ -subcompact.Let E ∈ [ X ] ≤ τ . Put A = E ∩ D and B = E ∩ K . Choose C ⊂ D \ A suchthat | C | = | B | . Denote D = D \ ( A ∪ C ). Let d ∈ D and let D ∗ be a5ne-point compactification of D \ { d } , where d is not-isolated point of thecompact space D ∗ . Put X = D ∗ ∪ K . Let ψ be a bijection between C and B . Construct a condensation ϕ : X \ ( A ∪ B ) → D ∗ ∪ K by the followingrule ϕ ( x ) = ( ψ ( x ) , x ∈ C,x, x ∈ ( K \ B ) ∪ D . The continuous function ϕ be extended to the continuous function f : X → D ∗ ∪ K where for fix k ∈ Kf ( x ) = ψ ( x ) , x / ∈ ( A ∪ B ) ,x, x ∈ B,k, x ∈ A. Note that X cannot be condensed onto an a τ - space. Suppose f : X → T is a condensation from the space X onto an a τ -space T . Since | f ( D ) | ≤ τ + , there is W ⊂ T \ f ( D ) such that W is open-closed in f ( K ) ( f ( K )is homeomorphic to K ). Hence, W is an extremally disconnected compactspace without isolated points. Let S be a countable subset of W . We showthat T \ S cannot be condensed onto a compact space. Indeed, let g : T \ S → B be a condensation where B is compact. Then T \ W is compact, hence, g ( T \ W ) is compact, and B \ g ( T \ W ) is locally compact. By Theorem in[23], the space B \ g ( T \ W ) (and, hence, W \ S ) admits a condensation ontoa compactum. This contradicts of Theorem 1.1.
2. Main resultsProposition 2.1.
Let X admits a condensation onto a strictly a τ -space forsome τ ≥ ℵ . Then X is strictly a τ -subcompact.Proof. Let f : X → Y be a condensation from a space X onto a strictly a τ -subcompact space Y . Take any S ∈ [ X ] ≤ τ . Then there is a condensation h : Y \ f ( S ) → K where K is compact, such that h can be extended to acontinuous map e h : Y → K . Note that e h ◦ f is a continuous extension over X of the condensation h ◦ ( f ↾ ( X \ S )) : X \ S → K .In 1970 S.Mr´ o wka [20] generalized the class of dyadic spaces defining theclass of polyadic spaces (= the continuous images of the products of the onepoint compactifications of discrete spaces).6n paper [17] W.Kulpa and M.Turza´ n ski introduced the class of weaklydyadic spaces.Let T be an infinite set. Denote a Cantor cube by D T := { p : p : T → { , }} . For s ⊂ T and p ∈ D T we shall use thefollowing notation G s ( p ) := { f ∈ D T : f ↾ s = p ↾ s and p − (0) ⊂ f − (0) } . Definition 2.2. ([17]) • A subset X ⊂ D T is said to be an ω -set iff for each p ∈ X there existsan s ⊂ T such that | s | ≤ ω and G s ( p ) ⊂ X . • A space Y is said to be a weakly dyadic space if Y is a continuous imageof a compact ω -set in D T .The class of all weakly diadic spaces contains the class of all centeredspaces in sense of Bell [8] which in turn, contains the class of all polyadicspaces. Kulpa and Turza´ n ski proved that a weakly dyadic space is a strictly a -space (Lemma 2 and Theorem in [17]). Corollary 2.3.
Suppose that X admits a condensation onto a weakly diadicspace. Then X is a strictly a -space. Theorem 2.4.
