math.GN 2011-10-06 0

Non-separable h-homogeneous absolute F_{sigma delta}-spaces and G_{delta sigma}-spaces

Denote by Q(k) a \sigma-discrete metric weight-homogeneous space of weight k. We give an internal description of the space Q(k)^\omega. We prove that the Baire space B(k) is densely homogeneous with respect to Q(k)^\omega if k > \omega. Properties of…

Authors: Sergey Medvedev

NON-SEP ARABLE h -HOMOGENEOUS ABSOLUTE F σδ -SP A CES AND G δσ -SP A CES SERGEY MEDV E DEV Abstract. Denote by Q ( k ) a σ -discrete metric weigh t-homoge neous space o f w eight k . W e give an internal des c ription of the space Q ( k ) ω . W e prove that the Baire space B ( k ) is densely homogeneous with resp ect to Q ( k ) ω if k > ω . Prop erties of some non-separ able h -homogeneous absolute F σδ -sets and G δσ -sets are inv estig ated. Intr oduction All to p ological spaces under discussion are metrizable and strongly zero-dimensional (i.e., Ind X = 0). The aim of this pap er is to c haracterize the space Q ( k ) ω for k > ω (see Theorem 2.2). Com bining this result with g eneral theorems ab out h -homogeneous spaces, we describe some non-separable h - homogeneous absolute F σδ -sets and G δσ -sets. W e deal mainly with the no n- separable case, b ecause the similar theorems for separable spaces are alr eady obtained. Ab out one h undred y ears ago, an arithmetical example of a strictly F σδ -subset of the space N of irrationals w as constructed b y Baire [Ba]. Let us recall his metho d. F or ev ery n -tuple t ∈ ω n , where n ∈ ω , tak e a p erfect subset X t of N . Supp o se, for a ll t ∈ ω n and i ∈ ω , X t ˆ i is a nowhe re dense closed subset of X t and the union ∪{ X t ˆ i : i ∈ ω } is dense in X t . Put E n = ∪{ X t : t ∈ ω n } . Then the in tersection E = ∩{ E n : n ∈ ω } is a strictly F σδ -subset of N (see [Ba]). The set E , whic h is obta ined in suc h a w ay , is called a c anonic al eleme nt of class 3 . This notion w as in tro duced b y Lusin. Keldysh (see [K1], [Ke]) show ed that any tw o canonical elemen ts of class 3 are homeomorphic. Clearly , ev ery canonical elemen t o f class 3 is a space of first category . Sierpinski observ ed that the prop ert y of b eing an absolute F σδ -set is in ternal. Consider a separable space X . Let { X t : t ∈ ω n , n ∈ ω } b e a family consisting o f closed subsets of X suc h that X = ∪{ X i : i ∈ ω } and ev ery X t = ∪{ X t ˆ i : i ∈ ω } . Suppose, for eve ry χ ∈ ω ω and an arbitrary p oint x n ∈ X χ ↾ n , the sequence { x n : n ∈ ω } con v erges t o a p o in t x ∈ X . Sierpinski [Si] prov ed that under these conditions the space X is an absolute F σδ -set. In a sense, the Sierpinski result is a con v erse o f the Baire theorem. Using the Sierpinski metho d, v an Engelen [E1] obtained a top ological c haracterization of the space Q ω , where Q is t he space o f rationals. Namely , he show ed that ev ery separable absolute F σδ -set of first catego ry , whic h is no where a bsolute G δσ , is homeomorphic t o Q ω . This 2010 Mathematics Su bje ct Classific ation. Primary 54H05, 5 4F65; Sec ondary 5 4E52 , 03E15. Key wor ds and phr ases. h - homogeneous spa ce, se t o f firs t ca tegory , F σδ -set, G δσ -set. 1 2 S. V. MEDVEDEV implies that Q ω is a