The Lindelцf Number is /u-Invariant .pdf IntroductionAll 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) = {feCp(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, /(X) is the Lindelцf 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 BciY, we define the image of A as a set p(A) = \J{p(x): x e A}, and preimage of B as a setp~x (B) = {x ё X : p( x) П B ф 0} .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 Ythere exist xeX such that yep(x).1. Concept of the supportDefinition 1.1. Let X be a topological space. A linear subspace Ac:RX 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 Ac:RX 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 CpX), 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 г-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 calledthe 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 < 8, then supps xcsuppe x, therefore supp x = j[ supp17n x. It isяеЛГnot 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 x^ker 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, 8 > 0, and let KcY be a finite subset. We introduce into consideration the quantitya(x,K,8) = sup |h(g' )(x) - h(g'' )(x)|, (2) where the supremum is taken over all g', g' 'eB such that |g' (y) - g'' (y)| < 8 for all yeK. This definition was introduced by S.P.Gul ' ko in [3]. We also definea(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 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 8 > 0 for 8>0 in (2). To prove this, we need the followingLemma 1.4. If a(x,K,0) < да, then a(x,K,8) < да for all 8 > 0.Proof. Fix xeX and finite KcY such that a(x,K,0) < да. We prove that the function 8 н a(x, K, 8) is continuous at the point 0. Let 8 > 0. Since h is an uniform homeomorphism, there exist a finite set K' cY and 8 > 0 such that for all g', g' 'eB we have the implication(|g'(y) - g'' (y)| < 8 for all yeK')=>|h(g)(x) - h(g'' )(x)| < s. Let g , g 'eB and |g' (y) - g'' (y)| < 8 for all yeK. Since B is a suitable subspace, there is a function ge B such that( ) Jg'(y), y e K;Ig (У), У e K' \ K.Then |g(y) - g'' (y)| < 8 for all yeK, hence, |h(g)(x) - h(g'' )(x)| < s. Now by the triangle inequality we obtain|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 x^ker A, and it also implies that the set-valued mapping x н supp x from X \ker A onto•·\ker B is surjective.•·Lemma 1.6. The set suppe x is finite for any s > 0.•·Proof. Let x^ker A and s > 0. Since h is an uniform homeomorphism, there exist a finite set KcY and 8 > 0 such that for all g', g' 'eB we have the implication (|g' (y) -g'' (y)| < 8 for all yeK)=>|h(g')(x) - h(g'' )(x)| < s. Let us show that suppe xcK. Fix y0 in•·\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, K (x),0) 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 y0 is any s-essential point for x0. By definition, for the neighborhood G ofy0 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)| 0. There exists 8 > 0, such that Ke(x)csupps x.Proof. Fix a point _y0eKЈ(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 80 > 0 such that|h(g )(x) - h(g'' )(x)| > s+8o. (5) Let us show that y0 e suppS(] x. Choose a neighborhood U of y0 that does not meetK', 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)| 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 80 > 0 (depending on z and s) such that KE (z) с suppS(] z с supp z , i.e., supp znG^0, hence, tp-1(G) is open, and themapping 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 Ke(x). Let functions g', g''eB coincide on the set supp x. By Lemma 1.8, Ke(x)csupp x, therefore, |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. Puthh* hh*suppE x = suppE x, supp x = supp x, suppE y = supp;, 7 y, supp y = suppv 7 y.2. Main resultDefinition 2.1. Two Tychonoff spaces X and Y are said to be fU-equivalent if thereexists an uniform homeomorphism h:Cp(Y)^Cp(X) such that supph x and supph y arefinite 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 /(XX) = /(Y).For the proof we need some notions.Definition 2.3. Let tp:X -> Y be a finite-valued, surjective, lower semicontinuous mapping ofX to Y. For ф and any UcYwe put ф*(Ц) = {xeX: ф(х)сЦ}. 