Let X = Z L ( L { X α : α ∈ A } ) , where | A | = τ , X α is an a τ - space, | X α | > τ for every α ∈ A and Z is compact. Then X is almost a τ -subcompact.Proof. Let S ∈ [ X ] ≤ τ . Without loss of generality we can assume that S ⊆ Z .Otherwise, for every α ∈ A there is a condensation ϕ α from X α \ S onto acompactum. Then we can consider the space Z L ( L { ϕ ( X α \ S ) : α ∈ A } )where the restriction ϕ ↾ ( X α \ S ) = ϕ α for every α ∈ A .Note that the space L { X α : α ∈ T } is locally compact for any T ⊂ A , and, by Parhomenko’s Theorem [23], it admits a condensation onto acompactum. Thus, if | S | < | A | , then we can consider the space P = Z L ( L { X α : α ∈ B } ) where B ⊂ A and | B | = | S | . Let S = { s α : α ∈ B } .Further, we suppose that X = P and | S | = | A | .We prove that X \ S is subcompact. For any α ∈ A we choose a point p α ∈ X α . On the set X \ S define the topology τ ′ , topologize X \ S by lettingsets:1. V x = ( Ox ∪ ( ∪{ X α : s α ∈ Ox } )) ∩ ( X \ S ) where Ox is a neighborhoodof x in X such that Ox ∩ { p α : α ∈ A } = ∅ , be basic neighborhood of a point x ∈ ( X \ S ) \ { p α : α ∈ A } ; 7. V p α = ( Op α ∪ Os α ) ∪ ( ∪{ X β : s β ∈ Os α \ s α } ) ∩ ( X \ S ) where Op α ⊆ X α is a neighborhood of p α in X and Os α is a neighborhood of s α in X , be basic neighborhood of a point p α for each α ∈ A .Note that ( X \ S, τ ′ ) is a Hausdorff space and X \ S admits a condensationonto ( X \ S, τ ′ ). It remains to prove that ( X \ S, τ ′ ) is compact.We show that any infinite set M ⊆ X \ S has a complete accumulationpoint in topology τ ′ .Perhaps | M ∩ Z | = | M | . The set M has a complete accumulation point z in the compact space Z . If z / ∈ S , then z is a complete accumulationpoint of M in ( X \ S, τ ′ ); if z ∈ S , then z = s α for some α ∈ A . Any basicneighborhood of p α includes some neighborhood Os α of s α and, hence, thepoint p α is a complete accumulation point of M in the topology τ ′ .Further, we assume that | M ∩ Z | < | M | .If there is an index α such that | M ∩ X α | = | M | , then a complete accumu-lation point x of the set M ∩ X α will be a complete accumulation point of M in( X \ S, τ ′ ). If there is no such index, then |{ α : M ∩ X α = ∅}| = | M | . The set { s α : M ∩ X α = ∅} has the same cardinality as M . The set { s α : M ∩ X α = ∅} has a complete accumulation point x in Z . If x ∈ Z \ S , then x will be acomplete accumulation point of M in ( X \ S, τ ′ ) because V x includes all X α for s α ∈ Ox for any neighborhood Ox of x . If x = s α , then the basicneighborhood V p α includes all X α for s α ∈ Os α for any neighborhood Os α and, hence, p α is a complete accumulation point of the set M .Note that in the previous theorem we also proved the following proposi-tion. Proposition 2.5.
Let X = Z L ( L { X α : α ∈ A } ) , where X α is a non-empty compact space for each α ∈ A , | A | = τ , Z is compact and S ∈ [ Z ] ≤ τ .Then X \ S is subcompact. Definition 2.6.
A space X is called almost a <τ -subcompact , if for any C ∈ [ X ] <τ there is a condensation of X \ C onto a compactum. Theorem 2.7.
Let X α be a non-empty compact space for any α ∈ A , where | A | = τ . Then X = L { X α : α ∈ A } is almost a <τ -subcompact.Proof. Let S ∈ [ X ] <τ . Then the set of indexes B = { α ∈ A : X α ∩ S = ∅} has the cardinality τ . The set X = L { X α : α ∈ A \ B } is a locallycompact space. By the theorem of Parkhomenko [23], there is a condensation ϕ : X → Z from the space X onto a compact space Z . By Proposition 2.5,the space Z L ( L { X α : α ∈ B } ) \ ϕ ( S ) is subcompact.8e recall the definition of P τ - product of spaces for τ ≥ ℵ . Let { X λ : λ ∈ Λ } be a family of topological spaces. Let X = Q { X λ : λ ∈ Λ } be theCartesian product with the Tychonoff topology. Take a point p = ( p λ ) λ ∈ Λ ∈ X . For each x = ( x λ ) λ ∈ Λ ∈ X , let Supp ( x ) = { λ ∈ Λ : x λ = p λ } . Thenthe subspace P τ ( p ) = { x ∈ X : | Supp ( x ) | ≤ τ } of X is called P τ - product { X λ : λ ∈ Λ } about p . The subspace { x ∈ X : | Supp ( x ) | < ℵ } of X is called σ - product . In 1959, H.H. Corson [14] introduced the definitions of P ℵ − products and σ -products and studied these spaces.Further, we study the property of the subcompactness of P τ - ( σ -) prod-ucts of compacta for any τ ≥ ℵ . Proposition 2.8.
