Functionally $\sigma$-discrete mappings and a generalization of Banach's theorem

0.18 MB·English

by  Olena Karlova

Functionally σ-discrete mappings
and a generalization of Banach’s theorem
Olena Karlova
Department of Mathematical Analysis, Faculty of Mathematics and Informatics, Chernivtsi National University,
Kotsyubyns’koho str., 2, Chernivtsi, Ukraine
5
1
0
2

n Abstract
a
J We present σ-strongly functionally discrete mappings which expand the class of σ-discrete map-

3
pings and generalize Banach’s theorem on analytically representable functions
1

Keywords: σ-discrete mapping, σ-strongly functionally discrete mapping, Lebesgue
]
N classification of functions, Borel classification of sets
G 2000 MSC: Primary 54C50, 54H05; Secondary 26A21
.
h
t
a
1. Introduction
m
[
Let X be a topological space and A be a family of subsets of X. We define classes Fα(A) and
1 G (A) in the following way: F (A) = A, G (A) = {X \A : A ∈ A} and for all 1 ≤ α < ω let
v α 0 0 1
1
∞
0
9 Fα(A) = An : An ∈ Gβ(A), n = 1,2,... ,
2 nn\=1 β[<α o
0
∞
.
1
G (A) = A : A ∈ F (A), n = 1,2,... .
0 α n n β
5 nn[=1 β[<α o
1
v: If A is the collection of all /functionally/ closed subsets of X, then elements of Fα(A)or Gα(A) are
i called sets of the α’th /functionally/ multiplicative class or sets of the α’th /functionally/ additive
X
class, respectively; elements of the family F (A)∩G (A) are called /functionally/ ambiguous sets
r α α
a
of the class α.
A mapping f : X → Y between topological spaces belongs to the α’th /functionally/ Lebesgue
class, if the preimage f−1(V) of any open set V ⊆ Y is of the α’th /functionally/ additive class
α in X. The collection of all mappings of the α’th /functionally/ Lebesgue class we denote by
H (X,Y) /K (X,Y)/. Notice that H (X,Y) = K (X,Y) for any perfectly normal space X and
α α α α
any topological space Y.
By C(X,Y) we denote the class of all continuous mappings between X and Y.
Let Φ (X,Y) = H (X,Y) and for all 1 < α < ω the symbol Φ (X,Y) stands for the collection
1 1 1 α
of all mappings between X and Y which are pointwise limits of sequences of mappings from
Φ (X,Y). The next result is the classical Banach’s theorem [1].
β
β<α
S
Theorem A. Let X be a metric space, Y be a metric separable space and 0 < α < ω . Then
1
(i) Φ (X,Y) = H (X,Y), if α < ω ,
α α 0
Email address: [email protected](Olena Karlova)
(ii) Φ (X,Y) = H (X,Y), if α ≥ ω .
α α+1 0
R. Hansell in [4] introduced the concept of σ-discrete mapping as a convenient tool for the
investigation ofmappings withvalues innon-separablespaces. Amapping f : X → Y is σ-discrete
∞
if there exists a family B = B of subsets of X such that every family B is discrete in X and
n n
n=1
S
the preimage f−1(V) of any open set V in Y is a union of sets from B. Class of all σ-discrete
mappings between X andY is denotedby Σ(X,Y). In[5, Theorem 7] Hansell proved the following
generalization of Banach’s theorem.
Theorem B. Let X be a perfect space, Y be a metric space and 0 < α < ω . Then
1
(i) Φ (X,Y)∩Σ(X,Y) = H (X,Y)∩Σ(X,Y), if α < ω ,
α α 0
(ii) Φ (X,Y)∩Σ(X,Y) = H (X,Y)∩Σ(X,Y), if α ≥ ω .
