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ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2008 Математика и механика № 2(3)
УДК 515.1
A.V. Arbit THE LINDELÖF NUMBER IS /«-INVARIANT
Two Tychonoff spaces X and Y are said to be /-equivalent (u-equivalent) if Cp(X) and Cp(Y) are linearly (uniformly) homeomorphic. N.V.Velichko proved that the Lindelöf property is preserved by the relation of /-equivalence. A. Bouziad strengthened this result and proved that the Lindelöf number is preserved by the relation of /-equivalence. In this paper the concept of the support different variants of which can be founded in the papers of S.P. Gul'ko and O.G. Okunev is introduced. Using this concept we introduce an equivalence relation on the class of topological spaces. Two Tychonoff spaces X and Y are said to be fu-equivalent if there exists an uniform homeomorphism h: Cp(Y) ^ Cp(X) such that supph x and supp x are finite sets for all xeX and ye Y. This is an intermediate relation between relations of u- and /-equivalence. In this paper it has been proved that the Lindelöf number is preserved by the relation of fu-equivalence.
Keywords: u-equiva/ence; Linde/öf number; Function spaces; Set-va/ued mappings.
Introduction
All spaces below are assumed to be Tychonoff. RX is a space of all real-valued functions on X, Cp(X) is a space of all real-valued continuous functions on X equipped with the topology of pointwise convergence.
Cp(X |F) = {/eCp(X ): f (x) = 0 for all xeF}, where F is a subset of X. The restriction of a function f to a subset A is denoted by f |A. The cardinality of a set A is denoted by |A|. K0 is the countable cardinal, l(X) is the Lindelöf number of X. Fin A is a family of all finite subsets of a set A. For a set-valued mapping p: X ^ Y and sets AcX and BcY, we define the image of A as a set p(A) = У {p(x): x e A}, and preimage of B as a set
p~x (B) = {x e X : p( x) П B Ф0} .A set-valued mapping p: X ^ Y is called lower semi-continuous if a preimage of every open subset of Y is open in X. It is called surjective if for any ye Y there exist xeX such that yep(x).
1. Concept of the support
Definition 1.1. Let X be a topological space. A linear subspace AcRX is called suitable if for any point xeX, its open neighborhood Ox, and two functions f, f'eA there exist a function feA such that f (x) = f " (x) and f (x') = f ' (x') for all x'eX \Ox. A point xeX is said to be a zero-point of a family AcRX if fx) = 0 for all feA. Denote by ker A the set of all zero-points of a family A.
The examples of suitable subspaces are Cp(X), Cp(X |F), where F is a closed subset of a space X, and then ker Cp(X |F) = F. Following definitions are analogous to the definitions introduced by O.G.Okunev in [4] for ¿-equivalent spaces.
Definition 1.2. Let X, Y be topological spaces, A, B - suitable subspaces of spaces RX and Ry, respectively, and let h:B ^ A be an uniform homeomorphism such that the image of the zero-function 0YeB under h is the zero-function 0X eA. Fix a point xeX
and e > 0. We call a point ye Y s-essential for x (under h) if for any open neighborhood Oy of y there exist functions g, g'eB, coinciding on the set Y \Oy and satisfying the following inequality:
|h(g)(x) - h(g'' )(x)| > s. (1)
Furthermore, we say that a point y is s-inessential for x if it is not s-essential for x, and call the set of all points that are s-essential for x the s-support of x (under h) and denote it by supp^ x. The union of s-supports of x (under h) over all positive s is called the support of x (under h) and is denoted by supph x. If h is fixed, then we write suppe x (supp x, respectively).
Remark 1.3. If xeker A, then supp x = 0.
It is clear that if s < S, then supps xcsuppe x, therefore supp x = U supp17n x. It is
not difficult to verify that suppe x is a closed set. Then we have the following two properties of the support:
(i) suppe x is a nonempty finite subset of Y for any s > 0 (if xiker A);
(ii) supp:X ^ Y is a countable-valued, surjective, lower semicontinuous mapping.
