Hartman-Mycielski functor of non-metrizable compacta

HAR TMAN-MYCIELSKI FUNCTOR OF NON-METRIZABLE COMP A CT A T aras Radul and Du ˇ san Repo v ˇ s A bs tr act . W e inv estigate some topological prop erties of a normal functor H int ro- duced earlier by Radul which is a certain functorial compactification of the Hartman- Mycielski construction H M . W e show that H is op en and find the condition when H X is an absolute retract homeomorphic to the Tyc hono v cube. 1. Intr o duction The general theor y of functor s a c ting on the catego r y C omp of compact Hausdo rff spaces (compacta) and contin uous mapping s was founded by Shchepin [Sh2]. He describ ed so me elementary prop erties o f such functors and defined the notion of the normal functor whic h has b ecome v ery fruitful. The classes of all normal and weakly nor ma l f unctors include many cla ssical constructions: the h yp erspace exp , the space of probability measures P , the sup erextension λ , the space of h yp erspaces of inclusion G , and many other functor s (cf. [FZ] a nd [TZ]). Let X b e a space and d an admissible metric on X b ounded b y 1. By H M ( X ) we shall de no te the s pace of all maps from [0 , 1) to the space X s uch that f | [ t i , t i +1 ) ≡ const , for some 0 = t 0 ≤ · · · ≤ t n = 1, with r espec t to the following metric d H M ( f , g ) = Z 1 0 d ( f ( t ) , g ( t )) dt, f , g ∈ H M ( X ) . The construction of H M ( X ) is k no wn as the Ha rtman-Mycielski constructio n [HM]. F or every Z ∈ C omp consider H M n ( Z ) = n f ∈ H M ( Z ) | ther e exist 0 = t 1 < · · · < t n +1 = 1 with f | [ t i , t i +1 ) ≡ z i ∈ Z, i = 1 , . . . , n o . Let U b e the unique uniformity of Z . F o r every U ∈ U and ε > 0 , let < α, U, ε > = { β ∈ H M n ( Z ) | m { t ∈ [0 , 1 ) | ( α ( t ) , β ( t ′ )) / ∈ U } < ε } . The sets < α, U, ε > form a base o f a compact Ha usdorff topo logy in H M n Z . Given a map f : X → Y in C omp , define a map H M n X → H M n Y by the for m ula H M n F ( α ) = f ◦ α . Then H M n is a nor mal functor in C omp (cf. [TZ; 2.5.2]). 1991 Mathematics Subje ct Classific ation . 54B30, 57N20. Key wor ds and phr ases. Hartman-Mycielski construction, absolute retract, Ty khonov cub e, normal functor. The research wa s supp orted by the Slo ven ian-Ukr ainian grant SLO-UKR 06-07/04. Ty p eset by A M S -T E X 1 2 T ARAS RADUL A ND DU ˇ SAN R EPO V ˇ S F or X ∈ C omp we consider the space H M X with the top ology describ e d above. In general, H M X is not compact. Zarichn yi ha s a sked if there exists a normal functor in C omp which contains all functors H M n as subfunctors (see [TZ]). Such a functor H was co nstructed in [Ra]. It was shown in [RR] that H X is homeomorphic to the Hilb ert cub e for ea c h no n-degenerated metrizable compactum X . W e inv estigate s ome top olog ical prop erties of the spac e H X for no n- metrizable compacta X . The main r esults of this pap er a re: Theorem 1.1 . H f is op en if and only if f is an op en map. Theorem 1 .2. H X is an absolute r etr act if and only if X is op enly gener ate d c omp actu m of weight ≤ ω 1 . Theorem 1 .3. H X is home omorphi c to T ychonov cub e if and only if X is an op enly gener ate d χ -homo gene ous c omp actum of weight ω 1 . 