Hartman-Mycielski functor of non-metrizable compacta
aa r X i v : . [ m a t h . GN ] A p r HARTMAN-MYCIELSKI FUNCTOROF NON-METRIZABLE COMPACTA
Taras Radul and Duˇsan Repovˇs
Abstract.
We investigate some topological properties of a normal functor H intro-duced earlier by Radul which is a certain functorial compactification of the Hartman-Mycielski construction HM . We show that H is open and find the condition when HX is an absolute retract homeomorphic to the Tychonov cube.
1. Introduction
The general theory of functors acting on the category C omp of compact Hausdorffspaces (compacta) and continuous mappings was founded by Shchepin [Sh2]. Hedescribed some elementary properties of such functors and defined the notion ofthe normal functor which has become very fruitful. The classes of all normal andweakly normal functors include many classical constructions: the hyperspace exp ,the space of probability measures P , the superextension λ , the space of hyperspacesof inclusion G , and many other functors (cf. [FZ] and [TZ]).Let X be a space and d an admissible metric on X bounded by 1. By HM ( X ) weshall denote the space of all maps from [0 ,
1) to the space X such that f | [ t i , t i +1 ) ≡ const , for some 0 = t ≤ · · · ≤ t n = 1, with respect to the following metric d HM ( f, g ) = Z d ( f ( t ) , g ( t )) dt, f, g ∈ HM ( X ) . The construction of HM ( X ) is known as the Hartman-Mycielski construction [HM].For every Z ∈ C omp consider HM n ( Z ) = n f ∈ HM ( Z ) | there exist 0 = t < · · · < t n +1 = 1with f | [ t i , t i +1 ) ≡ z i ∈ Z, i = 1 , . . . , n o . Let U be the unique uniformity of Z . For every U ∈ U and ε >
0, let < α, U, ε > = { β ∈ HM n ( Z ) | m { t ∈ [0 , | ( α ( t ) , β ( t ′ )) / ∈ U } < ε } . The sets < α, U, ε > form a base of a compact Hausdorff topology in HM n Z .Given a map f : X → Y in C omp , define a map HM n X → HM n Y by the formula HM n F ( α ) = f ◦ α . Then HM n is a normal functor in C omp (cf. [TZ; 2.5.2]). Mathematics Subject Classification . 54B30, 57N20.
Key words and phrases.
Hartman-Mycielski construction, absolute retract, Tykhonov cube,normal functor.The research was supported by the Slovenian-Ukrainian grant SLO-UKR 06-07/04.Typeset by
AMS -TEX TARAS RADUL AND DUˇSAN REPOVˇS
For X ∈ C omp we consider the space HM X with the topology described above.In general,
HM X is not compact. Zarichnyi has asked if there exists a normalfunctor in C omp which contains all functors HM n as subfunctors (see [TZ]). Such afunctor H was constructed in [Ra]. It was shown in [RR] that HX is homeomorphicto the Hilbert cube for each non-degenerated metrizable compactum X .We investigate some topological properties of the space HX for non-metrizablecompacta X . The main results of this paper are: Theorem 1.1. Hf is open if and only if f is an open map. Theorem 1.2. HX is an absolute retract if and only if X is openly generatedcompactum of weight ≤ ω . Theorem 1.3. HX is homeomorphic to Tychonov cube if and only if X is anopenly generated χ -homogeneous compactum of weight ω .