To prove these properties, we note some results of S.P.Gul ' ko [3]. Let X, Y be topological spaces, A, B - suitable subspaces of spaces RX and RY, respectively, and let h:B ^ A be an uniform homeomorphism such that the image of the zero function 0Y eB under h is the zero function 0X eA. Let xeX, S > 0, and let KcY be a finite subset. We introduce into consideration the quantity
a(x,K,S) = sup |h(g)(x) - h(g' )(x)|, (2)
where the supremum is taken over all g', g' 'eB such that |g' (y) - g'' (y)| < S for all yeK. This definition was introduced by S.P.Gul ' ko in [3]. We also define
a(x,K,0) = sup |h(g )(x) - h(g' ')(x)|, (3)
where the supremum is taken over all g', g' 'eB coinciding on K (if K is empty, then the supremum is taken over all g', g''eB). It is obvious that if 0<Si<S2, then a(x,K,Si)^a(x,K,S2), and if KicK2cY, then a(x,K2,S)<a(x, Ki,S) for all S>0. In [3], it was proved that for all xeX \ker A there exist a nonempty finite subset K(x)cY such that
(1) a(x,K(x),S) < x for any S > 0,
(2) a(x,K' ,S) = x for any proper subset K' of K(x) and any S > 0.
For all xe ker A we put K(x) = 0. We now prove that the set K(x) has a stronger property which we get substituting S > 0 for S>0 in (2). To prove this, we need the following
Lemma 1.4. If a(x,K,0) < x, then a(x,K,S) < x for all S > 0.
Proof. Fix xeX and finite KcY such that a(x,K,0) < x. We prove that the function 8 ^ a(x, K, 8) is continuous at the point 0. Let S > 0. Since h is an uniform homeomorphism, there exist a finite set K' cY and S > 0 such that for all g, g 'eB we have the implication
(|g(y) - g' (y)| < S for all yeK')^|h(g')(x) - h(g'' )(x)| < s.
Let g , g 'eB and |g' (y) - g'' (y)| < S for all yeK. Since B is a suitable subspace, there is a function geB such that
( . Ig'(y), y e K; (4)
g(y) = W ^ (4)
Ig (y), y e K' \ K.
Then |g(y) - g' (y)| < S for all y eK', hence, |h(g)(x) - h(g'' )(x)| < s. Now by the triangle inequality we obtain
|h(g')(x) - h(g ' )(x)|<|h(g )(x) - h(g)(x)|+|h(g)(x) - h(g' )(x)| < a(x,K,0)+s.
Passing to the supremum over all g, g 'eB such that |g (y) - g' (y)| < S for all yeK we have inequality a(x,K,S)<a(x,K,0)+s, which implies that the function 8 ^ a(x, K, 8) is continuous at the point 0. ■
For any xeX put a(x) = a(x,K(x),0). Now we can rewrite the properties of the setvalued mapping x ^ K(x) in new notations.
(K1) If g, g 'eB and g |k(x) = g' |k(x), then |h(g)(x) - h(g' )(x)|<a(x).
(K2) For any proper subset K' of K(x) and any real b there exist functions g, g'eB such that g |k = g ' |k and |h(g )(x) - h(g' )(x)| > b.
Besides, this mapping suijectively maps the space X \ker A onto Y \ker B (see [3]),
i.e., for any ye Y \ker B there exist xeX \ker A such that yeK(x). Now we shall prove the properties (i) and (ii) of the support suppe x.
Lemma 1.5. K(x)csuppe x for any s > 0.
Proof. Let xiker A. Fix a point j0eK(x) and s > 0. We shall show that y0 is s-essential for x. Put a = max(s,a(x)), K' = K(x)\{y0}. By the property (K2), there exist functions g, g'eB such that g|K = g' |K and |h(g)(x) - h(g'' )(x)| > 2a. Then there is a neighborhood U of y0 that does not meet K'. Choose a function geB such that g|Y \^ = g|y\u and g(yo) = g' (yo). Then we have g|*w = g ' |*w and |h(g)(x) - h(g' )(x)| < a(x) <a. By the triangle inequality we obtain that |h(g)(x) - h(g' )(x)| > a>s. Besides, g coincides with g on the set Y \U. By definition this means that y0 is s-essential for x. ■
So, lemma 1.5 implies that the set suppe x is nonempty for any s > 0 and any xiker A, and it also implies that the set-valued mapping x ^ supp x from X \ker A onto
Y \ker B is surjective.
Lemma 1.6. The set suppe x is finite for any s > 0.