α α+1 0
In this paper we develop technique from [5] and [3] and prove an analogue of Theorem B for
an arbitrary topological space X. To do this we introduce classes of mappings Σf(X,Y). Namely,
α
∞
a mapping f : X → Y belongs to Σf(X,Y), where 0 ≤ α < ω , if there exist a family B = B
α 1 n
n=1
S
∞
of functionally ambiguous sets of the class α in X and a family U = U of functionally open
n
n=1
S
subsets of X, where U = (U : B ∈ B ), such that every family U is discrete in X, B ⊆ U for
n B n n B
every B ∈ B andthepreimagef−1(V)ofanyopensetV inY isaunionofsets fromB. Properties
n
of this class are studied in Section 2. Let us observe that the class Σf(X,Y) coincides with the
α
class of all σ-discrete mappings of the α’th Lebesgue class in case X is a perfectly normal space
and Y is a metric space. Auxiliary technical propositions are gathered in Section 3. The forth
section contains three approximation lemmas which are crucial in the proof of the main theorem.
In Section 5 we present classes Λ which are close to classes Φ : let Λ (X,Y) = Σf(X,Y), and
α α 1 1
for all 1 < α < ω let Λ (X,Y) be the collection of all mappings between X and Y which are
1 α
pointwise limits of sequences of mappings from Λ (X,Y). The theorem below is the main
β
β<α
S
result of the paper.
Theorem C. Let X be a topological space, Y be a metric space and 0 < α < ω . Then
1
(i) Λ (X,Y) = Σf(X,Y), if α < ω ,
α α 0
(ii) Λ (X,Y) = Σf (X,Y), if α ≥ ω .
α α+1 0
An example at the end of the fifth section shows that the assertion on X in Theorem B is
essential.
2. Properties of σ-strongly functionally discrete mappings
Definition 1. A family A = (A : i ∈ I) of subsets of a topological space X is called
i
1. discrete, if every point of X has an open neighborhood which intersects with at most one
set from A;
2. strongly discrete, if there exists a discrete family (U : i ∈ I) of open subsets of X such that
i
A ⊆ U for every i ∈ I;
i i
3. strongly functionallydiscreteor, briefly, sfd-family,ifthereexistsadiscretefamily(U : i ∈ I)
i
of functionally open subsets of X such that A ⊆ U for every i ∈ I;
i i
4. well strongly functionally discreteofwell sfd-family, if thereexist discrete families(U : i ∈ I)
i
of functionally open sets and (F : i ∈ I) of functionally closed sets such that A ⊆ F ⊆ U
i i i i
for every i ∈ I.
Notice that (4) ⇒ (3) ⇒ (2) ⇒ (1) for any X; a topological space X is collectionwise normal
if and only if every discrete family in X is strongly discrete; if X is normal then (2) ⇔ (3).
Definition 2. Let P be a property of a family of sets. A family A is called a σ-P-family if
∞
A = A , where every family A has the property P.
n n
n=1
S
Definition 3. AfamilyB ofsetsofatopologicalspaceX iscalleda baseforamappingf : X → Y
if the preimage f−1(V) of an arbitrary open set V in Y is a union of sets from B.
Definition 4. If a mapping f : X → Y has a base which is a σ-P-family, then we say that f is
a σ-P mapping.
The collection of all σ-P mappings between X and Y we will denote by
• Σ(X,Y), if P is a property of discreteness;
• Σ∗(X,Y), if P is a property of a strong discreteness;
• Σf(X,Y), if P is a property of a strong functional discreteness.
By Σf(X,Y) /Σf∗(X,Y)/ we denote the collection of all mappings between X and Y which
α α
has a σ-sfd base of functionally ambiguous /multiplicative/ sets of the class α in X.
Remark 1. For any spaces X and Y the following relations holds:
1. Σf∗(X,Y) ⊆ Σf(X,Y)m if 0 ≤ β < α < ω ;
β α 1
2. Σf∗(X,Y) = Σf (X,Y)m if 0 ≤ β < ω ;
β β+1 1
3. Σf(X,Y) ⊆ C(X,Y).
0
Let us observe that every continuous mapping f : X → Y is σ-strongly functionally discrete
if either X, or Y is a metric space, since every metric space has σ-sfd base of open sets. Clearly,
every mapping with values in a second countable space is σ-sfd. In [4] Hansell proved that any
Borel measurable mapping f : X → Y is σ-discrete if X is a complete metric space and Y is a
metric space.