canonical elemen t of class 3. Moreo ve r, v an Engelen [E1 ] pro ve d that the Can tor set C is densely homogeneous with resp ect to Q ω . In [E2] v an Engelen describ ed all homogeneous Borel sets whic h are either an F σδ -subsets or a G δσ -subsets of the Cantor set C , but not b oth (see also [E3] and [E5]). In the pap er w e use (and mo dify to the non-separable case) the ab ov e theorems and a tec hnique from Saint Ray mond [SR], v an Mill [Mi], v an Engelen [E1 ], [E2], and Ostrov sky [O2]. The pap er is organized as fo llo ws: in Section 1 w e pro v e some lemmas to obtain a go o d represen tation of an absolute F σδ -set (see Lemma 1.10). In Section 2 w e giv e a description of the space Q ( k ) ω for k > ω (see Theorem 2.2) . In Section 3 w e in v estigate the pro ducts Q ( τ ) × Q ( k ) ω and their dense complemen ts in the Baire space B ( τ ), where ω ≤ k ≤ τ . 1. Not a t ion and some lemmas F or all undefined terms and notation see [En]. X ≈ Y means that X and Y are homeomorphic spaces. A clop en set is a set whic h is b oth closed and op en. A strongly zero-dimensional space X is called h -homo gene ous (or str ongly homo gene ous , see [Mi], [E4]) if ev ery nonempt y clop en subset of X is homeomorphic to X . Ev ery h -homo g eneous space is homogeneous, but the con ve rse do es not hold. A space X is called weigh t-ho m o gene ous if fo r ev ery nonempt y open subset U ⊆ X w e ha v e w ( U ) = w ( X ) . F or a cardinal k put E k = { X : X is a weigh t-homogeneous space of w eigh t k and Ind X = 0 } . Clearly , ev ery h -homogeneous space X ∈ E k for k = w ( X ). Let P b e a top ological prop ert y . A space X is nowher e P if no nonempt y op en subset of X has prop ert y P . L et P 1 and P 2 b e top ological prop erties; we write X ∈ P 1 + P 2 if X = A ∪ B , where A has prop ert y P 1 and B has pro p ert y P 2 . Let U b e a fa mily of subsets of a metric space ( X,  ). Put ∪ U = ∪{ U : U ∈ U } and mesh U = sup { diam( U ) : U ∈ U } . Denote by [ U ] the family { cl X U : U ∈ U } . If f : X → Y is a mapping, then w e write f ( U ) = { f ( U ) : U ∈ U } . Let k ≥ ω . In [M1] w e prop osed t o consider the space Q ( k ) as a no n-separable analogue of w eigh t k for the space Q o f ratio na l n um b ers. By definition, Q ( k ) is a σ -pro duct of ω copies of the discrete space D of cardinality k with a basic p oin t (0 , 0 , . . . ) ∈ k ω , i.e., Q ( k ) = { ( x 0 , x 1 , . . . ) ∈ k ω : ∃ n ∀ m ( m ≥ n ⇒ x m = 0) } . Clearly , Q ( ω ) ≈ Q . A top olo gical description of the space Q ( k ) is give n b y the follo wing theorem (see [M1]). Theorem 1.1. L et X b e a σ -disc r ete metric sp ac e of weight k that is homo gene ous with r esp e c t to wei g h t. Then X is home omorphic to Q ( k ) . Corollary 1.2. The sp ac e Q ( k ) is h -homo gene ous for ev e ry c ar dinal k . Lemma 1.3. [O1] L et A i b e a nowher e dense close d subset of a me tric str ongly zer o-dimensional sp ac e ( X i ,  i ) , i ∈ { 1 , 2 } . L et X 1 \ A 1 and X 2 \ A 2 b e home omorphic h -homo gene ous sp ac es. If F σ δ - AND G δσ -SP A CES 3 f 0 : A 1 → A 2 is a home omorphism, then ther e exists a home omorphism f : X 1 → X 2 such that the r estriction f | A 1 = f 0 . Lemma 1.4. L et A i b e a nowher