Let X be P τ -product { X β : β ∈ B } of non-trivial com-pacta X β of density at most τ and | B | ≥ τ + . Then X is not subcompact.Proof. On the contrary, let f : X → K be a condensation from X onto acompact space K . Since X is a τ - bounded space (the closure of any set ofcardinality at most τ is compact), then the space K is a τ - bounded space.Consider two cases:(1) d ( K ) > τ . Since K is a τ -bounded space, there is a family { K α : α < τ + } of compact subsets of K , such that K α ⊂ K β for α < β and d ( K α ) ≤ τ . For each K α , there is a compact subset Y α of X such that f ( Y α ) ⊇ K α . Since Y α depends on τ coordinates in P τ -product X and α < τ + , then S α<τ + Y α ⊂ Y ⊂ X , where Y is P τ -product of compacta X β ( α ) ,where Y α ⊆ X β ( α ) and d ( X β ( α ) ) ≤ τ for each α < τ + .Note that βY = Q { X β ( α ) : α < τ + } where βY is the Stone- ˇ C ech com-pactification of the space Y [6]. Since X β ( α ) is a compact space of density atmost τ for each α < τ + , then, by the Hewitt- Marczewski-Pondiczery the-orem (Theorem 2.3.15 in [15]), the space βY has a density at most τ . Thecondensation f ↾ Y : Y → f ( Y ) can be extended to the continuous function h : βY → f ( Y ). It follows that f ( Y ) is of density at most τ and it containsan increasing transfinite sequence { K α : α < τ + } . Contradiction.(2) d ( K ) ≤ τ . Let S be a dense subset of the space K and | S | ≤ τ .Consider the subset f − ( S ) of X . Since each point x ∈ f − ( S ) depends on τ coordinates α , then the set f − ( S ) also depends on τ coordinates { α s : s ∈ L } , | L | ≤ τ . Hence, f − ( S ) ⊂ Z = Q { X α s : s ∈ L } . Note that Z is acompact space of density at most τ and f ( Z ) = K . Since X = Z and f isan injective mapping, we get a contradiction.9 orollary 2.9. An uncountable P -product of non-trivial metrizable com-pacta is not subcompact. Question 1.
Let X be P τ -product { X β : β ∈ B } of non-trivial compacta X β of density (or weight) ≥ τ + for each β ∈ B and | B | ≥ τ + . Will X be asubcompact space? Theorem 2.10.
Let X be an infinite σ -product of compacta. Then X is notsubcompact.Proof. Since X is an infinite σ -product of compacta, it is a countable unionof compacta with an empty interior ( X is a space of the first category) and,hence, X cannot be condensed onto a compactum (a compact space is a spaceof the second category).Note that the inverse limit of an inverse system of (strictly) a -spaces maynot be an a -space. Indeed, let X be a compact space that is not an a -space.Since X is a subset of I α , where I = [0 ,
1] and α = w ( X ), the space X can berepresented as the inverse limit of an inverse system of S = { I β , π σβ , ω } , where β ≤ σ , β, σ ∈ ω , π σβ : π σ ( X ) → π β ( X ) and π η : X → I η is the projectionfor η ∈ ω . Note that I η is a metrizable compact space (an a -space) for each η ∈ ω , but X = lim ← S is not an a -space.The following theorem was proved in [30](Theorem 1). Theorem 2.11.
Let { X α : α ∈ Λ } be a family of non-empty Tychonoffspaces, w ( X α ) ≤ τ for each α ∈ Λ and | Λ | = 2 τ . Then L { X α : α ∈ Λ } admits a condensation onto I τ . Recall that the i -weight iw ( X ) of a space X is the smallest infinite car-dinal number κ such that X can be mapped by a one-to-one continuousmapping onto a space of the weight not greater than κ . Theorem 2.12.