Lemma 1. Let 0 ≤ α < ω , X be a topological space, (U : i ∈ I) be a locally finite family of
1 i
functionally open sets in X, (B : i ∈ I) be a family of sets of the α’th functionally additive
i
/multiplicative/ class in X such that B ⊆ U for every i ∈ I. Then the set B = B is of the
i i i
i∈I
S
α’th functionally additive /multiplicative/ class α in X.
Proof. For α = 0 we consider the case each B is functionally closed and take a continuous
i
function f : X → [0,1] such that B = f−1(0) and X \ U = f−1(1), i ∈ I. Then the function
i i i i i
f(x) = minf (x) is continuous and B = f−1(0).
i
i∈I
Assume that our proposition is true for all 0 ≤ ξ < α and prove it for ξ = α. If α is a limit
ordinal then we take an increasing sequence of ordinals (α )∞ which converges to α. If α = β+1
n n=1
then we put α = β for every n ∈ N.
n
Let B be a set of the α’th functionally additive class α and (B )∞ be a sequence of sets of
i i,n n=1
∞
the α ’th functionally multiplicative classes such that B = B . By the inductive assumption
n i i,n
n=1
S
the set F = B belongs to the α ’th functionally multiplicative class for every n. Hence, the
n i,n n
i∈I
S
∞
set B = B = F is of the α’th functionally additive class.
i n
i∈I n=1
S S
Now assume that B belongs to the α’th functionally multiplicative class for every i ∈ I and
i
∞
take a sequence (B )∞ of sets of the α ’th functionally additive classes such that B = B .
i,n n=1 n i i,n
n=1
T
Notice that each set G = B ∩ U is of the α ’th functionally additive class, G ⊆ U and
i,n i,n i n i,n i
∞
B = G . Then G = G belongs to the α ’th functionally additive class for every n.
i i,n n i,n n
n=1 i∈I
T S
∞
Hence, the set B = G if of the α’th functionally multiplicative class.
n
n=1
T
Corollary 2. For any 0 ≤ α < ω a union of an sfd-family of sets of the α’th functionally additive
1
/multiplicative/ class in a topological space is a set of the same class.
Lemma 3. Let X be a topological space and f ∈ Σf(X,Y). Then f has a σ-sfd base B which is
a union of a sequence of well sfd-families.
∞
Proof. Let B′ = B′ be a base for f, where B′ is an sfd-family for every n ∈ N. For all
n n
n=1
n ∈ N and B ∈ B′ wSe take a functionally open set U and a sequence of functionally closed sets
n B,n
∞
(F )∞ such that the family (U : B ∈ B′ ) is discrete, B ⊆ U and U = F for
B,m m=1 B,n n B,n B,n B,m
m=1
every B ∈ B′. For all n,m ∈ N we put S
∞
B = (B ∩F : B ∈ B′) and B = B .
n,m B,m n n,m
n,[m=1
It is easy to see that each B is well sfd-family and B is a base for f.
n,m
Theorem 4. Let 0 < α < ω , X be a topological space and Y be a topological space with a
1
σ-disjoint base. Then
K (X,Y)∩Σf(X,Y) ⊆ Σf(X,Y).
α α
∞
Proof. Let f ∈ K (X,Y) ∩Σf(X,Y). According to Lemma 3 there exists a base B = B
α m
m=1
S
for f such that each B = (B : i ∈ I ) is well sfd-family. For all m and i ∈ I we take a
m i,m m m
functionally open set U and a functionally closed set F in X such that B ⊆ F ⊆ U
i,m i,m i,m i,m i,m
and the family (U : i ∈ I ) is discrete.
i,m m
∞
Consider a σ-disjoint base V = V of open sets in Y. Since f ∈ K (X,Y), for every V ∈ V
n α
n=1
S
there exists a sequence (A )∞ of sets of functionally multiplicative classes < α in X such that
k,V k=1
∞
f−1(V) = A . For m,n,k ∈ N we put
k,V
k=1
S
B = (F ∩A : i ∈ I ,V ∈ V and B ⊆ f−1(V)).
m,n,k i,m k,V m n i,m
Notice that each family B consists of functionally ambiguous sets of the class α and is strongly
m,n,k
functionally discrete in X, since the family B is strongly functionally discrete and for any
m
nonempty set B ∈ B there is at most one set V ∈ V such that B ⊆ f−1(V). Let
i,m m n i,m
∞
B = B .