e dense clos e d subset of an h -homo gene ous sp ac e ( X i ,  i ) , i ∈ { 1 , 2 } . L et g : X 1 → X 2 b e a home omorphi s m such that g ( A 1 ) = A 2 . Then for a ho m e o- morphism f 0 : A 1 → A 2 and ε > 0 s atisfying  2 ( g | A 1 , f 0 ) < ε ther e exists a home omorphism f : X → X such that the r estriction f | A 1 = f 0 and  2 ( g , f ) < ε . Pr o of . F or X 1 = X 2 = C , where C is the Can tor set, the lemma w as prov ed b y v an Engelen [E1, Theorem 2.2] (see also [Mi]). Supp ose X 1 and X 2 are not compact. Put δ = ε −  2 ( g | A 1 , f 0 ) > 0 . T ak e a discrete clopen co v er U of X 1 b y sets o f diameter less than δ / 4. Let U 2 = { U 2 t : t ∈ T } b e a discrete clop en cov er of X 2 suc h that U 2 refines g ( U ) and mesh U 2 < δ / 4. Put U 1 t = g − 1 ( U 2 t ). Then U 1 = { U 1 t : t ∈ T } is a discrete clop en co v er of X 1 and mesh U 1 < δ / 4. Let T ∗ = { t ∈ T : U 1 t ∩ A 1 6 = ∅} . Put V i t = U i t ∩ A i , where i ∈ { 1 , 2 } and t ∈ T ∗ . The fa mily V 2 of nonempt y in tersections { V 2 s ∩ f 0 ( V 1 t ) : s ∈ T ∗ , t ∈ T ∗ } forms a discrete clop en (relativ e to A 2 ) co v er of A 2 . Hence, t he fa mily V 1 = { f − 1 0 ( V 2 ) : V 2 ∈ V 2 } is a discrete clop en (relative to A 1 ) co v er of A 1 . F or ev ery V 1 ∈ V 1 there exists exactly one t ∈ T ∗ suc h that V 1 ⊆ V 1 t ⊂ U 1 t . Then V 2 = f 0 ( V 1 ) ⊆ V 2 s ⊂ U 2 s for some (unique) s ∈ T ∗ . Since Ind X 1 = 0 and V 1 t is closed in U 1 t , t here exists a retra ction r 1 t : U 1 t → V 1 t . The set V 1 is clop en in V 1 t ; this implies that W 1 = ( r 1 t ) − 1 ( V 1 ) is a clop en subs et of U 1 t and W 1 ≈ X 1 . Similarly , there exis ts a retraction r 2 s : U 2 s → V 2 s , and W 2 = ( r 2 s ) − 1 ( V 2 ) ≈ X 2 . The set V i is now here dense a nd closed in W i for i ∈ { 1 , 2 } . Therefore, W 1 \ V 1 and W 2 \ V 2 are homeomorphic h -homogeneous spaces. By virtue of Lemma 1.3 there exists a homeomorphism f W 1 : W 1 → W 2 suc h that the restriction f W 1 | V 1 = f 0 and f W 1 ( V 1 ) = V 2 . By construction, diam( W 2 ) < δ / 4. T ake tw o p oin ts a ∈ V 1 and x ∈ W 1 . Then  2 ( g ( x ) , f W 1 ( x )) ≤  2 ( g ( x ) , g ( a )) +  2 ( g ( a ) , f W 1 ( a )) +  2 ( f W 1 ( a ) , f W 1 ( x )) ≤ ≤ δ / 4 +  2 ( g ( a ) , f 0 ( a )) + δ / 4 ≤  2 ( g | A 1 , f 0 ) + δ / 2 < ε. Let us construct the mapping f : X 1 → X 2 . T ake a p oin t x ∈ X 1 . If x ∈ U 1 t for t ∈ T \ T ∗ , then we define f ( x ) = g ( x ). If x ∈ U 1 t , where t ∈ T ∗ , then r 1 t ( x ) ∈ V 1 ⊆ V 1 t for some V 1 ∈ V 1 . Put f ( x ) = f W 1 ( x ) pro vided W 1 = ( r 1 t ) − 1 ( V 1 ). In b oth cases  2 ( g ( x ) , f ( x )) < ε . Clearly , f is a homeomorphism and f extends f 0 .  Lemma 1.5. L et A i b e a nowher e dense clos e d subset of an h -homo gene ous sp ac e ( X i ,  i ) , i ∈ { 1 , 2 } . Supp ose we have a home omorphism g : X 1 → X 2 such that g ( A 1 ) = A 2 . Then for a home omorphism f 0 : A 1 → A 2 and ε > 0 satisfying  1 ( g − 1 | A 2 , f − 1 0 ) +  2 ( g | A 1 , f 0 ) < ε ther e exists a hom e omorphism f : X 1 → X 2 such that the r estriction f | A 1 = f 0 and  1 ( g − 1 , f − 1 ) +  2 ( g , f ) < ε . 