Let { X α : α ∈ Λ } be a family of non-empty spaces, iw ( X α ) ≤ τ for each α ∈ Λ and | Λ | = 2 τ . Then L { X α : α ∈ Λ } is a < τ -subcompact.Proof. Let S ∈ [ X ] < τ where X = L { X α : α ∈ Λ } . Since iw ( X α ) ≤ τ , foreach α ∈ Λ, there is a condensation X α onto a Tychonoff space Y α of theweight ≤ τ . Let Y = L { Y α : α ∈ Λ } . Then there is a condensation f : X → Y . Since w ( Y α \ f ( S )) ≤ τ for each α ∈ Λ and |{ α : Y α \ f ( S ) = ∅}| = 2 τ ,then, by Theorem 2.11, Y \ f ( S ) admits a condensation onto I τ .10ecall that the absolute aX of a topological space X is the set of con-verging ultrafilters and the natural map π X : aX → X simply assigns thelimit to each ultrafilter [24]. Example 2.13.
There is a compact space X with χ ( x, X ) > τ (or πχ ( x, X ) >τ ) for each x ∈ X such that it is not a τ -space.Proof. Let Z be a compact space such that χ ( z, Z ) > τ ( πχ ( z, Z ) > τ ) foreach z ∈ Z . Let X = aZ be the absolute of Z . Then X is an extremallydisconnected compact space and χ ( x, X ) > τ ( πχ ( x, X ) > τ ) for each x ∈ X .By Theorem 1.1, X is not a τ -space. Indeed, it is sufficient to consider X \ S ,where S ∈ [ X ] ≤ τ . Example 2.14.
There are a compact non- a τ -space X and an irreduciblecontinuous mapping of X onto I τ .Proof. Let X = a I τ be the absolute of the space I τ . The natural map π I τ : X → I τ is an irreducible continuous surjection. Then, by Theorem 1.1, thespace X \ C where C ∈ [ X ] ω is not subcompact. Theorem 2.15.
Let X = L { X α : α ∈ Λ } S { ξ } where X α is a compactspace for each α ∈ Λ and a basic neighborhood of ξ contains of all but finitelymany sets X α . The space X is an (strongly) a τ -space if and only if X α is an(strongly) a τ -space for each α ∈ Λ .Proof. ( ⇒ ) Let X be a (strictly) a τ -space for some cardinal number τ , β ∈ Λand S ∈ [ X β ] ≤ τ . Since X is a (strictly) a τ -space, there is a condensation f : X \ S → Z from X \ S onto a compact space Z . Then the compact space X \ X β is homeomorphic to the compact space f ( X \ X β ). It follows that thespace f ( X β \ S ) = Z \ f ( X \ X β ) is locally compact and, by Parhomenko’sTheorem [23], it admits a condensation onto a compact space.( ⇐ ) Let X α be an (strictly) a τ -space for each α ∈ Λ and some cardinalnumber τ . Let S ∈ [ X ] ≤ τ . For each α ∈ Λ there exists a condensation f α : X α \ S → K α , where K α is compact.1. ξ ∈ S . Then X \ S admits a condensation f onto a locally compactspace Y = L { K α : α ∈ Λ } so that f ↾ ( X α \ S ) = f α for each α ∈ Λ. ByParhomenko’s Theorem [23], Y admits a condensation onto a compactum.2. ξ / ∈ S . Then X \ S admits a condensation f onto a compact space Z = L { K α : α ∈ Λ } L { p } so that f ↾ ( X α \ S ) = f α for each α ∈ Λ and f ( ξ ) = p , where a basic neighbourhood of the point p contains of all butfinitely many sets K α . 11 emark 2.16. The previous theorem is not true in the class of (strictly) a τ -subcompact spaces. In [27], it was considered a space X such that X L X admits a consensation onto a metrizable compact space (moreover, X L X is strictly a τ -subcompact), but X is not subcompact. Theorem 2.17.
Let X = L { X α : α ∈ Λ } S { ξ } , where X α is an (strictly) a τ -space for each α ∈ Λ , ξ / ∈ S { X α : α ∈ Λ } and X is Hausdorff. Then X is an (strictly) a τ -space.Proof. Note that the space X admits a condensation onto Y = L { X α : α ∈ Λ } S { ξ } , where a basic neighborhood of ξ contains of all but finitely manysets X α . By Theorem 2.15 and Proposition 2.1, the space Y is an (strictly) a τ -space.In [12] (Theorem 13), it is proved the following theorem. Theorem 2.18.