Proof. Let xiker A and s > 0. Since h is an uniform homeomorphism, there exist a finite set KcY and S > 0 such that for all g, g 'eB we have the implication (|g (y) -g' (y)| < S for all yeK)^|h(g' )(x) - h(g'' )(x)| < s. Let us show that suppe xcK. Fix y0 in
Y \K. Then there is a neighborhood U of y0 that does not meet K. Choose functions g, g'eB coinciding on the set Y \U. Then they coincide on K, hence, |h(g)(x) -h(g ' )(x)| < s. By the definition this means that y0 is s-inessential for x, i.e., y0£suppe x. Thus, suppe xcK. ■
Property (i) of the support is proved. For the proof of property (ii) we introduce into consideration the set Ke(x)cY for any xeX \ ker A and any s > 0, satisfying the following properties:
(KE0) Ke (x) is finite and nonempty;
(KE1) a(x, Ke (x),0)<s;
(KE2) a(x,K' ,0) > s for any proper subset K' of Ke(x).
Such a set we could obtain from the set K from the previous proof, decreasing this set while it satisfies point (KE2). There can be several sets, satisfying properties (KE1) and (KE2), then we denote by Ke (x) anyone of them. The following lemma is an analogous to result obtained by O.G.Okunev [4] for ¿-equivalence.
Lemma 1.7. Let AcCp(X), x0eX \ker A, s > 0, G is open subset of Y such that suppe x0 nG/0. Then there is an open neighborhood U of x0 such that Ke (x)nG^0 for all x from U.
Proof. We may assume that suppe x0 nG = {y0}, where y is any s-essential point for x0. By definition, for the neighborhood G of y there exist functions g, g 'eB coinciding on Y \G such that |h(g)(x0) - h(g'' )(x0)| > s. Put U = {xeX: |h(g)(x) - h(g'' )(x)| > s}; then U is an open neighborhood of x0. Let us check that Ke(x)nG^0 for all x from U. Assume the converse. Let xe U be a point such that Ke(x)nG = 0. Then g coincides with g ' on Ke(x). Therefore |h(g )(x) - h(g'' )(x)|<s, A contradiction with xe U. ■
From the result obtained by O.G.Okunev for t-equivalence it follows that the setvalued mapping suppe has a weaker property then the lower semicontinuity. In our case this property of suppe implies the lower semicontinuity of the mapping supp. To prove this fact we shall need
Lemma 1.8. Let xeX \ker A, s > 0. There exists S > 0, such that Ke(x)csupps x. Proof. Fix a point _y0eK£(x). Put K = Ke(x)\(y0}. By definition of Ke(x), there exist functions g, g'eB coinciding on K' such that |h(g)(x) - h(g'' )(x)| > s. There exists S0 > 0 such that
|h(g)(x) - h(g' )(x)| > s+S0. (5)
Let us show that y0 e suppSo x. Choose a neighborhood U ofy that does not meet
K, and a function geB coinciding with g on Y \U such that g(y0) = g' (y0). Then g coincides with g on Ke(x), hence, |h(g')(x) - h(g)(x)|<s. From this and inequality (5) it follows that |h(g)(x) - h(g)(x)| > S0. But g coincide with g on Y \U, consequently, y is S0-essential for x. Let ' s enumerate all the points of the set Ke(x) = {yi,...,yn}, and for anyy chose S; so that y is Sressential for x. Put S = min{S;: i<n}; then Ke(x)csupps x. ■
Theorem 1.9. If AcCp(X), then the set-valued mapping supp:X ^ Y is lower semi-continuous.
Proof. Put ^(x) = supp x. We need to show that for any nonempty open set GcY it' s preimage ^-1(G) = {xeX: ^(x)n G^0} is open in X. Let GcY is any nonempty open set such that ^-1(G)^0, and let xe^-1(G). Then there exists s > 0 such that suppe xnG^0. By Lemma 1.7 there exists an open neighborhood U of x such that KE(z)nG/0 for all z from U. By Lemma 1.8, for all zeX and s > 0 we can find S0 > 0 (depending on z and s) such that KE (z) c suppS|j z c supp z , i.e., supp znG^0, hence, ^-1(G) is open, and the
mapping supp is lower semicontinuous. ■
Besides, the set supp x has the following property.
Theorem 1.10. Let xeX.
(a) If two functions g, g 'eB coincide on the set supp x, then h(g' )(x) = h(g'' )(x).
(b) If F is a closed subspace of Y such that h(g' )(x) = h(g'' )(x) for any two functions g, g 'eB coinciding on F, then supp xcF.