4 S. V. MEDVEDEV Pr o of . F or d 1 =  1 ( g − 1 | A 2 , f − 1 0 ) and d 2 =  2 ( g | A 1 , f 0 ) there exist ε 1 and ε 2 suc h that ε 1 + ε 2 = ε , d 1 < ε 1 , and d 2 < ε 2 . Let δ = min { ε 1 − d 1 , ε 2 − d 2 } . T ak e the mapping f : X 1 → X 2 that w as obtained in the pro of of Lemma 1.4. It can easily b e c hec k ed that f is the required homeomorphism.  W e iden tify cardinals with initial o rdinals; in particular, ω = { 0 , 1 , 2 , . . . } . Denote b y Λ the unique 0 - tuple, i.e., k 0 = { Λ } . Let k 1 = k . If t ∈ k n , then lh( t ) = n is a length of t and t ˆ t n = ( t 0 , . . . , t n − 1 , t n ) ∈ k n +1 for t n ∈ k . Let k <ω = ∪{ k i : i ∈ ω } . F or ev ery infinite cardinal k let B ( k ) = k ω b e the Baire space of w eigh t k ( see [En]). F or a p oin t x = ( x 0 , x 1 , . . . ) ∈ k ω w e denote b y x ↾ n the n -tuple ( x 0 , x 1 , . . . , x n − 1 ). F or an n -tuple t ∈ k n w e denote b y B t ( k ) the Baire interv al { x ∈ k ω : x ↾ n = t } . A set Y ⊂ X is a Souslin set ( or an analytic set [Ha], or an A - set ) in a space X if, for some collection { F ( t ↾ n ) : n ∈ ω , t ∈ ω ω } of closed subsets of X , we hav e Y = ∪{ T { F ( t ↾ n ) : n ∈ ω } : t ∈ ω ω } . Y is called a c o-Souslin set in a space X if X \ Y is a Souslin set in X . If A and B are t w o families of subs ets o f X , w e sa y (see [Ha]) that B is a b ase for A if each member o f A is a union of mem b ers from B . A σ -discr ete b ase is a base which is a lso a σ -discrete family . A mapping f : X → Y is called c o - σ -disc r ete if the image of eac h discrete family in X has a σ -discrete base in Y . If A is an F σ -subset of X , then we write A ∈ F σ ( X ). Similarly , w e intro duce families G σ ( X ), F σδ ( X ), and so forth. A space Y is an absolute G δ -set if for ev ery homeomorphic em bedding f : Y → X of Y in a space X the image f ( Y ) ∈ G σ ( X ). Denote by G δ , F σδ , and G δσ the families o f absolute G δ -sets, absolute F σδ -sets, a nd absolute G δσ -sets, resp ective ly . F or a space X define the family LF ( X ) = { Y : eac h p oin t y ∈ Y lies in some clop en neigh b orho o d that is homeomorphic to a close d subset of X } . A space Y ∈ σ LF ( X ) if Y = ∪{ Y i : i ∈ ω } , where eac h Y i ∈ LF ( X ) and Y i is a closed subset of Y . W e say that X is lo c al ly of weight ω . Denote b y X the Ba ire space B ( k ). Replacing F b y F ∩ A , w e may assume that F ⊆ A , where the bar denotes the closure in X . Clearly , F \ A is a Souslin set in X . Let us sho w that w ( F \ A ) = k . Indeed, if w ( F \ A ) < k , then the b oundary of A in F has w eigh t 0 there exis ts i ∈ ω suc h that ( n + i + 1 ) − 1 < ε . F r om (a5) it follows that the ε -ball ab out y con tains the nonempty op en set A ∩ W ( t ˆ i, α ˆ α n +1 ) f o r some α n +1 ∈ k . By construction, W ( t ˆ i, α ˆ α n +1 ) ∩ Y = ∅ ; hence, Y is nowh ere dense in A . Theorem 1.8 is pro v ed.  