0 m,n,k
m,[n,k=1
We show that B is a base for f. Fix V ∈ V and verify that
0
∞
f−1(V) = (F ∩A ).
i,m k,V
m[,k=1 i∈[Im
Bi,m⊆f−1(V)
Since A ⊆ f−1(V) for every k, the set in the right side of the equality is contained in f−1(V).
k,V
On the other hand, if x ∈ f−1(V) then x ∈ A for some k. Moreover, B is a base for f,
k,V
consequently, there are m and i ∈ I such that x ∈ B ⊆ f−1(V). Then x ∈ F .
m i,m i,m
Proposition 5. Let 0 < α < ω , X and Y be topological spaces. Then
1
Σf(X,Y) ⊆ K (X,Y)∩Σf(X,Y).
α α
Proof. Let f ∈ Σf(X,Y). Clearly, f ∈ Σf(X,Y). We show that f ∈ K (X,Y). Let V be an
α α
∞
open set in Y and B = B be a base for f such that each family B is strongly functionally
m m
m=1
S
discrete in X and consists of functionally ambiguous sets of the class α. Then there exists a
subfamily B ⊆ B such that f−1(V) = B . For every m ∈ N we denote B′ = (B ∈ B : B ∈
V V m V
Bm). Corollary 2 implies that every set BSm = Bm′ belongs to the α’th functionally additive class
∞
in X. Moreover, f−1(V) = B . Hence, f S∈ K (X,Y).
m α
m=1
S
Theorem 4 and Proposition 5 imply
Theorem 6. Let 0 < α < ω , X be a topological space and Y be a space with a σ-disjoint base.
1
Then
K (X,Y)∩Σf(X,Y) = Σf(X,Y).
α α
Proposition 7. Let 0 ≤ α < ω , X, Y and Z be topological spaces, f ∈ Σf(X,Y), g ∈ Σf(Y,Z)
1 α α
and let h : X → Y ×Z is defined by
h(x) = (f(x),g(x))
for every x ∈ X. Then h ∈ Σf(X,Y ×Z).
α
∞ ∞
Proof. Let B = B and B = B be σ-sfd bases of functionally ambiguous sets of the
f n,f g m,g
n=1 m=1
class α for f and g,Srespectively. For aSll n,m ∈ N we put
B = {B ∩B : B ∈ B ,B ∈ B }.
n,m f g f n,f g m,g
∞
It is easy to see that B = B is a σ-sfd base for h which consists of functionally ambiguous
n,m
n,m=1
S
sets of the class α in X.
Definition 5. We say that a family (A : i ∈ I) is a partition of a space X if X = A and
i i
i∈I
S
A ∩A = ∅ for i 6= j.
i j
Proposition 8. Let 0 ≤ α < ω , (X : n ∈ N) be a partition of a topological space X by
1 n
functionally ambiguous sets of the class α, (f )∞ be a sequence of mappings from Σf(X,Y) and
n n=1 α
f(x) = f (x) if x ∈ X for some n. Then f ∈ Σf(X,Y).
n n α
Proof. Let B be a σ-sfd base for a mapping f which consists of functionally ambiguous sets
n n
of the class α in X. Let
∞
B = (B ∩X : B ∈ B ).
n n
n[=1
It is easy to see that B is a σ-sfd base for f which consists of functionally ambiguous sets of the
α’th class.
3. Auxiliary facts on functionally measurable sets
The proofs of the next two lemmas are completely similar to the proofs of Theorem 2 from [6,
p. 350] and Theorem 2 from [6, p. 357].