Proof. (a) Let s > 0. Fix Kg(x). Let functions g , g e.B coincide on the set supp x. By Lemma 1.8, Ke(x)csupp x, therefore, |h(g' )(x) - h(g'' )(x)|<s. Since s is arbitrary, we obtain h(g )(x) = h(g' )(x).
(b) Assume the converse. Let ye (supp x)\FV0. There exists s > 0 such that yesuppe x. Let U is an open neighborhood of y that does not meet F . Then there are g , g'eB coinciding on Y \U such that |h(g)(x) - h(g'')(x)| > s. But in this case g coincides with g ' on F, whence h(g' )(x) = h(g'' )(x). This contradiction proves the theorem. ■ The concept of the support can be generalized.
Definition 1.11. If h:B ^ A is an arbitrary uniform homeomorphism we shall define a mapping h : B^ A by the formula h (g) = h(g) - h(0Y) for all geB. Then h is also an
uniform homeomorphism and h*(0Y) = 0X. Put
h h* h h*
suppe x = suppe x, supp x = supp x,
h-1 (h*)-1 h-1 (h*)-1
suppE y = supp^ 7 y, supp y = suppv ' y.
2. Main result
Definition 2.1. Two Tychonoff spaces X and Y are said to be fU-equivalent if there exists an uniform homeomorphism h:Cp(Y)^Cp(X) such that supph x and supph y are finite sets for all xeX and ye Y.
The main result of the paper is following.
Theorem 2.2. If X and Y are fU-equivalent then /(X) = /(Y).
For the proof we need some notions.
Definition 2.3. Let y:X^ Y be a finite-valued, surjective, lower semicontinuous mapping of X to Y. For y and any UcY we put y*(U) = (xeX: y(x)cU}.
We denote by T the family of all open subsets of Y.
Definition 2.4. A set-valued mapping G : T ^ X is said to be y-extractor (simply extractor) if the following conditions hold:
S(1) y*(U)cG(U) for any UeT;
S(2) For any increasing consequence (U„)„eN, UneT such that
X = un G (Un) (6)
keN n>k
the following equality holds:
Y = U Un. (7)
The complement of G(U) to X we denote by F(U) and the set-valued mapping
F: T ^ X we call y-co-extractor (simply co-extractor).
The concept of extractor was introduced by A.Bouziad in [1].
Let be an open cover of Y closed with respect to finite unions. Fix any infinite cardinal t. Let us introduce some notations. Put [U]T = {U': U' c U, |U'| < t}.
We say that the set AcX has a type FT in X or A is FT-subset ofX, where t is a cardinal, if A can be represented as a union of a family ^of closed subsets so that |F| < t . By T'i we denote the family of all subsets of X that have a type FT. Denote by Xx the family of all subsets A of X such that /(A)<t.
Let /(X)<t. Then T'i cXx. Define the mapping
U :FinFT ^ [U]T, U = U(F), F e FinFT, (8)
which will be called y-constructor from ^ (simply constructor). For this aim we define
a number p(x) = |y(x)| for any xeX. For any FcX we put p(F) = min(p(x): xeF}. The number p(F) is said to be level of F. Further, for any UeT and keN put = (xeX: |y(x)nU| > k}. Since y is lower semicontinuous, we see that U[k] is open in X for any keN (it can be empty for k > 2).
Suppose F = {FFn} c FT. Put Fa = P| Fi for any Ac(1, ... , n}, A^0 and put
ieA
F = {Fa ^ 0 :0 ^ A c {1,______, n}}, i.e., F is the family of all nonempty intersections of
elements from F. Let F e F and k = p(F). It is not difficult to verify that the family U[k] = {U[k] : U g U} is an open cover of F which Lindelof number does not exceed t, hence it contains a subcover {U[k] : U g Uf } of F, where UF c U such that |Uf| < t . Put U(F) = U (U UF). Obviously, U(F) e [U]T, and if Ft c F2, then U(Ft) c U (F2).
FeF
The constructor (8) is defined. Note one property of the constructor.
(*) For any F e FinFT, nonempty F e F , and xeF the following inequality holds:
|<p(x) n U(F)| >p(F). (9)
Definition 2.5. An open cover U of Y is said to be T-trivial if it has subcover which cardinality does not exceed t. Otherwise this cover is said to be T-nontrivial.