Lemma 1.9. L et k > ω . L et F b e an F σ -subset of the Bair e sp ac e B ( k ) and A ⊂ F . Supp ose B ( k ) \ A is a Souslin set in B ( k ) , A is of first c ate gory, and A is nowhe r e σ LW ( 0 ther e exists a family A = { A ( i, α ) : i ∈ ω , α ∈ k } of subsets of A such that (b1) A = ∪ A and mesh A < ε ; (b2) e ach A ( i, α ) is of first c ate gory and nowher e σ LW ( 0 the set B i \ S { B j : j < i } is op en in B i ; it ma y b e empty . T ak e a disjoint σ -discrete (in B ( k )) co v er γ i of B i \ S { B j : j < i } by clop en (in B i ) sets suc h that mesh( γ i ) < ε . Then the family γ = { V ∈ γ i : i ∈ ω } is σ - discrete in B ( k ). It consists of pairwise disjoin t closed 10 S. V. MEDVEDEV subsets of diameter les s tha n ε . Hence, γ = { V ( i, α ) : i ∈ ω , α ∈ k i } , where eac h k i ≤ k and eac h family { V ( i, α ) : α ∈ k i } is discrete in B ( k ). If k i < k fo r some i , w e take a set V ( i, α ) and separate V ( i, α ) in to k nonempt y disjoin t clop en (in V ( i, α )) subsets. This can b e done, b ecause k > ω and V ( i, α ) is no where of w eigh t ω . L et A b e an F σδ -subset of the Bair e sp ac e B ( k ) such that A is of first c at e gory and A is nowher e σ LW ( ω . L et X 1 and X 2 b e d ense F σδ -subsets of the B a i r e sp ac e B ( k ) such that X 1 and X 2 ar e sets of first c ate g ory and nowher e σ LW ( q and j ∈ { 1 , 2 } , where ˜ s = s ↾ (lh( s ) − 1) . Fix s ∈ ω q n +1 . W e distinguish tw o cases. Case 1. Let s = ˜ s ˆ 0 . W e can construct the required sets and functions as ab ov e for t he step q = n + 1. Case 2. Let s = ˜ s ˆ t a nd t ≥ 1. F or eac h i < t w e ha v e ν ( ˜ s ˆ i ) ≤ n ; hence, there exists a unique num ber ι ∈ ω suc h that the tuple ˜ s ˆ i ˆ ι ∈ ω q +1 n +1 . By the inductiv e assumption, the sets Z ∗ j ( ˜ s ˆ i ) and t he homeomorphism f ˜ s ˆ i ˆ ι n +1 : Z ∗ 1 ( ˜ s ˆ i ) → Z ∗ 2 ( ˜ s ˆ i ) are already defined for any j ∈ { 1 , 2 } and i < t suc h t ha t  1  ( f ˜ s ˆ i ˆ ι n +1 ) − 1 , f − 1 n   Z ∗ 2 ( ˜ s ˆ i )  +  2  f ˜ s ˆ i ˆ ι n +1 , f n   Z ∗ 1 ( ˜ s ˆ i )  < ( n + 1) − 1 . Using (d3), w e observ e that the sets Z ∗ j ( ˜ s ˆ i ) , where 0 ≤ i < t , are pairwise disjoin t. F ro m (d4) it f o llo ws that Y j ( t ) = S { Z ∗ j ( ˜ s ˆ i ) : i < t } is a no where dense closed subset of Z ∗ j ( ˜ s ). 14 S. V. MEDVEDEV F or a p oint x ∈ Z ∗ 1 ( ˜ s ˆ i ) put g t n +1 ( x ) = f ˜ s ˆ i ˆ ι n +1 ( x ). Then t he mapping g t n +1 : Y 1 ( t ) → Y 2 ( t ) is a homeomorphism. F ro m (d8) and (d9) it follows that f n ( Y 1 ( t )) = Y 2 ( t ). By construction,  1  ( g t n +1 ) − 1 , f − 1 n   Y 2 ( t )  +  2  g t n +1 , f n   Y 1 ( t )  < ( n + 1) − 1 . Fix α ∈ k q − 1 . By (d1), Z j ( ˜ s, α ) is a clop en subset of X j  ϕ q − 1 n − t − 1 ,j ( ˜ s, α ) , ψ q − 1 n − t − 1 ,j ( ˜ s, α )  for j ∈ { 1 , 2 } . Cho ose the smallest num b er m ˜ s,α,j suc h that the set Z j ( ˜ s, α ) \  X ∗ j ( ϕ q n − t,j ( ˜ s, α ) ˆ m ˜ s,α,j ) \ Y j ( t )  is nonempt y . Using (d7) and (d3), w e see that the last set is clop en in X ∗ j ( ϕ q n − t,j ( ˜ s, α ) ˆ m ˜ s,α,j ). T ake a discrete clop en (in Z j ( ˜ s, α )) cov er τ α,j of Z j ( ˜ s, α ) suc h that (e1) mesh( τ α,j ) < (2 n + 2) − 1 ; (e2) ev ery U ∈ τ α,j in tersects at most one set of the family  X j  ϕ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ m ˜ s,α,j , ψ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ α q  : α q ∈ k  ; (e3) for ev ery U ∈ τ α,j if U ∩ Y j ( t ) 6 = ∅ , then U ∩  X ∗ j  ϕ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ m ˜ s,α,j  \ Y j ( t )  = ∅ . According to (d8), we may assume that τ α, 2 = f n ( τ α, 1 ). Let τ ∗ α, 1 = { U ∈ τ α, 1 : U ∩  X ∗ 1 ( ϕ q − 1 n − t − 1 , 1 ( ˜ s, α ) ˆ m ˜ s,α, 1 ) \ Y 1 ( t )  6 = ∅ or f n ( U ) \  X ∗ 2  ϕ q − 1 n − t − 1 , 2 ( ˜ s, α ) ˆ m ˜ s,α, 2  \ Y 2 ( t )  6 = ∅} . Using we ight-homogeneit y of X ∗ 1  ϕ q − 1 n − t − 1 , 1 ( ˜ s, α ) ˆ m ˜ s,α, 1  , w e can index the family τ ∗ α, 1 as { U 1 ( γ ) : γ ∈ k } . Put τ ∗ α, 2 = f n ( τ ∗ α, 1 ) = { U 2 ( γ ) = f n ( U 1 ( γ )) : γ ∈ k } . Since f n ( Y 1 ( t )) = Y 2 ( t ), for U ∈ τ α, 1 w e hav e U ∩ Y 1 ( t ) 6 = ∅ ⇐ ⇒ f n ( U ) ∩ Y 2 ( t ) 6 = ∅ . T ogether with (e3), this implies that f n ( U ) ∩ Y 2 ( t ) = ∅ if U ∈ τ ∗ α, 1 . Similar ly , f − 1 n ( U ) ∩ Y 1 ( t ) = ∅ if U ∈ τ ∗ α, 2 . Fix a γ ∈ k . F or j ∈ { 1 , 2 } c ho ose the smallest n um b er l γ ,j ∈ ω suc h that the set Z j ( s, α ˆ γ ) = U j ( γ ) \  X j ( ϕ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ l γ ,j , ψ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ α γ ,j ) \ Y j ( t )  is no nempt y for some α γ ,j ∈ k . Then the tuples ϕ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ l γ ,j = ϕ q n +1 ,j ( s, α ˆ γ ) ∈ ω q and ψ q − 1 n − t − 1 ,j ( ˜ s, α ) ˆ α γ ,j = ψ q n +1 ,j ( s, α ˆ γ ) ∈ k q are assigned to ( s, α ˆ γ ). As ab ov e, w e hav e ν ( ϕ q n +1 ,j ( s, α ˆ γ )) ≥ n + 1 . F σ δ - AND G δσ -SP A CES 15 By construction, Z j ( s, α ˆ γ ) is a nowhe re dense closed subse t of Z j ( ˜ s, α ) and Z j ( s, α ˆ γ ) is a no where dense closed subset of Z j ( ˜ s, α ). Since Z 1 ( s, α ˆ γ ) ≈ B ( k ) ≈ Z 2 ( s, α ˆ γ ), by Lemma 1.3, there exists a homeomorphism h s n +1 ,α ˆ γ : U 1 ( γ ) → U 2 ( γ ) suc h that h s n +1 ,α ˆ γ  Z 1 ( s, α ˆ γ )  = Z 2 ( s, α ˆ γ ) . Clearly , t he combination mapping h s n +1 ,α = ∇{ h s n +1 ,α ˆ γ : γ ∈ k } : ∪ τ ∗ α, 1 → ∪ τ ∗ α, 2 is a homeomorphism suc h tha t h s n +1 ,α  ∪{ Z 1 ( s, α ˆ γ ) : γ ∈ k }  = ∪{ Z 2 ( s, α ˆ γ ) : γ ∈ k } . By construction, the se t ∪ τ ∗ α,j is clop en in Z j ( ˜ s, α ) , Y j ( t ) ∩ S τ ∗ α,j = ∅ , and Y j ( t ) ∩ Z 1 ( ˜ s, α ) is a now here dense closed subset of Z j ( ˜ s, α ) for eac h j ∈ { 1 , 2 } and α ∈ k q − 1 . F or j ∈ { 1 , 2 } put Z ∗ j ( s ) = ∪{ Z j ( s, α ˆ γ ) : α ∈ k q − 1 , γ ∈ k } . Then Z ∗ j ( s ) = ∪{ Z j ( s, α ˆ γ ) : α ∈ k q − 1 , γ ∈ k } . Let τ ∗ j = ∪{ S τ ∗ α,j : α ∈ k q − 1 } . The n the set ∪ τ ∗ j is clop en in Z ∗ j ( ˜ s ), Y j ( t ) ∩ S τ ∗ j = ∅ , and Y j ( t ) ∩ Z ∗ j ( ˜ s ) is a now here dense clos ed subset of Z ∗ j ( ˜ s ). W e define the homeomorphism h s n +1 : ∪ τ ∗ 1 → ∪ τ ∗ 2 b y the r ule h s n +1 ( x ) = h s n +1 ,α ( x ) if x ∈ ∪ τ ∗ α, 1 for some α ∈ k q − 1 . By virtue of Lemma 1.5 the homeomorphism g t n +1 : Y 1 ( t ) → Y 2 ( t ) can b e extended t o a homeomorphism ˜ g t n +1 : Z ∗ 1 ( ˜ s ) \ [ τ ∗ 1 → Z ∗ 2 ( ˜ s ) \ [ τ ∗ 2 . Then the com binatio n mapping f s n +1 = h s n +1 ∇ ˜ g t n +1 : Z ∗ 1 ( ˜ s ) → Z ∗ 2 ( ˜ s ) is a homeomorphism. One can c hec k that the follo wing inequalit y  1  ( f s n +1 ) − 1 , f − 1 n   Z ∗ 2 ( ˜ s )  +  2  f s n +1 , f n   Z ∗ 1 ( ˜ s )  < ( n + 1) − 1 holds. Note that the sets Z ∗ j ( ˜ s ) and Z ∗ j ( ˜ r ) are disjoin t for differen t tuples s and r f r om ω q n +1 . This completes the induction o n q for s ∈ ω q n +1 . The set ω 1 n +1 con tains a unique tuple s = ( n ). Since ˜ s = Λ , w e ha v e Z ∗ j (Λ) = X j and Z ∗ j (Λ) = B j for j ∈ { 1 , 2 } . Then t he mapping f ( n ) n +1 : Z ∗ 1 (Λ) → Z ∗ 2 (Λ) is the required homeomorphism f n +1 : B 1 → B 2 . The induction on n is finished. Let us construct a homeomorphism f : B 1 → B 2 with f ( X 1 ) = X 2 . Consider a p oin t x ∈ B 1 . The sequenc e { f n ( x ) : n ∈ ω } is a Cauc h y sequence in a complete metric space B 2 . Hence, it con v erges to a p oint y ∈ B 2 ; put f ( x ) = y . F rom (d10) it follo ws that the seque nce { f n } is uniformly conv ergen t to t he mapping f . According to [En, Theorem 4.2.19], f is con tin uous. Similarly , the mapping g = lim n →∞ f − 1 n : B 2 → B 1 is contin uous. 16 S. V. MEDVEDEV F or a p oin t y ∈ B 2 and n ∈ ω let z = f − 1 n ( y ); then  2 ( f ◦ f − 1 n ( y ) , y ) = lim i →∞  2 ( f i ◦ f − 1 n ( y ) , f n ◦ f − 1 n ( y )) = lim i →∞  2 ( f i ( z ) , f n ( z )) ≤ ( n + 1) − 1 . Therefore, lim n →∞  2 ( f ◦ f − 1 n ( y ) , y ) = 0. But lim n →∞ ( f ◦ f − 1 n ) = f ◦ g , so, f ◦ g : B 2 → B 2 is the iden tit y mapping. In the same w ay , g ◦ f : B 1 → B 1 is the iden tit y mapping. Hence , f − 1 = g and f : B 1 → B 2 is a homeomorphism. Let us sho w that f ( X 1 ) ⊆ X 2 . T ak e a p oin t x ∈ X 1 , then { x } = ∩{ X 1 ( χ ↾ n, ξ ↾ n ) : n ∈ ω } for some χ ∈ ω ω and ξ ∈ k ω . F rom (d2), (c1), and (c4) it follo ws that there ex ist τ ∈ ω ω and ϑ ∈ k ω suc h tha t f n ( x ) ∈ X 2 ( τ ↾ n, ϑ ↾ n ) fo r eac h n ∈ ω . Using (c7), w e get ∩{ X 2 ( τ ↾ n, ϑ ↾ n ) : n ∈ ω } = { y } for some y ∈ X 2 . The condition (c6) of Lemma 1.10 implies that f ( x ) = lim n →∞ f n ( x ) = y ∈ X 2 . Similarly , f − 1 ( X 2 ) ⊆ X 1 . Th us, f ( X 1 ) = X 2 .  Theorem 2.2. L et X b e a n absolute F σδ -set of first c ate gory such that Ind X = 0 , X is nowh er e σ LW ( ω . Clearly , t he space Q ( k ) ω is o f first category , Q ( k ) ω ∈ F σδ , and Q ( k ) ω ∈ E k . Let us v erify that Q ( k ) ω is no where σ LW ( n 0 } ⊂ U 0 . Eviden tly , Z 0 ≈ Q ( k ) ω . Since Z 0 ∩ F 1 is lo cally of w eight n 1 } ⊆ U 1 . Con tin uing in this w ay , we obta in the p o int q = { q 0 , q 1 , . . . } ∈ U whic h b elongs to U \ [ { F i ∪ G i : i ∈ ω } . F σ δ - AND G δσ -SP A CES 17 Hence, V 6 = ∪{ F i S G i : i ∈ ω } , a con tradiction. Thus, Q ( k ) ω is nowhere σ LW ( ω the corollary follow s from Theorems 2.1 and 2.2.  Corollary 2.4. L et X 1 and X 2 b e dense subsets of the B air e sp ac e B ∗ ( k ) such that B ∗ ( k ) \ X 1 ≈ B ∗ ( k ) \ X 2 ≈ Q ( k ) ω . Then X 1 ≈ X 2 . Pr o of . By Coro llary 2.3 there exists a homeomorphism f : B ∗ ( k ) → B ∗ ( k ) suc h that f ( B ∗ ( k ) \ X 1 ) = B ∗ ( k ) \ X 2 . Then f ( X 1 ) = X 2 .  Corollary 2.5. L et F b e an F σδ -subset of the Bair e sp ac e B ∗ ( k ) . T h en F × Q ( k ) ω ≈ Q ( k ) ω and F is home omorphic to a close d subset of Q ( k ) ω . Pr o of . One can v erifies that the pro duct X = F × Q ( k ) ω satisfies the conditions of Theorem 2.2. Then X ≈ Q ( k ) ω . Clearly , F is homeomorphic to a closed subset of F × Q ( k ) ω ≈ Q ( k ) ω .  