Let X = Q { X α : α < τ } be product of non-trivial com-pacta. Let f : X → Y be a continuous surjection such that the cardinality | f π − A π A x | > ℵ for any A ∈ [ τ ] <ω and x ∈ X . Then Y is a strictly a -space. An amplification of this theorem is the replacement of the condition | f π − A π A x | > ℵ with the condition f π − A π A x = f x . Theorem 2.19.
Let X = Q { X α : α < τ } be product of an infinite numberof non-trivial compacta. Let f : X → Y be a continuous surjection such that f π − A π A x = f x for any A ∈ [ τ ] <ω and x ∈ X . Then Y is a strictly a -space.Proof. Let x ∈ X and A ∈ [ τ ] <ω . We show that the set f π − A π A x is un-countable. On the contrary, suppose that f π − A π A x is countable. Thenthe compact space P = π − A π A x = S { π − A π A x ∩ f − y : y ∈ f π − A π A x } isthe sum of countable collection of compacta π − A π A x ∩ f − y . So there is y ∈ f π − A π A x for which Int P ( π − A π A x ∩ f − y ) = ∅ . There is a basic openset U = Q { U α : α ∈ B } × Q { X α : α ∈ τ \ B } where B is finite and U α is open for all α ∈ B . U ∩ π − A π A x = ∅ , f ( U ∩ π − A π A x ) = y . Then f π − A ∪ B π A ∪ B x ′ = y = f x ′ for any point x ′ ∈ U ∩ π − A π A x , which contradictsthe condition of the theorem. We proved that | f π − A π A x | > ℵ and, thus, allthe conditions of Theorem 2.18 are met. Corollary 2.20.
Let X be dyadic compact and χ ( x, X ) > m for each x ∈ X .Then X is a strictly a m -space. 12 roof. By Theorem 2.18 and Theorem 11 in [12].In ([12], Theorem 7), it is proved that a product X = Q { X α : α < τ } ofnon-trivial compacta X α is a strictly a τ -space.If we require that every X α be a subcompact space, we get the followingproposition. Proposition 2.21.
The product X = Q { X α : α < τ } of non-trivial subcom-pact spaces X α is strictly a τ -subcompact.Proof. Since X α admits a condensation onto a compact space K α for each α < τ , it is sufficient to note that X admits a condensation onto the compactspace K = Q { K α : α < τ } . By Theorem 7 in [12], K is a strictly a τ -space.By Proposition 2.1, X is a strictly a τ -subcompact space.In particular, we obtain that an infinite product of non-trivial subcom-pacts is a strictly a -subcompact space.Note that a continuous image of an infinite product of non-trivial sub-compact spaces may be not subcompact. Example 2.22.
Let X = N × Q { D α : α < ω } , where D α = { , } is discretefor each α < ω . Consider a condensation f : X → Y where Y = Q × Q { D α : α < ω } .The space Y can be represented in the form: Y = S {{ p } × C : p ∈ Q } where C = Q { D α : α < ω } is the Cantor cube. Thus, Y is the countablesum of compacta with an empty interior ( Y is a space of the first category)and, hence, Y cannot be condensed onto a compactum (a compact space isa space of the second category).In ([12], Theorem 5), it is proved that the product of a compact spaceand a metrizable compact space without isolated points is a strictly a - space. Proposition 2.23. If X admits a condensation onto a metrizable compactspace without isolated points and Y is subcompact, then X × Y is a strictly a -subcompact space.Proof. Let ϕ : X → X be a condensation from X onto a metrizable compactspace X without isolated points and suppose ψ : Y → Y is a condensationfrom Y onto a compact space Y . f : X × Y → X × Y where f ( x, y ) =13 ϕ ( x ) , ψ ( y )) is a condensation from X × Y onto a compact space X × Y without isolated points. By Theorem 5 in [12], X × Y is a strictly a -space.By Proposition 2.1, X × Y is a strictly a -subcompact space. Question 2.
Will the product of a (metrizable) compact space and astrictly a τ -subcompact space be a strictly a τ -subcompact space?In [12] (Proposition 4), it is proved the following result. Proposition 2.24.
There are ordered compacta X and Y such that X × Y is not an a -space. Question 3.
Suppose that a strictly a τ -space Z is represented as Z = X × Y , where τ ≥ ℵ . Will at least one of the multipliers be a strictly a τ -space? Acknowledgement
The authors would like to thank the referee for care-ful reading and valuable comments and suggestions. The work was performedas part of research conducted in the Ural Mathematical Center.
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