2. Construction of H and its connection with the functor ofprobability measures P Let X ∈ C omp . By CX we denote the Banach space of all continuous functions ϕ : X → R with the usual sup-norm: k ϕ k = sup {| ϕ ( x ) | | x ∈ X } . We denote thesegment [0 ,
1] by I .For X ∈ C omp let us define the uniformity of HM X . For each ϕ ∈ C ( X ) and a, b ∈ [0 ,
1] with a < b we define the function ϕ ( a,b ) : HM X → R by the followingformula ϕ ( a,b ) = 1( b − a ) Z ba ϕ ◦ α ( t ) dt. Define S HM ( X ) = { ϕ ( a,b ) | ϕ ∈ C ( X ) and ( a, b ) ⊂ [0 , } . For ϕ , . . . , ϕ n ∈ S HM ( X ) define a pseudometric ρ ϕ ,...,ϕ n on HM X by theformula ρ ϕ ,...,ϕ n ( f, g ) = max {| ϕ i ( f ) − ϕ i ( g ) | | i ∈ { , . . . , n }} , where f, g ∈ HM X . The family of pseudometrics P = { ρ ϕ ,...,ϕ n | n ∈ N , where ϕ , . . . , ϕ n ∈ S HM ( X ) } , defines a totally bounded uniformity U HMX of HM X (see [Ra]).For each compactum X we consider the uniform space ( HX, U HX ) which is thecompletion of ( HM X, U HMX ) and the topological space HX with the topologyinduced by the uniformity U HX . Since U HMX is totally bounded, the space HX iscompact.Let f : X → Y be a continuous map. Define the map HM f : HM X → HM Y by the formula
HM f ( α ) = f ◦ α , for all α ∈ HM X . It was shown in [Ra] that themap
HM f : (
HM X, U HMX ) → ( HM Y, U HMY ) is uniformly continuous. Hencethere exists the continuous map Hf : HX → HY such that Hf | HM X = HM f .It is easy to see that H : C omp → C omp is a covariant functor and HM n is asubfunctor of H for each n ∈ N .Let us remark that the family of functions S HM ( X ) embed HM X in the productof closed intervals Q ϕ ( a,b ) ∈ S HM ( X ) I ϕ ( a,b ) where I ϕ ( a,b ) = [min x ∈ X | ϕ ( x ) | , max x ∈ X | ϕ ( x ) | ]. Thus, the space HX is the closure of the image of HM X . We denote by
ARTMAN-MYCIELSKI FUNCTOR OF NON-METRIZABLE COMPACTA 3 p ϕ ( a,b ) : HX → I ϕ ( a,b ) the restriction of the natural projection. Let us remark thatthe function Hf could be defined by the condition p ϕ ( a,b ) ◦ Hf = p ( ϕ ◦ f ) ( a,b ) for each ϕ ( a,b ) ∈ S HM ( Y ).It is shown in [RR] that HX is a convex subset of Q ϕ ( a,b ) ∈ S HM ( X ) I ϕ ( a,b ) .Define the map e : HM X × HM X × I → HM X by the condition that e ( α , α , t )( l ) is equal to α ( l ) if l < t and α ( l ) in the opposite case for α , α ∈ HM X , t ∈ I and l ∈ [0 , HM X with the uniformity U HMX and I with the natural metric. The map e : HM X × HM X × I → HM X is uniformlycontinuous [RR].Hence there exists the extension of e to the continuous map e : HX × HX × I → HX . It is easy to check that e ( α, α, t ) = α , for each α ∈ HX .We recall that P X is the space of all nonnegative functionals µ : C ( X ) → R with norm 1, and taken in the weak* topology for a compactum X (see [TZ] or[FZ] for more details). Recall that the base of the weak* topology in P X consistsof the sets of the form O ( µ , f , . . . , f n , ε ) = { µ ∈ P X | | µ ( f i ) − µ ( f i ) | < ε forevery 1 ≤ i ≤ n } . Hence we can consider
P X as a subspace of the product of closedintervals Q ϕ ∈ C ( X ) I ϕ where I ϕ = [min x ∈ X | ϕ ( x ) | , max x ∈ X | ϕ ( x ) | ]. We denote by π ϕ : P X → I ϕ the restriction of the natural projection.For each ( a, b ) ⊂ (0 ,
1) we can define a map rX ( a,b ) : HX → P X by formula π ϕ ◦ rX ( a,b ) = p ϕ ( a,b ) . It is easy to check that rX ( a,b ) is well defined, continuousand affine map.We define as well a map iX : P X → HX by the formula p ϕ ( a,b ) ◦ iX = π ϕ . Wehave that rX ( a,b ) ◦ iX = id P X , hence rX ( a,b ) is a retraction for each ( a, b ) ⊂ (0 , rX (0 , we denote simply by rX .Let us remark that r ( a,b ) : H → P is a natural transformation. (It means thatfor each map f : X → Y we have P f ◦ rX ( a,b ) = rY ( a,b ) ◦ Hf .) The same propertyis valid for i : P → H .