3. Some homogeneous F σδ - and G δσ -subset s of B ( k ) In this Section w e giv e top ological c haracterizations of the pro ducts Q ( τ ) × Q ( k ) ω and their dense complemen ts A I I τ ,k in the Baire space B ( τ ), where ω ≤ k ≤ τ . W e also in ve stigate the pro ducts Q ( τ ) × A I I k ,k . All these spaces are h - homogeneous. Notice that this list of h - homogeneous F σδ - a nd G δσ -subsets of B ( τ ) is not complete. Theorems 3.10 and 3.23 are Hurewicz-t yp e theorems for F σδ -subsets of the Baire space B ( k ). T o establish these results w e shall use the follo wing statemen ts. Theorem 3.1. [M2, Theorem 3.5] L et Y b e an h -homo gene ous sp ac e and X b e a w eight- homo gene ous sp ac e of w eight τ . Supp ose X is o f first c ate gory, X ∈ σ LF ( Y ) , and every nonempty clop en subset of X c ontains a nowher e dense close d c o p y of Y . Then X is home o- morphic to Q ( τ ) × Y . Theorem 3.2. [M2, Theorem 3.13] or [M4 , Theorem 4 ] Supp ose X an d Y ar e h -hom o gene ous sp ac es of first c ate gory such that w ( X ) = w ( Y ) , Y ∈ σ LF ( X ) , and X ∈ σ LF ( Y ) . Then X is home omorphic to Y . The last theorem implies the following corollary . Corollary 3.3. L et X b e an h -ho mo gene ous sp ac es of fi rst c ate gory and w ( X ) ≥ k . Then X ≈ Q × X ≈ Q ( k ) × X . Theorem 3.4. L et ω < k ≤ τ . L et the Bair e sp ac e B ( k ) b e densely homo gene ous with r esp e ct to an h -hom o gene ous sp ac e Y . Then the Bair e sp ac e B ( τ ) is dense ly homo gene ous with r esp e ct to the pr o duct Q ( τ ) × Y . 18 S. V. MEDVEDEV V a n Engelen (see [E4, Theorem 4.7]) prov ed that if the Cantor set C is densely homogeneous with resp ect to a n h -homogeneous space A , then C is densely homogeneous with resp ect to the pro duct Q × A . Theorem 3.4 generalize this result for the non-separable case. It w as announced in [M3 ] and prov ed in [M2] (see [M2, Lemma 5.3]). Definition 3.5. In this Section w e shall often denote the space Q ( k ) ω b y M I k . Let us in tro duce the space A I I k . Fix a dense cop y A of Q ( k ) ω in B ∗ ( k ) and put A I I k = B ∗ ( k ) \ A . Let A I k = Q × A I I k and M I I k = B ∗ ( k ) \ Y , where Y is a dense cop y of A I k in B ∗ ( k ). Notice that the spaces M I k and A I k are of first category . The spaces M I I k and A I I k con tain dense to p ologically complete subspaces. Clearly , M I k and M I I k are absolute F σδ -sets. The spaces A I k and A I I k are absolute G δσ -sets. F urt hermore, each o f the spaces M I k , M I I k , A I k , and A I I k is a w eight-homogeneous space of w eigh t k . Theorem 3.6. L et X b e an absolute G δσ -set such that Ind X = 0 , X is nowher e F σδ \ σ LW ( ω . By Definition 3.5, M I I k = B ( k ) \ Y , where Y is a dense cop y of A I k in B ( k ). Em b ed X densely in B ( k ). As ab o ve , we obtain that B ( k ) \ X is no where F σδ \ σ LW ( ω . Consider t w o dense copies X 1 and X 2 of M I I k in B ( k ). As in the pro of of Theorem 3.1 8 w e obtain that B ( k ) \ X 1 ≈ B ( k ) \ X 2 ≈ A I k . By T heorem 3.9 with k = τ , the Baire space B ( k ) is densely homogeneous with resp ect to the space Q ( k ) × A I k ≈ A I k . Hence, there exists a homeomorphism f : B ( k ) → B ( k ) suc h that f ( B ( k ) \ X 1 ) = B ( k ) \ X 2 . Then f ( X 1 ) = X 2 .  The follo wing theorem is a mo dification of Theorem 3.10. Theorem 3.23. L et X b e an F σδ -subset of the Bair e sp ac e B ∗ ( k ) and X / ∈ σ LW (

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