2. Construction of H and i ts connection with the functor of probability m easures P Let X ∈ C omp . By C X we denote the Banach space of all co n tinuous functions ϕ : X → R with the usua l sup-nor m: k ϕ k = s up {| ϕ ( x ) | | x ∈ X } . W e denote the segment [0 , 1] by I . F or X ∈ C omp let us define the uniformity of H M X . F o r ea c h ϕ ∈ C ( X ) a nd a, b ∈ [0 , 1] with a < b we define the function ϕ ( a,b ) : H M X → R by the following formula ϕ ( a,b ) = 1 ( b − a ) Z b a ϕ ◦ α ( t ) dt. Define S H M ( X ) = { ϕ ( a,b ) | ϕ ∈ C ( X ) and ( a, b ) ⊂ [0 , 1) } . F or ϕ 1 , . . . , ϕ n ∈ S H M ( X ) define a pseudometric ρ ϕ 1 ,...,ϕ n on H M X by the formula ρ ϕ 1 ,...,ϕ n ( f , g ) = max {| ϕ i ( f ) − ϕ i ( g ) | | i ∈ { 1 , . . . , n }} , where f , g ∈ H M X . The family of pseudometrics P = { ρ ϕ 1 ,...,ϕ n | n ∈ N , where ϕ 1 , . . . , ϕ n ∈ S H M ( X ) } , defines a tota lly b ounded unifor mit y U H M X of H M X (see [Ra]). F or each compac tum X we consider the uniform space ( H X , U H X ) which is the completion of ( H M X , U H M X ) and the top ological space H X with the top olog y induced by the uniformity U H X . Since U H M X is totally bo unded, the spac e H X is compact. Let f : X → Y be a contin uous map. Define the map H M f : H M X → H M Y by the formula H M f ( α ) = f ◦ α , for all α ∈ H M X . It w as shown in [Ra] that the map H M f : ( H M X , U H M X ) → ( H M Y , U H M Y ) is uniformly co n tinuous. Hence there exists the co n tinuous map H f : H X → H Y such that H f | H M X = H M f . It is eas y to see that H : C omp → C omp is a cov ariant functor and H M n is a subfunctor o f H for ea c h n ∈ N . Let us remark that the family o f functions S H M ( X ) em b ed H M X in the product of closed interv als Q ϕ ( a,b ) ∈ S H M ( X ) I ϕ ( a,b ) where I ϕ ( a,b ) = [min x ∈ X | ϕ ( x ) | , ma x x ∈ X | ϕ ( x ) | ]. Thus, the space H X is the clos ure of the image of H M X . W e denote by HAR TM AN-MYCIELSKI FUNCTOR OF N ON-METRIZABLE COMP A CT A 3 p ϕ ( a,b ) : H X → I ϕ ( a,b ) the r estriction of the natura l pro jection. Let us remark that the function H f co uld be defined by the condition p ϕ ( a,b ) ◦ H f = p ( ϕ ◦ f ) ( a,b ) for each ϕ ( a,b ) ∈ S H M ( Y ). It is shown in [RR] that H X is a conv ex subset of Q ϕ ( a,b ) ∈ S H M ( X ) I ϕ ( a,b ) . Define the map e 1 : H M X × H M X × I → H M X by the condition tha t e 1 ( α 1 , α 2 , t )( l ) is equal to α 1 ( l ) if l < t and α 2 ( l ) in the o pposite case for α 1 , α 2 ∈ H M X , t ∈ I and l ∈ [0 , 1). W e co nsider H M X with the uniformity U H M X and I with the natura l metric. The map e 1 : H M X × H M X × I → H M X is unifor mly contin uous [RR]. Hence ther e exists the extension of e 1 to the con tinuous map e : H X × H X × I → H X . It is easy to chec k that e ( α, α, t ) = α , for each α ∈ H X . W e reca ll that P X is the space of all nonneg ativ e functionals µ : C ( X ) → R with nor m 1 , a nd taken in the weak* topo logy for a compactum X (see [TZ] or [FZ] for mor e details ). Recall that the base of the weak* top ology in P X consists of the sets of the form O ( µ 0 , f 1 , . . . , f n , ε ) = { µ ∈ P X | | µ ( f i ) − µ 0 ( f i ) | < ε for every 1 ≤ i ≤ n } . Hence we can consider P X as a subspace of the pro duct of closed int erv als Q ϕ ∈ C ( X ) I ϕ where I ϕ = [min x ∈ X | ϕ ( x ) | , ma x x ∈ X | ϕ ( x ) | ]. W e denote by π ϕ : P X → I ϕ the r estriction of the natural pro jection. F or each ( a, b ) ⊂ (0 , 1) w e can define a map r X ( a,b ) : H X → P X b y formula π ϕ ◦ rX ( a,b ) = p ϕ ( a,b ) . It is ea sy to chec k that r X ( a,b ) is well de fined, contin uous and affine map. W e define as well a map iX : P X → H X b y the for m ula p ϕ ( a,b ) ◦ iX = π ϕ . W e hav e that rX ( a,b ) ◦ iX = id P X , hence rX ( a,b ) is a retra ction fo r each ( a, b ) ⊂ (0 , 1). The map r X (0 , 1) we denote simply by r X . Let us r e mark that r ( a,b ) : H → P is a natura l transfor mation. (It means that for ea c h map f : X → Y we hav e P f ◦ rX ( a,b ) = r Y ( a,b ) ◦ H f .) The same pr oper t y is v alid for i : P → H . 3. Op eness of the functor H A subset A ⊂ H X is called e - c onvex if e ( α, β , t ) ∈ A , for ea ch α , β ∈ A and t ∈ I . If, a dditionally , A is conv ex, we call A H - c onvex . W e suppo se that f : X → Y is a contin uous surjective map b etw een co mpacta during this section. The pr o ofs of the next three lemmas ar e easy checking o n H M X which is a dense subset of H X . Lemma 3.1. F or e ach µ , ν ∈ H X and t ∈ [0 , 1] we have e ( H f ( µ ) , H f ( ν ) , t ) = H f ( e ( µ, ν , t )) . Lemma 3 .2. Consider any ν ∈ H X and a , b , c ∈ R such that 0 ≤ a < c < b ≤ 1 . Then we have p ϕ ( a,b ) ( ν ) = c − a b − a p ϕ ( a,c ) ( ν ) + b − c b − a p ϕ ( c,b ) ( ν ) for e ach ν ∈ H X . Lemma 3. 3. L et t ∈ (0 , 1) and ( a, b ) ⊂ (0 , 1) . F or e ach µ , ν ∈ H X 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. L et A b e a close d H -c onvex su bset of H X and ν / ∈ A . Th en ther e exist ϕ ∈ C ( X ) and ( a, b ) ⊂ (0 , 1) such t hat. p ϕ ( a,b ) ( ν ) < p ϕ ( a,b ) ( µ ) for e ach µ ∈ A . Pr o o f. Supp ose the contrary . W e can for each µ ∈ A cho ose ψ µ ∈ S H M ( X ) such that p ψ µ ( ν ) < p ψ µ ( µ ). Since A is compact, there exist µ 1 , . . . , µ n ∈ A such that for ea c h µ ∈ A ther e exists i ∈ { 1 , . . . , n } such that p ψ µ i ( ν ) < p ψ µ i ( µ ). By Lemma 3.2 we ca n choos e a family of in terv als { ( a i , b i ) } k i =1 such that b i ≤ a i +1 and fo r 4 T ARAS RADUL A ND DU ˇ SAN R EPO V ˇ S each i ∈ { 1 , . . . , k } a family of function ϕ 1 ( a i ,b i ) , . . . , ϕ n i ( a i ,b i ) ∈ S H M ( X ) s uc h that for eac h µ ∈ A there exist i ∈ { 1 , . . . , k } and l ∈ { 1 , . . . , n i } suc h that p ϕ l ( a i ,b i ) ( ν ) < p ϕ l ( a i ,b i ) ( µ ). Consider the set K = { µ ∈ A | p ϕ l ( a i ,b i ) ( µ ) ≤ p ϕ l ( a i ,b i ) ( ν ) for ea c h i ∈ { 2 , . . . , k } and l ∈ { 1 , . . . , n i }} . Then K is a compact co n vex subset o f A and for each µ ∈ A there exis ts l ∈ { 1 , . . . , n 1 } such that p ϕ l ( a 1 ,b 1 ) ( ν ) < p ϕ l ( a 1 ,b 1 ) ( µ ). Then r X ( a 1 ,b 1 ) ( K ) is a conv ex compact subs et of P