Arch. Math., Vol. 60, 543-545 (1993) 0003-889X/93/6006-0543 $ 2.10/0 9 1993 Birkhduser Verlag, Basel A theorem of insertion and extension of functions for normal spaces By M. LOPEZ-PELLICER,S. ROMAGUERAand A. GUTII~RREZ In this paper we use the technique of C~-binary relations in order to prove two characterizations of normality in terms of insertion between certain F~ and G~ subsets. They allow us to deduce the main result of the paper, a theorem of insertion and extension of functions, which has as easy corollaries the Katetov-Tong's theorem (proved by Katetov in [2] and [3] and by Tong in [5]) and the Tietze-Uryshon's theorem ([6], 15.8). Along the paper, N represents the real line with its usual topology, q~ = {q,},,G ~ is the countable set of rational numbers, and N is the set of the natural numbers. By function we mean real function; the upper (lower) semicontinuous functions are abbreviated to u.s.c. (1.s.c.) functions. We do not suppose any separation axiom for normal spaces. By A and .4 we denote, respectively, the interior and the closure of A. D e fi n i t i o n. Let X be a set, P(X) is the collection of subsets of X and cg ~ P(x) is a family which is stable under countable unions. A binary relation 0 in P(X) is called a C~-binary relation if: a) Given two finite subsets J and N' of P(X) such that A ~ B whenever A e sd and B ~ ~ , there is C 6 cg such that AQC and CoB for every A e ~4 and B e -~. b) GivenD, E,F, G E P ( X ) , i f D c E , E ~ F a n d F c G , t h e n D o G a n d E c F . Lemma 1. Let 0 be a off-binary relation in P(X). Let d - {A,},~= 1 and ~ = {B,}~= 1 two collections of elements of P(X) and let A, B ~ P(X) such that 0 A. c A, B ~ ~ B., n-1 n=l A, OB and A OB,, for every n ~ N. Then there is C ~ c# such that A nOC and CoB n for every nE~. P r o o f. We shall determine, by induction, two sequences {C ,}n= ~ 1 and {Dn},% 1 of elements of Cg such that AmoCm, C~oD, and D, oB,, for each m, n ~ N. Then C = 0 Cn verifies the lemma. "= ~ In fact, by applying the cg-interpolation to {A~} and {B}, we obtain C~ E cg such that AzQCzoB; in the same way, by taking {A, C~} and {B1}, there is 01 e cg such that AoO~, CloD1 and D1QBl. Let us suppose Ck, D~ ~ c~, k = 1, ..., n - 1 such that Aio Ci, C~QB,C~oDj and DjQBjfor i,j = 1..... n - 1. By cg-interpolation between {An} and {B, D1,..., D,_ t}, we determine 544 M. LaSPEZ-PELLICER,S, ROMAGUERAand A~ GUTLgRREZ ARCH,MATH. C, e ~ such that A~o C~, C~o~Band C, oDi forj = 1, ..., n - 1oBy c4-interpolation between {A, C~ ..... C~} and {B~}, we determine D~ e ~ such that AoD~, C~o~D~for i = i;...:, n, and: DnQB~. Therefore, we have obtained by induction the sequences {C.}2;~ and { D } ~ previously supposed. Theorem 2. (Theorem of insertion between F~ and Ga). A space (X, T) is normal if, and only if, given a F~ subset F and a G~ subset G so that F ~ G and F c ~, then there is a subset H such that F ~ I2I ~ H c G. P r o o f. For the necessity, let F = 0 C,~ and G = (~ O. subsets of X Verifying the ~t=l n=l hypothesis. By taking ~f = T and the binary relation A~B if, and only if,/1 ~/}, for each A, B e P(X), we have that Q is a %binary relation. Let ~r = {C.}[~j, ~_= {O~}.=~, A = F and B = G. By Lemma 1, there isan open set H s u c h that F c H c H c G. The sufficiency is obvious, since every closed subset of X is F. and every open subset o f X i s G~. Corollary 3. A space (X, T) is normal if, and only if, given {F(q.)/q. e ~}, a .family of F~ subsets of X, and {G(q.)/q. ~ ~}, a family of G~ subsets of X, such that, for every m, n ~ N, a) b) F(q.) c G(q.) and F(qn) c d(q.) whenever q. e ~, and F(q.,) c F(q.) and G(q,.) ~ G(q.) whenever q,. < q., then there is a sequence {H(q.)/q.e~} of subsets of X such that F ( q . ) c f t ( q . ) H(q.) c G(q.) for every n ~ N and H(qm) c I~I(q.) when q,. < q.. Proof. For the necessity, Theorem2 gives us a subset H(qi) such that F(ql) ~ / t ( q l ) ~ H(qi) c G(ql). Let us suppose determined, by induction; the required sets H(q.), n I, "'" , m. By Theorem 2 applied to F = F(qm+ ~) u ( ~ ~r H (q.)/q~ < q,.§ ! and 1 < n < m}) and G = G(q~+i) c~ ( (-~ {121(q.)/q,.+~ < q. and 1 < n _< m}), we obtain a subset H(q.,+ ~) such that F c lgI(q,.+~) ~ H(q.,+ ~) ~ G; this set verifies the required conditions. Then, by induction, we determine the sequence {H(q.)/q. ~ if).}. The sufficiency is obvious. By [1] and [4] it is well-known that a function h defined in the topological space X is determined by a countable covering {H (q,,)/q. e 11~}of X with volt intersection such that h- 1 (1 _ co, q,[) c H (q,) c h- i (] _ co, q,]) for each n e N; it is said that h is the function determined by {H (q~)/q~ e ~}, By other hand, h is 1.s.c. if and only if H (q~) ~ H(q~); h is u.s.c, if and only if H(qm) c / ~ ( % ) and h is continuous if and only if H(q~) c / ~ (qn) whenever q , , < q , since h - l ( l - o o , r[)=~{I21(%)/q,<r} and h - l ( ] - c o , r]) = c~ {H(q,)/q, > r} for every reN.. Lastly, given another function g, determined by {G(q,)/q, ~ ~}, g 6 h if and only if H(qm) c G(q,) when q,, < q,. Now we can prove the following theorem for insertion and extension of functions~ Theorem 4. A space (X, T) is normal if and only if given the functions g < f defined in X, being 9 u-s.c-: in X , f 1.s.c. in X and f continuous in the closed subset C c X, Vol. 60, 1993 Insertion and extension of functions 545 there is a function h such that h is continuous in X, g < h ~ f and h (x) = f (x) for every x~C. P r o o f. F o r proving the sufficiency, if A is closed in X and B is an open neighbourh o o d of A, we m a y take the characteristic function of B as f, the characteristic function of A as # and C = O. Now, let (X, T) be a n o r m a l space. F o r every q, e ~ , let F (%) = f - 1 (] _ ~ , q,[), which is F~, and G (q.) = g - i (] _ 0% q.]) c~ [ f - i (] _ oo, q.]) u (X - C)] = c~{g- 1 (] - o% qm[)/ q. < qm} c~ [c~ { f -1 (] _ 0% q., D c~ C/q. < q.,} w (X - C)], which is Ga since f is continuous in C. The families {F (q.)}.~= 1 and {G (q.)}[=l verify the hypothesis of Corollary 3 because F (q.) c f - 1 (] _ 0% q.]) and f - 1 (] _ 0% r D u (X - C) is open in X. The family {H (q.)}.~= ~, provided by C o r o l l a r y 3, determines a continuous function h with g < h < f and h (x) = f (x) when x e C, since f-i(]_ 0% q,[)c~ C c H(q,)c~ C c f - ~ ( ] - 0% q,])c~C. Corollary 5. (Katetov-Tong's theorem). Given two functions g ~ f defined in the normal space (X, T), with 9 u.s.c, and f 1.s.c., there is a continuous function h defined in X such that g <=h <=f. P r o o f. It follows from T h e o r e m 4 by taking C = 0. Corollary 6. (Tieze-Urysohn's theorem). Let f be a continuous function defined in the closed subspace C of a normal space (X, T), such that If(x)[ <=M for each x ~ C. Then there is a continuous function h defined in X such that h (x) = f (x) for every x ~ C and [h(x)[ ~ M for every x ~ X. P r o o f. It follows from Theorem 4 with the functions f i ( x ) = ~ f ( x ) when x e C (M and when x e X - C g-----M. References [1] E BROOKS,Indefinite Cut sets for real functions. Amer. Math. Monthly 78, 1007-1010 (1971). [2] M. KATETOV,On real-valued functions in topological spaces. Fund. Math, 38, 85-91 (1951). [3] M. KATETOV,Correction to "On real-valued functions in topological spaces". Fund. Math. 40, 203-205 (1963). [4] M. H. STONE, Boundedness properties in function lattices. Canad. J. Mat. 1, 176-186 (1949). [5] H. TONG, Some characterizations of normal and perfectly normal spaces. Duke Math. J. 19, 289--292 (1952). [6] S. WILLARI),General topology. Massachusetts 1970. Eingegangen am2.4.1992 Anschrift der Autoren: Manuel L6pez-Pellicer Depto. Matemfitica Aplicada E.T.S.I. Agr6nomos Universidad Polit~cnica Camino de Vera s/n 46022-Valencia Spain Archiv der Mathematik 60 Salvador Romaguera Depto. Matemfitica Aplicada E.T.S.I. Caminos, Canales y Puertos Universidad Polit6cnica Camino de Vera s/n 46022-Valencia Spain Angel Guti6rrez Depto. Didfictica de la Matem/ttica Universidad de Valencia Alcalde Reig s.n. 46006-Valencia Spain 35