3. Openess of the functor H A subset A ⊂ HX is called e - convex if e ( α, β, t ) ∈ A , for each α , β ∈ A and t ∈ I . If, additionally, A is convex, we call A H - convex .We suppose that f : X → Y is a continuous surjective map between compactaduring this section. The proofs of the next three lemmas are easy checking on HM X which is a dense subset of HX . Lemma 3.1.
For each µ , ν ∈ HX and t ∈ [0 , we have e ( Hf ( µ ) , Hf ( ν ) , t ) = Hf ( e ( µ, ν, t )) . Lemma 3.2.
Consider any ν ∈ HX and a , b , c ∈ R such that ≤ a < c < b ≤ .Then we have p ϕ ( a,b ) ( ν ) = c − ab − a p ϕ ( a,c ) ( ν ) + b − cb − a p ϕ ( c,b ) ( ν ) for each ν ∈ HX . Lemma 3.3.
Let t ∈ (0 , and ( a, b ) ⊂ (0 , . For each µ , ν ∈ HX and ϕ ∈ C ( X ) we have p ϕ ( a,b ) ( e ( µ, ν, t )) = p ϕ ( a,b ) ( µ ) if b ≤ t and p ϕ ( a,b ) ( e ( µ, ν, t )) = p ϕ ( a,b ) ( ν ) if t ≤ a . Lemma 3.4.
Let A be a closed H -convex subset of HX and ν / ∈ A . Then thereexist ϕ ∈ C ( X ) and ( a, b ) ⊂ (0 , such that. p ϕ ( a,b ) ( ν ) < p ϕ ( a,b ) ( µ ) for each µ ∈ A .Proof. Suppose the contrary. We can for each µ ∈ A choose ψ µ ∈ S HM ( X ) suchthat p ψ µ ( ν ) < p ψ µ ( µ ). Since A is compact, there exist µ , . . . , µ n ∈ A such thatfor each µ ∈ A there exists i ∈ { , . . . , n } such that p ψ µi ( ν ) < p ψ µi ( µ ). By Lemma3.2 we can choose a family of intervals { ( a i , b i ) } ki =1 such that b i ≤ a i +1 and for TARAS RADUL AND DUˇSAN REPOVˇS each i ∈ { , . . . , k } a family of function ϕ a i ,b i ) , . . . , ϕ n i ( a i ,b i ) ∈ S HM ( X ) such thatfor each µ ∈ A there exist i ∈ { , . . . , k } and l ∈ { , . . . , n i } such that p ϕ l ( ai,bi ) ( ν )
Let f : X → Y be a map such that the map Hf : HX → HY is open. Let us show that the map P f is open. Consider anyopen set U ⊂ P X and µ ∈ U . Then ( rX ) − ( U ) is an open set in HX and iX ( µ ) ∈ ( rX ) − ( U ) Since Hf is an open map, Hf (( rX ) − ( U )) is open in HY and Hf ( iX ( µ )) ∈ Hf (( rX ) − ( U )). Since r is a natural transformation, we have Hf (( rX ) − ( U )) ⊂ ( rY ) − ( P f ( U )). We have iY ( P f ( µ )) = Hf ( iX ( µ )) or P f ( µ ) ∈ ( iY ) − ( Hf (( rX ) − ( U ))) ⊂ ( iY ) − (( rY ) − ( P f ( U ))) = P f ( U ). Since( iY ) − ( Hf (( rX ) − ( U ))) is open, the map P f is open. Hence f is open as well[DE].Now let a map f : X → Y be open. Let us suppose that Hf is not open. Thenthere exists µ ∈ HX , a net { ν α , α ∈ A} ⊂ O ( Y ) converging to ν = Hf ( µ ) anda neighborhood W of µ such that ( Hf ) − ( ν α ) ∩ W = ∅ for each α ∈ A . Since HM ( Y ) is a dense subset of HY , we can suppose that all ν α ∈ HM ( Y ) . Since HX is a compactum, we can assume that the net A α = ( Hf ) − ( ν α ) converges inexp( HX ) to some closed subset A ⊂ HX . It is easy to check that A ⊂ ( Hf ) − ( ν )and µ / ∈ A . By the Lemma 3.5 all the sets A α are H -convex. It is easy to seethat A is H -convex as well. Since µ / ∈ A , there exists by Lemma 3.4 ϕ ∈ C ( X )and ( a, b ) ⊂ (0 ,