Lemma 9. Let 0 < α < ω and X be a topological space. Then for any disjoint sets A,B ⊆ X of
1
the α’th functionally multiplicative class there exists a functionally ambiguous set C of the class
α such that A ⊆ C ⊆ X \B.
Lemma 10. If A is a functionally ambiguous set of the (α+1)’th class in a topological space X,
where α is a limit ordinal, then there exists a sequence (A )∞ of functionally ambiguous sets of
n n=1
classes < α such that
∞ ∞ ∞ ∞
A = A = A . (1)
n+k n+k
n[=1k\=0 n\=1k[=0
The definition of sfd-family easily implies the following fact.
Lemma 11. Let A ,..., A be sfd-families of subsets of a topological space X, A = A for
1 n k k
k = 1,...,n and the family (Ak : k = 1,...,n) is strongly functionally discrete. Then thSe family
n
A = A is strongly functionally discrete.
k
k=1
S
Lemma 12. Let 0 < α < ω , A be a disjoint σ-sfd family of functionally additive sets of the α’th
1
class in a topological space X. Then for any A ∈ A there exists an increasing sequence (DA)∞
n n=1
∞
of functionally ambiguous sets of the class α such that A = DA and the family (DA : A ∈ A)
n n
n=1
is strongly functionally discrete for every n ∈ N. S
If α = β +1, then every set DA can be chosen from the β’th functionally multiplicative class.
n
∞
Proof. Let α = 1 and A = A , where A is an sfd-family of sets of the first functionally
k k
k=1
S
additive class and ∪A ∩ ∪A = ∅ for k 6= j. For every A ∈ A we take an increasing sequence
k j
∞
(BA)∞ of functio(cid:0)nally(cid:1)clos(cid:0)ed se(cid:1)ts such that A = BA. Now for all A ∈ A and n ∈ N we put
n n=1 n
n=1
S
BA, if A ∈ A for k ≤ n,
FA = n k
n (cid:26) ∅, if A ∈ A for k > n.
k
Then (FA : A ∈ A) = (BA : A ∈ A ) for every n. Since every family B = (BA : A ∈ A ) is
n n k k n k
k≤n
S
strongly functionally discrete, the set B = B is functionally closed. Moreover, B ∩B = ∅
k k k m
for all k 6= m. Since (Bk : k = 1,...,n) is anSsfd-family, Lemma 11 implies that (FnA : A ∈ A) is
∞
also sfd-family. Moreover, A = FA for every A ∈ A.
n
n=1
S
Assume that the assertion of lemma is true for all 1 ≤ ξ < α and verify it for ξ = α. Consider
a disjoint sequence of sfd-families A which consist of sets of the α’th functionally additive class.
k
∞
For every A ∈ A we take an increasing sequence (BA)∞ such that A = BA. We may assume
n n=1 n
n=1
S
that every set BA belongs to the α ’th functionally multiplicative class, where α = β for every
n n n
n ∈ N if α = β+1, and (α )∞ is an increasing sequence of ordinals such that α = supα if α is
n n=1 n
a limit ordinal.
Fix n ∈ N and for every k = 1,...,n we denote B = (BA : A ∈ A ) and B = ∪B . Since
k,n n k k,n k,n
B ,...,B are mutually disjoint sets of the α ’th functionally multiplicative class, Lemma 9
1,n n,n n
implies that there exist mutually disjoint functionally ambiguous sets C ,...,C of the class α
1,n n,n n
∞
such that B ⊆ C for every k = 1,...,n. By the inductive assumption C = C and
k,n k,n k,n k,n,m
m=1
for every m ∈ N the family (C : k = 1,...,n) is strongly functionally discrete aSnd consists of
k,n,m
functionally ambiguous sets of the class α .
n
Now for all n,m ∈ N and A ∈ A we put
BA ∩C , if A ∈ A for k ≤ n,
DA = n k,m,n k
n,m (cid:26) ∅, if A ∈ A for k > n.
k
Then (DA : A ∈ A) = (BA ∩C : A ∈ A ).