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 ф-extractor (simply extractor) if the following conditions hold: S(1) ф*(U)сG(U) for any UeT;S(2) For any increasing consequence (U„)„eN, UneTsuch thatX = UP G (U„) (6)keN n>kthe following equality holds:Y = U Un. (7)яеЛГThe complement of G(U) to X we denote by F(U) and the set-valued mapping F :T - X we call ф-co-extractor (simply co-extractor).The concept of extractor was introduced by A.Bouziad in [1].Let ЧЛ be an open cover of Y closed with respect to finite unions. Fix any infinite cardinal т. Let us introduce some notations. Put [U]T = {(J U': U' с U, |U'| < т}.We say that the set AcXhas a type FT in X or A is FT-subset ofX, where т is a cardinal, if A can be represented as a union of a family f of closed subsets so that |F| 2).Suppose F = {FlFn} с FT. Put FA = ff Fi for any Ac{1, ... , n}, A^0 and putF = {FA ф0 :0 ф A с {1,...,n}}, i.e., F is the family of all nonempty intersections ofelements from F. Let F e F and k = p(F). It is not difficult to verify that the family is an open cover of F which Lindelцf number does not exceed т,hence it contains a subcover {U[k] : U e UF} of F, where UF с U such that |Uf| 0. It follows from definition, that y is s-essential for x under h*. Therefore, supp^ x с supp^* x. Then, we shallprove that supp^* x с supp^ x if 0 < 8 < s. Let y be s-essential point for x under h* for some s > 0, and let 0 < 8 < s. Put s0 = (s - 8)/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 C^(Y) such thatgl|K = go' |k, (11) and a function g2eCp(Y) such that g2|7 \oy = gi|r \oy, g2|f = go'' |f . Theng2|K = go ' ' |k. (12)By (11), (12), and (10) we have|h*(go ' )(x) - h(gi)(x)| < so, |h*(go '' )(x) - h*(g2)(x)| < so,hence,|h(gi)(x) - h(g2)(x)| > |h*(go ' )(x) - h*(go'' )(x)| - |h*(go ' )(x) - h(gi)(x)| -- |h(g2)(x) - h*(go'' )(x)| > s - 2so = 8. Inclusion supp^* x с 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(C(Y)) = ОД and h-i(C(X)) = C(Y);2.h takes zero-function 0Ye Cp(Y) to zero-function 0Xe CP(XX);3..supph x and supph y are finite sets for all xeX and ye Y.Suppose that /(Y) > т to obtain a contradiction. In our terminology it means that there exists т-nontrivial open cover J of Y. We can assume without loss of generality that J is closed with respect to finite unions and ^cS, 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 ф = supph:X - Y. Note an important property of ф.(Ф) If g', g'' eRY and g' |ф(1) = g' '|ф(1), then h(g' )(x) = h(g'' )(x).For any AcY define the function eAeRY by the formula11, У 0 so that for any functionfeRX the following implication holds:( |f(x,)| < 8 for all ie{1, ... ,^}) => |h-i(f)(y)| < 1. Such a choice is possible because of the continuity of the mapping h_i and the condition h-i(0X) = 0Y. By condition (6), we can choose a number N such that x;eG(U„) for all n > N and ie{1, ... ,/?}, i.e., h(eUn)(xi) = 0 . Passing to the limit as и-да, we obtainh(eU)(xi) = 0 for all ie{1, ... ,/?}. Then from (13) we have |ey(y)| < 1, hence,yeU. This contradiction concludes the proof. ■Now denote by С the family of all functionally closed subsets of Y. Any functionally open subset V admits a decomposition V = j[ Fn where FneC and F„cF„+i for allneN. If there exists a decomposition satisfying the following condition:ф* (V)Vp*(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 т be an infinite cardinal, 4J.cS - an open, т-nontrivial cover of Y, closed with respect to finite unions, and {Ut}teT c J - a subfamily such that |Т| |(p(x)nU(3N)| > p(F) > k+1. Hence, (p(x)cVN+b and by condition S(2), it follows that x^F(VN+1), therefore, xЈF. This contradiction concludes the proof of statement (ST). In particular, inequality (16) involves that for any xe X there is a number n0 such that x^F(V„) for all n > n0, i.e., xeG(V„). In other words, equality (6) holds. So, by condition S(2), we have Y = [j Vn . Since V„e[rU]l for all neN, we seeneNthat the cover ЧЛ of Y is т-trivial, a contradiction. So, /(У)

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