1) such that p ϕ ( a,b ) ( µ ) < p ϕ ( a,b ) ( µ ) for each µ ∈ A . Considerany α ∈ A . Let { y , . . . , y s } = ν α ([0 , y i the point x i suchthat f ( x i ) = y i and ϕ ( x i ) = ϕ ∗ ( y i ). Define a map j : { y , . . . , y s } → { x , . . . , x s } by the formula j ( y i ) = x i and put µ α ( t ) = j ◦ ν α ( t ) for t ∈ [0 , µ be alimit point of the net µ α , then µ ∈ A . Since ϕ ( a,b ) ( µ α ) = ϕ ∗ ( a,b ) ( ν α ), we have p ϕ ( a,b ) ( µ ) = p ϕ ∗ ( a,b ) ( ν ) = p ( ϕ ∗ ◦ f ) ( a,b ) ( µ ) ≤ p ϕ ( a,b ) ( µ ). We have obtained thecontradiction and the theorem is proved. ARTMAN-MYCIELSKI FUNCTOR OF NON-METRIZABLE COMPACTA 5
4. Proofs
We will need some notations and facts from the theory of non-metrizable com-pacta. See [10] for more details.Let τ be an infinite cardinal number. A partially ordered set A is called τ - complete , if every subset of cardinality ≤ τ has a least upper bound in A . An in-verse system consisting of compacta and surjective bonding maps over a τ -completeindexing set is called τ -complete. A continuous τ -complete system consisting ofcompacta of weight ≤ τ is called a τ - system .As usual, by ω we denote the countable cardinal number.A compactum X is called openly generated if X can be represented as the limitof an ω -system with open bonding maps. Proof of Theorem 1.2.
It was shown in [RR] that HX is an absolute retract foreach metrizable compactum X . So, we can consider only non-metrizable case.Let X be an openly generated compactum of weight ≤ ω . By Theorem 1.1 thecompactum HX is also openly generated. Since weight of X (and HX [Ra]) is ≤ ω , HX is AE (0). Since HX is a convex compactum, HX is AR [Fe].Now, suppose HX ∈ AR . Since rX : HX → P X ia a retraction,
P X is an AR too. Then X is an openly generated compactum of weight ≤ ω [Fe]. The theoremis proved.By w ( X ) we denote the weight of the compactum X , by χ ( x, X ) the characterin the point x and by χ ( X ) the character of the space X . The space X is called χ - homogeneous if for each x, y ∈ X we have χ ( x, X ) = χ ( y, X ). We will usethe following characterization of the Tychonov cube I τ . An AR -compactum X ofweight τ is homeomorphic to I τ for an uncountable cardinal number τ if and onlyif X is χ -homogeneous [Sh1].Let x ∈ X . Define δ ( x ) ∈ HX by the condition p ϕ ( a,b ) ( δ ( x )) = ϕ ( x ) for each ϕ ( a,b ) ∈ S HM ( X ). Lemma 4.1.