Proof of theorem 2.2. Let h:Cp(Y) ^ CP(X) be an uniform homeomorphism such
that supph x and supph y are finite sets for all xeX and ye Y, and let /(X)<t. Every uniform homeomorphism between Cp-spaces can be extended to some uniform homeomorphism h* between the spaces of all functions (see [2]). The following lemma states that the mapping supph will not changes if we substitute the uniform homeomorphism h:Cp(Y) ^ Cp(X) for its extension h*:RY ^ RX.
Lemma 2.6. Let h:Cp(Y) ^ Cp(X) be an uniform homeomorphism and h*:RY ^ RX - its uniformly continuous extension. Then supph x = supph* x for any xeX.
Proof. Let y be £-essential point for x under h for some s > 0. It follows from definition, that y is s-essential for x under h*. Therefore, supp^ x c supp^* x. Then, we shall prove that supp^* x c supp^ x if 0 < S < s. Let y be s-essential point for x under h* for some s > 0, and let 0 < S < s. Put s0 = (s - S)/2. Let Oy be an open neighborhood of y There exist functions g0', g0''eRY, coinciding on the set Y \Oy and satisfying the following inequality:
|h*( go')(x) - h*( go'')(x)| > s.
Since h* is an uniform homeomorphism, there exist a finite set KcY and A > 0 such that for all g , g''eRY we have the implication
(|g'(y) - g''(y)| < A for allyeK) ^ |h*(g')(x) - h*(g'')(x)| < so. (10)
Put F = KnOy. There is a function gi e Cp(Y) such that
gi|K = go' K (11)
and a function g2e CP(Y) such that g2|Y \Oy = gi\r \Oy^ g2|F = go'' If . Then
g2k = go '' K (12)
By (11), (12), and (10) we have
|h*(g0 ')(x) - h(gl)(x)| < ^ |h*(g0 '' )(x) - h*(g2)(x)| < s0,
hence,
|h(gi)(x) - h(g2)(x)| > |h*(go ')(x) - h*(go'' )(x)| - |h*(go ')(x) - h(gl)(x)| -- |h(g2)(x) - h*(go '' )(x)| > s - 2so = S.
Inclusion supp^* x c supp^ x is proved. This completes the proof. ■
Further, we can assume without loss of generality that h is an uniform homeomor-phism from RY to RX satisfying following conditions:
1. h(Cp(Y)) = Cp(X) and h-1(Cp(X)) = Cp(Y);
2. h takes zero-function 0Ye Cp(Y) to zero-function 0Xe CP(X);
3. supph x and supph y are finite sets for all xeX and ye Y.
Suppose that /(Y) > t to obtain a contradiction. In our terminology it means that there exists T-nontrivial open cover U of Y. We can assume without loss of generality that U is closed with respect to finite unions and UcS, where S is a base of Y which consists of all functionally open subsets of Y. Family S is closed with respect to finite unions. Let y = supph:X ^ Y. Note an important property of y.
(O) If g', g'' eRY and g'|V(X) = g''|V(X), then h(g')(x) = h(g'' )(x).
For any AcY define the function eAeRY by the formula
JO, y e A;
eA (y) = \, „ , (13)
11, y £ A.
For every open set VeT put
G(V) = {xeX: h(eV)(x) = 0}, F(V) = {xeX: h(eV)(x)^0}.
Lemma 2.7. G is y-extractor.
Proof. Check that condition S(1) is fulfilled. Let VeT and xey*(V). Since y(x)e V, we have eV|9(I)=0, hence, by property (O), we get h(eV)(x) = 0, i.e., xeG(V).
It remains to check S(2). Let (U„)„eN, UneT, be any increasing consequence, satisfying condition (6). Assume that Y ^ |J Un . Put U = U Un . Let yeY \U. Choose a
neN neN
finite subset K = {x1, ... , x^}cX and S > 0 so that for any functionfeRX the following implication holds:
( |f(x)| < S for all ie{1, ... ,rt) ^ |h-1(f)(y)| < 1.
Such a choice is possible because of the continuity of the mapping h1 and the condition h-1(0X) = 0Y. By condition (6), we can choose a number N such that x;eG(U„) for all n > N and ie{1, ... ,_p}, i.e., h(eUn)(xi) = 0 . Passing to the limit as n^x, we obtain
h(eU)(xi) = 0 for all ie{1, ... ,_p}. Then from (13) we have |eU(y)| < 1, hence,yeU. This contradiction concludes the proof. ■
Now denote by C the family of all functionally closed subsets of Y. Any functionally open subset V admits a decomposition V = U Fn where FneC and F„cF„+1 for all
neN. If there exists a decomposition satisfying the following condition:
y* (V)\y*(F„)^0 for all neN, then we say that the subset V is adequate. This notion was introduced by A.Bouziad in [1].