n,m n k,m,n k
k≤n
S
Fix n,m ∈ N and for every k = 1,...,n we put
D = (BA ∩C : A ∈ A ).
k n k,m,n k
NoticethateveryfamilyD isstronglyfunctionallydiscreteandconsistsoffunctionallyambiguous
k
sets of the class α. Then the set D = D is functionally ambiguous of the class α for k =
k k
1,...,n. Moreover, Dk ∩ Dm = ∅ for aSll k 6= m. Since the family (Dk : k = 1,...,n) is
strongly functionally discrete, Lemma 11 implies that (DA : A ∈ A) is an sfd-family. Moreover,
n,m
∞
A = DA for every A ∈ A.
n,m
n,m=1
S
In case α = β +1 for all n,m ∈ N and A ∈ A we choose an increasing sequence (DA )∞
n,m,k k=1
∞
of functionally multiplicative class β such that DA = DA . Clearly, (DA : A ∈ A) is an
n,m n,m,k n,m,k
k=1
S
∞
sfd-family for all n,m,k and A = DA .
n,m,k
n,m,k=1
S
Lemma 13. Let 0 ≤ α < ω , X be a topological space, A be an σ-sfd family of sets of the α’th
1
functionally multiplicative class such that A = X. Then there exists a sequence (A )∞ of
n n=1
families of sets of the α’th functionally mulStiplicative class such that
∞
1. A ≺ A,
n
n=1
S
2. A ≺ A ,
n n+1
3. A is an sfd-family,
n
∞
4. A = X
n
n=1
S S
for every n ∈ N.
∞
Proof. Let A = B and let B be an sfd-family of sets of the α’th functionally multiplicative
k k
k=1
class. For every k ∈SN we put
∞
C = (B \ B : B ∈ B ) and C = C .
k j k k
j[<k[ k[=1
Then C is a disjoint σ-sfd family of sets of the (α +1)’th functionally additive class, C ≺ A and
C = X. By Lemma 12 for every C ∈ C there exists an increasing sequence (DC)∞ of sets of
n n=1
∞
Sthe α’th functionally multiplicative class such that C = DC and the family (DC : C ∈ C) is
n n
n=1
strongly functionally discrete for every n ∈ N. S
It remains to put A = ( DC : C ∈ C) for n ∈ N.
n k
k≤n
S
4. Approximation lemmas
Definition 6. We say that a sequence (f )∞ of mappings f : X → Y is stably convergent to
n n=1 n
f : X → Y and denote this fact by f −s→t f, if for every x ∈ X there exists n ∈ N such that
n 0
f (x) = f(x) for all n ≥ n .
n 0
st
If A ⊆ YX, then the symbol A stands for the set of all stable limits of sequences from A.
Lemma 14. Let X, Y be topological spaces, ((B : i ∈ I ))∞ be a sequence of sfd-families
i,m m m=1
of sets of the (α + 1)’th functionally ambiguous sets in X, α be a limit ordinal, let the family
(B = B : m ∈ N) be a partition of X, ((y : i ∈ I ))∞ be a sequence of points from Y
m i,m i,m m m=1
i∈SIm
and let f : X → Y is defined by
f(x) = y ,
i,m
st
if x ∈ B for some m ∈ N and i ∈ I . Then f ∈ Σf (X,Y) .
i,m m <α
Proof. Fix m ∈ N. Since B is functionally ambiguous set of the class (α + 1) in X by
m
Corollary 1, Lemma 10 implies that there exists a sequence (C )∞ of functionally ambiguous
m,n n=1
sets of classes < α such that
∞ ∞ ∞ ∞
B = C = C . (2)
m m,n+k m,n+k
n[=1k\=0 n\=1k[=0
Moreover, there exists a discrete family (U : i ∈ I ) of functionally open sets in X such that
i,m m
B ⊆ U for every i ∈ I .
i,m i,m m
For all m,n ∈ N we put
D = C \ C .