Let f : X → Y be an open map. Then Hf has a degenerate fiber ifand only if f has a degenerate fiber.Proof. Let f : X → Y be an open map such that there exists y ∈ Y with f − ( y ) = { x } , x ∈ X . Consider any µ ∈ HX with Hf ( µ ) = δ ( y ). Let us show that µ = δ ( x ).Consider any ϕ ( a,b ) ∈ S HM ( X ). Suppose that p ϕ ( a,b ) ( µ ) = ϕ ( x ). We can assumethat p ϕ ( a,b ) ( µ ) < ϕ ( x ). By [Ra, Lemma 1] there exists a function ψ ∈ C ( Y ) suchthat ψ ( y ) = ϕ ( x ) and ψ ◦ f ≤ ϕ . Then we have p ( ψ ◦ f ) ( a,b ) ( µ ) ≤ p ϕ ( a,b ) ( µ ) < ϕ ( x )and p ψ ( a,b ) ( δ ( y )) = p ψ ( a,b ) O ( f )( µ ) = p ( ψ ◦ f ) ( a,b ) ( µ ) < ϕ ( x ) = ψ ( y ). Hence we obtainthe contradiction. Thus, Hf has a degenerate fiber.Now, suppose f has no degenerate fiber. Consider any µ ∈ HY . Take any y ∈ supp µ ⊂ Y . Since f is an open map and f − ( y ) is not a singleton, wecan choose two closed subsets A , A ⊂ X such that f ( A ) = f ( A ) = Y and( A ∩ f − ( y )) ∩ ( A ∩ f − ( y )) = ∅ . Since the functor H preserves surjective maps[Ra], there exist µ ∈ H ( A ) and µ ∈ H ( A ) such that Hf ( µ ) = Hf ( µ ) = µ .Since y ∈ supp µ , there exist y ∈ supp µ ⊂ A and y ∈ supp µ ⊂ A such that f ( y ) = f ( y ) = y . Hence µ = µ and the lemma is proved. Lemma 4.2.
An openly generated compactum X of weight ω is χ -homogeneousif and only if HX is χ -homogeneous.Proof. Let HX be χ -homogeneous. Since the functor H preserves the weight [Ra], HX is an absolute retract such that χ ( µ, HX ) = ω for each µ ∈ HX . Then take TARAS RADUL AND DUˇSAN REPOVˇS any x ∈ X and suppose that there exists { U i | i ∈ N } a countable base of openneighborhoods. Consider a family of functions { ϕ i ∈ C ( X ) | i ∈ N } such that ϕ i ( x ) = 1, ϕ i | X \ U ⊂ = 0. Then the family of function { ϕ i ( a,b ) | i ∈ N ; a, b ∈ Q } define a countable base of neighborhoods of δ ( x ) in HX . We obtain a contradiction,hence X is χ -homogeneous of X .Now let X be a χ -homogeneous openly generated compactum of weight ω . Then χ ( X ) = ω [Ra, Lemma 4]. Suppose that there exists a point ν ∈ HX such that χ ( ν, HX ) < ω . Represent X as the limit space of an ω -system { X α , p α , A} withopen limit projections p α . There exists α ∈ A such that ( Hp α ) − ( Hp α ( ν )) = { ν } .By Lemma 4.2 there exists a point z ∈ X α such that p − α ( z ) = { x } , x ∈ X . Hence χ ( x, X ) < ω and we obtain the contradiction. The lemma is proved. Proof of Theorem 1.3.
The proof of the theorem follows from Theorem 1.2 andLemma 4.2.
References [DE] S. Ditor and L. Eifler,
Some open mapping theorems for measures , Trans. Amer. Math.Soc. (1972), 287–293.[Fe] V. V. Fedorchuk,
Probability measures in topology , Uspekhi Mat. Nauk (1991), 41–80.(Russian)[FZ] V. V. Fedorchuk and M. M. Zarichnyi, Covariant functors in categories of topologi-cal spaces , Results of Science and Technology, Algebra.Topology.Geometry , VINITI,Moscow, pp. 47–95. (Russian)[HM] S. Hartman and J. Mycielski, On the embedding of topological groups into connectedtopological groups , Colloq. Math. (1958), 167–169.[Ra] T. Radul, A normal functor based on the Hartman-Mycielski construction , Mat. Studii (2003), 201–207.[RR] T. Radul and D. Repovˇs, On topological properties of the Hartman-Mycielski functor ,Proc. Indian Acad. Sci. Math. Sci. (2005), 477–482.[Sh1] E. V. Shchepin,
On Tychonov manifolds , Dokl.Akad.Nauk USSR (1979), 551–554.(Russian)[Sh2] E. V. Shchepin,
Functors and uncountable powers of compacta , Uspekhi Mat. Nauk (1981), 3–62. (Russian)[TZ] A. Telejko and M. Zarichnyi, Categorical Topology of Compact Hausdorff Spaces , Lviv,VNTL, 1999, p. 263.
Department of Mechanics and Mathematics, Lviv National University, Univer-sytetska st.,1, 79602 Lviv, Ukraine.
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