Lemma 2.8. Let t be an infinite cardinal, UcS - an open, T-nontrivial cover of Y, closed with respect to finite unions, and {Ut}teT cU- a subfamily such that |T|<t. Then there is a family {Vs }seS c [U]K|j, closed with respect to finite unions, such that
1. |S|<t,
2. each set Vs is adequate,
3. u U c u V .
teT seS
Proof. Put U = U Ut . Since the cover <U is T-nontrivial, we have Y\U^0. Choose
teT
x^X such that y(xi)^U and a set Vie'U such that y(xi)cVi. Suppose xi, ... , xn and Vi, ... ,Vn are already chosen. The set Y \ (U uViu ... uVn) is nonempty, hence there is an element xn+ieX such that y(x„+i)^U uViu ... uVn. There exist V„+ie^ such that y(x„+i)cV„+i. We get two consequences (x„)„sN, xneX and (V„)„sN, V„e^ for all neN. Put V = U Vn . Let {Ws}seS be the family of all finite unions of sets from the family
neN
{Ut}teT. For each seS put V, = WsuV. It is clear that the family {Vs}seS c [U]K|j is closed with respect to finite unions, |S|<t, and U Ut c U Vs . Let us check that each V,
teT seS
is adequate. Let seS. Fix a decomposition (Fns )neN of Ws and a decomposition ()neN of Vk, keN. The consequence (G^)neN, where Gsn = F£ uF,1 u... uFn", is a required decomposition of Vs. Besides, we have (x„)„eNcy*(Vs) and x«+i ¿V*(GSn) for all neN. ■
Lemma 2.9. Let S be infinite set and {Vs}seS be a family of adequate functionally open subsets of Y, closed with respect to finite unions. Then F(|J Vs) is F-subset ofX,
seS
where t = |S|.
Proof. Put V = U Vs . Let ()neN be a decomposition of V, such that Fns e C and
seS
c F+ for all neN. For any natural n and seS choose a function gsn e C(Y) such
that
s — fi s I _ 1
gn Fs ~ 0 YWs _ 1'
Fn 1 s
For every xe9*(Vs), and every natural k put
Uk (x) = {x' e X{ (g{({)) (x')- h (gk+„(x,s))(x)| <11 k},
where n(x, s) is the least number n such that y(x) c FS. Then UjS (x) is an open neighborhood of xeX. Put
As = n U Uk (x), B, = {xeX: «P(x)n(V\V,) * 0}, A = f| (A uB).
keN xe9*(Vs) seS
We now prove that G(V) = A. Since the sets A, and Bs are Gg-subsets of X, it will be enough for the proof of the lemma. First we shall show that
F(V) c X\ A. (14)
Let x' eF(V). Then there exists s > 0 such that |h(eV)(x')| > s. Since y(x ') is a finite set and the family {Vs}seS closed with respect to finite unions, there exists seS such that
^(x')nVcV,, i.e., x'¿Bs. Note that, since, eV |9(x= eV |9(x.) by (®) we have
h(eV )(x') = h(eV)(x'). Since h(eV )(x') = |h(eV)(x')| > s and lim gsn = eV , there exists
natural N such that
|f (gn)(x')| >s for all n > N.
Choose a number k such that k > max{N,1/s}. We shall check that
x' i u US (x).
XEf*(Vs )
It will imply that x'iAs, and inclusion (14) will be proved. Let xey*(Vs). Note that,
since gk+n(x,s) |<p(x) = 0 , we have h(gk+n(X J) )(x) = 0 . Then
Ih (gk+ n(x,s) ) (x ') - h (gk+n(x,s) ) (x) = |h (gSk + n(x,s) ) (x ')\ > S ^ k ,
i.e., x' iUk (x). Inclusion (14) is proved.
Let us prove the inverse inclusion X \AcF(V). Since the sets Vs are adequate, we can assume that there decompositions satisfy the following property: y *(Vs )\ y * (Fns) ^ 0 for all neN. Let x'^A. Choose seS such that x'iAsuBs. Then we have y(x')n VcVs. Fix natural k such that x' g U Uk (x), natural m such that y(x') n V c F^n , and
XEf*(Vs )
x0ey*(Vs) such that y(x0) £ Fm . Then we have n(x0, s) > m and 9(x') n V = 9(x') n Vs c Fm c F+„(Xo,,).