m,n m,n k,n
k[<m
Notice that every D is a functionally ambiguous set of a class < α. Moreover,
m,n
∀x ∈ X ∃m ∈ N ∃n ∈ N ∀n ≥ n ) x ∈ D ). (3)
x x x mx,n
(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:0)
Indeed, if x ∈ X, then there exists a unique number m such that x ∈ B and x 6∈ B for all
x mx k
k 6= m . Then the equality 3) implies that there are numbers N ,...,N such that
x 1 mx
x 6∈ C if k < m and x ∈ C .
k,n x mx,n
n[≥Nk n≥\Nmx
Hence, for all n ≥ n = max{N ,...,N } we have x ∈ D .
x 1 mx mx,n
Let y be any point from {y : i ∈ I }. Fix n ∈ N and for all x ∈ X let
0 i,1 1
y , if x ∈ D ∩U for some m < n and i ∈ I ,
f (x) = i,m m,n i,m m
n (cid:26) y , otherwise.
0
Observe that f : X → Y is defined correctly, since the family (U : i ∈ I ) is discrete for every
n i,m m
m and the family (D : m < n) if disjoint.
m,n
We show that f ∈ Σf (X,Y). For m = 1,...,n−1 let B = (D ∩U : i ∈ I ) and let
n <α m m,n i,m m
n
B be a family which consists of the set X \ B . Clearly, B = B is σ-sfd family of
n m m
(cid:0) SmS<n (cid:1) mS=1
functionally ambiguous sets of classes < α. It follows from the definition of f that B is a base
n
for f .
n
It remains to prove that f −s→t f on X. Indeed, if x ∈ X, then there exist m ∈ N and i ∈ I
n m
such that x ∈ B ⊆ U . Then f(x) = y . It follows from (3) that there exists a number
i,m i,m i,m
n > m with x ∈ D for all n ≥ n . Then f (x) = y for all n ≥ n . Hence, f (x) = f(x) for
0 m,n 0 n i,m 0 n
all n ≥ n .
0
Lemma 15. Let α < ω be a limit ordinal, X be a topological space, Y be a metric space and
1
st
f ∈ Σf (X,Y). Then there exists a sequence of mappings f ∈ Σf (X,Y) which is uniformly
α+1 n <α
convergent to f on X.
Proof. Let B be a σ-sfd base for f which consists of functionally ambiguous sets of the class
(α+1) in X. For every n ∈ N we consider a covering U of Y by open balls of diameters < 1 and
n n
put
B = (B ∈ B : ∃U ∈ U | B ⊆ f−1(U)).
n n
Then B is a σ-sfd family of functionally ambiguous sets of the class (α+1), diam f(B) < 1 for
n n
every B ∈ B and X = ∪B for every n ∈ N.
n n
∞
Fix n ∈ N and let B = B , where B is an sfd-family of functionally ambiguous sets
n n,m n,m
m=1
S
of the class (α + 1). We put A = B and A = B \ B : B ∈ B for m > 1.
n,1 n,1 n,m n,k n,m
(cid:0) (cid:0)SkS<m (cid:1) (cid:1)
Notice that for every m ∈ N the set A = ∪A is functionally ambiguous of the class (α+1)
n,m n,m
and the family (A : m ∈ N) is a partition of X. For every A ∈ A we choose an arbitrary
m n,m
point yA ∈ f(A). We define a mapping f : X → Y by
n,m n
f (x) = yA ,
n n,m
st
if x ∈ A for some m ∈ N and A ∈ A . Then f ∈ Σf (X,Y) by Lemma 14.
n,m n <α
It remains to verify that (f )∞ converges uniformly to f. Indeed, if x ∈ X and n ∈ N, then
n n=1
f (x) = yA ∈ f(A) for some m ∈ N and A ∈ A . Since A ⊆ B for some B ∈ B ,
n n,m n,m n,m
1
d(f(x),f (x)) ≤ diamf(B) < ,
n
n
which completes the proof.