X which do esn’t c on tain r X ( a 1 ,b 1 ) ( ν ). Then there exists ψ 1 ∈ C ( X ) such that π ψ 1 ( rX ( a 1 ,b 1 ) ( ν )) < π ψ 1 ( η ) for ea c h η ∈ rX ( a 1 ,b 1 ) ( K ). Hence, for ea c h µ ∈ K we hav e p ψ 1 ( a 1 ,b 1 ) ( ν ) < p ψ 1 ( a 1 ,b 1 ) ( µ ). Pro ceeding, we obta in ψ 1 , . . . , ψ k ∈ C ( X ) s uch tha t for ea c h µ ∈ A there ex ists i ∈ { 1 , . . . , k } such that p ψ i ( a i ,b i ) ( ν ) < p ψ i ( a i ,b i ) ( µ ). By o ur supp osition we can choose µ i ∈ A for each i ∈ { 1 , . . . , k } such that p ψ i ( a i ,b i ) ( µ i ) ≤ p ψ i ( a i ,b i ) ( ν ). Put ξ 1 = µ 1 and ξ i +1 = e ( ξ i , µ i +1 , b i ) for i ∈ { 1 , . . . , k − 1 } . Since A is e -co n vex, ξ k ∈ A . B y Lemma 3 .3 we hav e p ψ i ( a i ,b i ) ( ξ k ) ≤ p ψ i ( a i ,b i ) ( ν ) for ea c h i ∈ { 1 , . . . , k } . Th us, we obtain a contradiction and the lemma is pr ov ed. The pro of of the next lemma fo llo ws from Lemma 3.1 and the fact that H f is affine map. Lemma 3 .5. ( H f ) − 1 ( ν ) is H -c onvex for e ach ν ∈ H Y . Let f : X → Y be a ma p and ϕ ∈ C ( X ). By ϕ ∗ we denote the function ϕ ∗ : Y → R defined by the for m ula ϕ ∗ ( y ) = inf ( ϕ ( f − 1 ( y )), y ∈ Y . It is known [DE] that if f is o p en then the function ϕ ∗ is contin uous. Pr o o f of The or em 1.1. Let f : X → Y b e a map such that the map H f : H X → H Y is op en. Let us show that the map P f is op e n. Consider any op en set U ⊂ P X and µ ∈ U . Then ( rX ) − 1 ( U ) is an o p en set in H X a nd iX ( µ ) ∈ ( r X ) − 1 ( U ) Since H f is an op en map, H f (( rX ) − 1 ( U )) is op en in H Y and H f ( iX ( µ )) ∈ H f (( r X ) − 1 ( U )). Since r is a na tural tra nsformation, we hav e H f (( r X ) − 1 ( U )) ⊂ ( rY ) − 1 ( P f ( U )). W e hav e iY ( P f ( µ )) = H f ( iX ( µ )) or P f ( µ ) ∈ ( iY ) − 1 ( H f (( r X ) − 1 ( U ))) ⊂ ( iY ) − 1 (( rY ) − 1 ( P f ( U ))) = P f ( U ). Since ( iY ) − 1 ( H f (( r X ) − 1 ( U ))) is op en, the map P f is open. Hence f is op en a s well [DE]. Now let a map f : X → Y b e op en. Let us supp ose that H f is not op en. Then there exists µ 0 ∈ H X , a net { ν α , α ∈ A} ⊂ O ( Y ) conv erging to ν 0 = H f ( µ 0 ) a nd a neighborho o d W o f µ 0 such that ( H f ) − 1 ( ν α ) ∩ W = ∅ for each α ∈ A . Since H M ( Y ) is a dense subset of H Y , w e can supp ose that all ν α ∈ H M ( Y ) . Since H X is a compactum, we can assume that the net A α = ( H f ) − 1 ( ν α ) co n verges in exp( H X ) to some c lo sed subset A ⊂ H X . It is easy to c heck that A ⊂ ( H f ) − 1 ( ν 0 ) and µ 0 / ∈ A . By the Lemma 3.5 all the s ets A α are H -conv ex. It is easy to s ee that A is H -conv ex a s well. Since µ 0 / ∈ A , there exists by Lemma 3.4 ϕ ∈ C ( X ) and ( a, b ) ⊂ (0 , 1 ) such that p ϕ ( a,b ) ( µ 0 ) < p ϕ ( a,b ) ( µ ) for each µ ∈ A . Consider any α ∈ A . Let { y 1 , . . . , y s } = ν α ([0 , 1)). Choo se for each y i the point x i such that f ( x i ) = y i and ϕ ( x i ) = ϕ ∗ ( y i ). Define a map j : { y 1 , . . . , y s } → { x 1 , . . . , x s } by the formula j ( y i ) = x i and put µ α ( t ) = j ◦ ν α ( t ) for t ∈ [0 , 1). Let µ b e a limit p o in t of the net µ α , then µ ∈ A . Since ϕ ( a,b ) ( µ α ) = ϕ ∗ ( a,b ) ( ν α ), w e hav e p ϕ ( a,b ) ( µ ) = p ϕ ∗ ( a,b ) ( ν 0 ) = p ( ϕ ∗ ◦ f ) ( a,b ) ( µ 0 ) ≤ p ϕ ( a,b ) ( µ 0 ). W e ha ve obtained the contradiction and the theor em is pr o ved. HAR TM AN-MYCIELSKI FUNCTOR OF N ON-METRIZABLE COMP A CT A 5 4. Pro ofs W e will need so me notations and facts from the theo r y of non-metrizable com- pacta. See [10] for mo re details. Let τ b e an infinite cardinal num ber. A partially o rdered set A is ca lled τ - c omplete , if every subs e t of cardinality ≤ τ has a least upp er b ound in A . An in- verse system consisting of compacta and surjective b onding maps ov er a τ -complete indexing set is called τ -complete. A con tinuous τ -complete system consisting of compacta of weight ≤ τ is called a τ - system . As usual, by ω we denote the co un table cardinal num b er. A co mpactum X is ca lle d op enly gener ate d if X ca n b e repr esen ted as the limit of an ω - s ystem with op en b onding maps. Pr o o f of The or em 1.2. It was shown in [RR] that H X is an absolute retract for each metrizable co mpactum X . So , we can consider only non-metriza ble case. Let X b e an op enly gener ated compactum of weight ≤ ω 1 . By Theor em 1.1 the compactum H X is a lso op enly generated. Since weigh t of X (and H X [Ra]) is ≤ ω 1 , H X is AE (0). Since H X is a conv ex compa ctum, H X is AR [F e]. Now, supp ose H X ∈ AR . Since r X : H X → P X ia a retraction, P X is an AR to o. Then X is an op enly generated compactum of weigh t ≤ ω 1 [F e]. The theorem is proved. By w ( X ) we denote the weigh t of the co mpactum X , by χ ( x, X ) the character in the po in t x and by χ ( X ) the ch ar acter of the s pace X . The space X is called χ - homo gene ou s if for eac h x, y ∈ X we have χ ( x, X ) = χ ( y , X ). W e will use the following characterization of the Tyc honov cub e I τ . An AR -compa ctum X o f weigh t τ is homeo morphic to I τ for a n uncountable cardinal num ber τ if and only if X is χ - homogeneous [Sh1]. Let x ∈ X . Define δ ( x ) ∈ H X by the condition p ϕ ( a,b ) ( δ ( x )) = ϕ ( x ) for each ϕ ( a,b ) ∈ S H M ( X ). Lemma 4.1. L et f : X → Y b e an op en map. Then H f has a de gener ate fib er if and only if f has a de gener ate fib er. Pr o o f. Let f : X → Y be a n open map such that ther e exists y ∈ Y with f − 1 ( y ) = { x } , x ∈ X . Consider any µ ∈ H X with H f ( µ ) = δ ( y ). Let us show that µ = δ ( x ). Consider a n y ϕ ( a,b ) ∈ S H M ( X ). Supp ose that p ϕ ( a,b ) ( µ ) 6 = ϕ ( x ). W e can assume that p ϕ ( a,b ) ( µ ) < ϕ ( x ). By [Ra , Lemma 1 ] there exists a function ψ ∈ C ( Y ) such that ψ ( y ) = ϕ ( x ) and ψ ◦ f ≤ ϕ . Then we hav e 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 obta in the co n tradiction. Thus, H f has a degenerate fib er. Now, s upp ose f has no degener ate fib e r . Consider a n y µ ∈ H Y . T ake any y ∈ s upp µ ⊂ Y . Since f is an op en map and f − 1 ( y ) is not a singleton, w e can choo s e t wo closed subsets A 1 , A 2 ⊂ X such that f ( A 1 ) = f ( A 2 ) = Y and ( A 1 ∩ f − 1 ( y )) ∩ ( A 2 ∩ f − 1 ( y )) = ∅ . Since the functor H pr eserves surjective maps [Ra], there exist µ 1 ∈ H ( A 1 ) and µ 2 ∈ H ( A 2 ) such that H f ( µ 1 ) = H f ( µ 2 ) = µ . Since y ∈ supp µ , there exist y 1 ∈ supp µ 1 ⊂ A 1 and y 2 ∈ supp µ 2 ⊂ A 2 such that f ( y 1 ) = f ( y 2 ) = y . Hence µ 1 6 = µ 2 and the lemma is prov ed. Lemma 4.2. An op enly gener ate d c omp actu m X of weight ω 1 is χ -homogene ous if and only if H X is χ -homogene ous. Pr o o f. Let H X b e χ -homog e neous. Since the functor H preser v es the w eight [Ra ], H X is a n abso lute retr act such that χ ( µ, H X ) = ω 1 for each µ ∈ H X . T hen take 6 T ARAS RADUL A ND DU ˇ SAN R EPO V ˇ S any x ∈ X a nd supp ose that there exists { U i | i ∈ N } a countable base of op en neighborho o ds. Cons ider 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 neighbor ho o ds of δ ( x ) in H X . W e obtain a co n tradiction, hence X is χ -homogene o us of X . Now let X be a χ - homogeneous op enly g enerated compactum o f w eight ω 1 . Then χ ( X ) = ω 1 [Ra, Le mma 4]. Suppose that there exis ts a p oin t ν ∈ H X suc h that χ ( ν, H X ) < ω 1 . Represent X as the limit spac e of an ω -system { X α , p α , A} w ith op en limit pr o jectio ns p α . Ther e exists α ∈ A such that ( H p α ) − 1 ( H p α ( ν )) = { ν } . By Lemma 4.2 there exists a p oint z ∈ X α such that p − 1 α ( z ) = { x } , x ∈ X . Hence χ ( x, X ) < ω 1 and we obtain the contradiction. The le mma is prov ed. Pr o o f of The or em 1.3. The pro of o f the theorem follows fr om Theorem 1.2 and Lemma 4.2 . References [DE] S. Ditor and L. Eifler, Some op en mapping the or ems for mea sur es , T rans. Amer. Math. Soc. 164 (1972), 287–293. [F e] V. V. F edorch uk, Pr ob ability me asur es in top olo gy , Uspekhi Mat. Nauk 46 (1991), 41–80. (Russian) [FZ] V. V. F edorch uk and M. M . Zarichn yi, Covariant funct ors in c ate gories of top olo gi- c al sp a c es , Results of Science and T ec hnology , Algebra.T op ology .Geometry 28 , VINITI, Mosco w, pp. 47–95. (Russian) [HM] S. Hartman and J. M ycielski, On the emb e dding of top olo gic al gr oups into c onne cte d top olo gica l g ro ups , Colloq. M ath. 5 (1958), 167–169. [Ra] T. Radul, A norma l functor b ase d on the Hartman-Mycielski c onstruction , Mat. Studii 19 (2003), 201–207. [RR] T. Radul and D. Rep o v ˇ s, On top olo gic al pr op ert ies of the Hartman-Mycielski functor , Pro c. Indian Acad. Sci. Math. Sci. 1 15 (2005), 477–482. [Sh1] E. V. Shc hepin, On T ychono v manifolds , Dokl.A k ad.Nauk USSR 246 (1979), 551–554. (Russian) [Sh2] E. V. Shc hepin, F u nctors and unc ountable p owers of co mp acta , Usp ekhi Mat. N auk 36 (1981), 3–62. (Russian) [TZ] A. T elejk o and M. Zarichn yi, Cate goric al T op olo gy of Comp act Hausdorff Sp ac es , Lviv, VNTL, 1999, p. 263. D epar tm en t of M e ch an ics an d M a t hem a t ic s, L vi v N atio na l U niv er sit y, U n ive r- sy te ts ka st .,1 , 7 9 6 02 L vi v, U k ra ine . E-mail addr ess : tarasradul@yah oo.co.uk In st it ut e F or M atem a t ics , P hy si cs a nd M e ch an ics , an d U ni ver si t y O f L ju blj an a, P .O .B ox 2 96 4 , Lj ub lja na , S l ove nia 1 0 01 E-mail addr ess : dusan.repovs@g uest.arnes.si