Put i = k+n(x0, s). Since x' iUk (x0), we have |h(gi)(x') - h(gi)(x0)| > 1/ k . Besides, h(gi)(x0) = 0. From this we obtain that |h(gst)(x')| > 1/ k. But since
ev Lx') = evs |<p(x') and evs Lx') = g/ Lx') , we have h(g/)(x') = h(ev)(x'), and, í'inallУ, |h(eV)(x')|>1/k, i.e., x'eF(V). The statement of the lemma is proved. ■
By 2.9 and 2.8 we have
Theorem 2.10. Let t be an infinite cardinal and UcS - an open, T-nontrivial cover of Y, closed with respect to finite unions, {Ut}teT c U - a subfamily such that |T| < t. Then there is Ve [U]T such that |J Ut c V and F(V) is FT-subset ofX.
teT
Now we have all the facts necessary to prove the result, formulated in the beginning. We shall construct by induction increasing consequence (V„)„sN, Vne[U]T such that
Y = U Vn . Simultaneously we shall construct the consequence CF)«eN, TneFin TT,
neN
such that Tn'cTn' for n' < n'' . For this aim we shall use the constructor U and the coextractor F defined above.
Put To = X. Pick a set Ve[U]T such that U(To)cV and F(Vi)eTT (this is possible by Theorem 2.10), and put Ti = {X, F(V1)}. Choose the set V2e[U]T so that V1uU(Ti)cV2 and F( V2)eTT . Suppose we have already defined the sets Ve [U]T and TeFin TT for all numbers i such that 1 < i < k, satisfying the following conditions:
1. F(Vi)eTT, 1 < i < k;
2. V,uU(TDc V+i, 1 < i < k - 1, where T = {X, F(Vi), ... , F(V)}, 1 < i < k.
Choose the set Vk+i e [U]T so that the following conditions hold:
VkUUC5)cVw and F(Vk+i)erT. (15)
Put f+i = {X, F(Vi), ... , F(Vk+i)}. The consequences (V„)„en, V„e[^]T and CT„)„eN,
T^eFin fT are already defined. Prove by induction over n the following statement:
(ST) For any natural n and any subset {j, ... ,jk}c{1, ... , n} such that F(Vj ) n ... n F(Vjk ) ¿0, the following inequality holds:
p(F(VJi ) n ... n F(VJk )) > k +1. (16)
For k = 1 it is enough to show that p(F(Vn))>2. For any xeX by inequality (9) we have |y(x)nV„| > |y(x)nVi| > |y(x)nU(To)| > p(X) > 1. Therefore, if p(x) = 1 for some xeX, then y(x)cV„, consequently by S(1) we have xgF(V„), and thus, p(F(Vn)) > 2.
Suppose statement (ST) is true for all numbers n such that 1<n<N. Let us prove that it is true for n = N+1. It suffices to show that for any subset {j, ... ,jk}c{1, ... , N} such that F = F (Vj ) n ... n F (Vj ) n F (VN+1 ) ^0, we have p(F) > k+2. Put
F ' = F(Vj ) n ... n F(Vj ), then F = FnF(VN+i). By the inductive assumption, we have p(F) > k+1. Assume that p(F) = k+1 to obtain a contradiction. Let xeF such that |y(x)| = k+1. Since F ' e Fn , we see that by condition (15) and inequality (9), it follows that |y(x)nVN+i| > |y(x)nU(fv)| > p(F) > k+1. Hence, y(x)cVN+i, and by condition S(2), it follows that x£F(VN+1), therefore, x^F. This contradiction concludes the proof of statement (ST). In particular, inequality (16) involves that for any xeX there is a number n0 such that xiF(V„) for all n > n0, i.e., xeG(Vn). In other words, equality (6) holds. So, by condition S(2), we have Y = |J Vn . Since V„e[^]T for all neN, we see
neN
that the cover <U of Y is T-trivial, a contradiction. So, l(Y)<T. Consequently, l(Y)< l(X). Analogously, l(X)< l(Y). Theorem is proved. ■
REFERENCES
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2. EngelkingR. General Topology (PWN, Warszawa, 1977).
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4. Okunev O. Homeomorphisms of function spaces and hereditary cardinal invariants // Topol. and its Appl. 1997. V. 80. P. 177 - 188.
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Статья принята в печать 07.07.2008 г.