Lemma 16. Let 0 < α < ω , X be a topological space, (Y,d) be a metric space, f : X → Y be a
1
mapping, A ,...,A be families of subsets of X such that
1 n
(i) A is an sfd-family of sets of the α’th functionally multiplicative class;
k
(ii) A ≺ A for k < n;
k+1 k
(iii) diam(f(A)) < 1 for all A ∈ A
2k+2 k
for every k = 1,...,n. Then there exists a mapping g ∈ Σf(X,Y) such that the inclusion x ∈ ∪A
α k
for k = 1,...,n implies the inequality
1
d(f(x),g(x)) < . (4)
2k
Proof. Let A = (A : i ∈ I ) and (U : i ∈ I ) be discrete families of functionally open sets
k i,k k i,k k
in X such that A ⊆ U for every i ∈ I and k = 1,...,n.
i,k i,k k
By Lemma 9 there exists a family (B : i ∈ I ) of functionally ambiguous sets of the class
i,1 1
α such that A ⊆ B ⊆ U . Since A ≺ A , for every i ∈ I there exists a unique j ∈ I
i,1 i,1 i,1 2 1 2 1
such that A ⊆ A . Notice that U ∩ B is a functionally ambiguous set of the class α
i,2 j,1 i,2 j,1
and applying Lemma 9 we obtain a functionally ambiguous set B of the class α such that
i,2
A ⊆ B ⊆ U ∩ B . Proceeding in this way we obtain a sequence ((B : i ∈ I ))n of
i,2 i,2 i,2 j,1 i,k k k=1
families of subsets of X such that
• B ⊆ U for every i ∈ I ;
i,k i,k k
• B is a functionally ambiguous set of the class α for every i ∈ I ;
i,k k
• for every k < n and for every i ∈ I there exists a unique j ∈ I such that
k+1 k
A ⊆ A , (5)
i,k+1 j,k
A ⊆ B ⊆ B . (6)
i,k+1 i,k+1 j,k
for all k = 1,...,n. Observe that for every k the set
B = B
k i,k
i[∈I
k
is functionally ambiguous of the class α according to Corollary 1.
We take any points y ∈ f(X) and y ∈ f(A ) for every k and i ∈ I . For all x ∈ X we put
0 i,k i,k k
g (x) = y .
0 0
Assume that for some k < n we have already defined mappings g ,...,g from Σf(X,Y) such
1 k α
that
g (x), if x ∈ X \B ,
g (x) = k−1 k (7)
k (cid:26) y , if x ∈ B for some i ∈ I .
i,k i,k k
We put
g (x), if x ∈ X \B ,
g (x) = k k+1
k+1 (cid:26) y , if x ∈ B for some i ∈ I .
i,k+1 i,k+1 k+1
Then g ∈ Σf(X,Y) by Lemma 8. Repeating inductively this process we obtain mappings
k+1 α
g ,...,g from Σf(X,Y) each of which satisfies (7).
1 n α
Now we prove that
1
d(g (x),g (x)) < (8)
k+1 k 2k+2
for all 0 ≤ k < n and x ∈ X. Indeed, if x ∈ X \ B , then g (x) = g (x) and
k+1 k+1 k
d(g (x),g (x)) = 0. Assume that x ∈ B for some i ∈ I and choose j ∈ I such that (5)
k+1 k i,k+1 k+1 k
and (6) holds. Then g (x) = y and g (x) = y . Since f(A ) ⊆ f(A ), y ∈ f(A ).
k+1 i,k+1 k j,k i,k+1 j,k i,k+1 j,k
Hence, d(g (x),g (x)) ≤ diam(f(A )) < 1 .
k+1 k j,k 2k+2
We put g = g and show that (4) holds. Let 1 ≤ k ≤ n and x ∈ ∪A . Then x ∈ A ⊆ B
n k i,k i,k
for some i ∈ I . It follows that g (x) = y and consequently
k k i,k
1
d(f(x),g (x)) ≤ diam(f(A )) < .
k i,k 2k+2
Taking into account (8) we obtain that
n−1
1 1 1
d(f(x),g (x)) ≤ d(f(x),g (x))+ d(g (x),g (x)) < + < .
n k i i+1 2k+2 2k+1 2k
Xi=k