Neubrunn, Tibor

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaTibor NeubrunnA generalized continuity and product spacesMathematica Slovaca, Vol. 26 (1976), No. 2, 97--99Persistent URL: http://dml.cz/dmlcz/136112Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1976Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slov., 26, 1976, No. 2, 97—99 A GENERALIZED CONTINUITY AND PRODUCT SPACES TIBOR NEUBRUNN A generalization of Kempisty's notion of quasicontinuity is known as somewhat continuity. The present paper shows tha t a separate somewhat continuity need not imply somewhat continuity, while a function / (x, y) qua-sicontinuous in one variable and somewhat continuous in the other variable is shown to be somewhat continuous. Kempisty's classical theorem on the quasicontinuity of separately quasicontinuous functions is obtained in a general setting as a corollary. The notion of a quasicontinuous function / : X -> Y where X, Y are topo-logical spaces was introduced for the case of Euclidean spaces by K e m p i s t y in [2]. In the general case (see e. g. [3]) / is said to be quasicontinuous at XQ e X i£ f"1 (V) n U contains a nonempty open set for any open sets U, V where x0 e U,f(x0) e V. I t is said to be quasicontinuous if it is quasicontinuous at any xo eX. The notion of a somewhat continuous function (see [1]) gen-eralizes the notion of quasicontinuity. / : X -> Y is said to be somewhat continuous if for any open f c 7 such that/~1(V) ^ 0 w e have int/_ 1(V) =£ 0. A theorem concerning quasicontinuity on product spaces, stating that separate quasicontinuity implies quasicontinuity, was given for the case of the function f(x, y) of two real variables by K e m p i s t y in [2]. JNT. F . G. Mar-t i n [3] has given a general version of Kempisty's theorem for the functions / : X x Y -» Z, where X is a Baire space, Y second countable and Z metric. If / is a function denned on the product space X X Y, we shall call an x — section for a given x e X the function fx defined on Y such tha t fx(y) = = f(x, y). The y — section fy for a given y e Y is defined analogouslv. Theorem 1. Let X be a Baire space, Y second contable and Z regular. Let f: X x Y ~> Z have all x — sections somewhat continuous and all y — sections quascicontinuous. Then f is somewhat continuous. Proof . Let / not be somewhat continuous. There exists G ^- 0 open such that /- i(G) -£ 0 and mtf-^G) = 0. Let (xo, yo) f~x(G). Choose G± open such tha t G\ <~ G, f(xo, yo) ~ G\. This is possible because of the regularity of Z. Owing to the quasicontinuity and hence somewhat continuity of fVo a t the point xo we have int fy](G\) -?-- 0. 97 Put U = int fy](Gi). For any x e U form fx\Gi). Since fx(y0) = f(x, yo) e Gi, we have fx1(Gi) i=- 0. The somewhat continuity of fz gives int fx^(Gi) =£ 0 for any x e U. Let {Vn} be a countable basis of the space Y. Define An as the set of all 00 x G U for which Vn <= i n t / ^ G i ) . Evidently \J An^U. n=l Let # <= U be any nonempty open set. Let us form 8 X Vn for given n. Because of the fact int f~x(G) = 0 there exists (x*, y*) e 8 X Vn such that f(x*, y*) $ G. Choose a neighbourhood G* of f(x*, y*) such that G*C\Gi = 0, Using the quasicontinuity of/,* at x* we obtain that there exists a nonempty set S' <= S such that f(x, y*) e G* for any x e S', hence f(x, y*) $ Gi.Thus y""^fx1(Gi). This implies Vn 4= ^(Gi), hence a; & An. Thus &' n An = 0. This means that -Aw is nowhere dense and the set U= [J Anis of the first category. This is n=l a contradiction. Theorem 2. Let X oe a Baire space, Y second countable and Z regular. Then a function f: X X Y -+Z quasicontinuous in each variable separately is quasicontinuous on X X Y. To prove the above Theorem we shall prove first the following. Lemma. A function f: X -» Y (X, Y arbitrary topological spaces) is quasi -continuous on X if and only if there exists a basis £8 of the space X such that for any element B e^ the restriction f/B is somewhat continuous. Proof . Necessity. Let Be@. Suppose that (f/B)~^ (V) ^ 0 for some V open. Then there exists x0 e B such that (f/B) (x0) e V. From the quasicon-tinuity of / at x0 it immediately follows that there exists a nonempty open set G <= B such that (f/B) (G) <= V. Hence int (f/B)~± (V) -^ 0. Sufficiency. Let x0 e X be any point, U an open set containing x0 and V an open set containing /(#0). Let B e & he such that x0 e B <= U. Consider the restriction//^. We have (f/B)~l (V) j± 0, hence int (f/B)-1 (V) -^ ^ 0 . Put G = int (f/B)-1 (V). Evidently ffc U and / (G) cz V. The quasi continuity of/ at #o is proved. Since x0 was arbitrary, the quasicontinuity of / on X follows. P r o o f of T h e o r e m 2. Let {Vn)n-i be a basis of Y and &l any basis of X . The collection of B X Vn, where n = 1, 2, . . . and 2? runs over ^?, is a basis of X X Y. Considering the restriction f/B X Vn, we see that it satisfies on each B X Vn the assumptions of Theorem 1. (B x Vn is considered with the relative topology). Hence f/B x Vn is somewhat continuous. Now the result follows from the lemma. 98 The following example shows that the somewhat continuity of f(x, y) in each variable separately does not imply the somewhat continuity of f(x, y) as a function of two variables. E x a m p l e . Define the functions fi,fz,fe, f* on <0, i) X <0, 1>, < i , 1> X <0, 1>, <0, 1> X < - i , 0) <0, 1) X <—1, — I) respectively. 1 if y is rational 0 if y is irrationa 0 if y is rational P u t f±(x, y) — I . .n . . , . i J v ' i j f (  if  is irrati al Mx, y) 11 if y is irrational ... . (1 if x is rational Mx'v) = (o if* is is irrational f ( . f 0 if a; is rational Functions / s , /6 , / ? , / s are defined on < - l , 0 ) X < | , 1>, < - l , 0) X <0, 1), < - l , i ) x < - l , 0) <—- \, 0) X <—1, 0) respectively, as follows Hx> y) ==/-(—x> —y), fe(x, y) =/3(—x, —y) fi(x,y) =/2(—a?, —.7), / s ( ^ 2/) = / i ( — » , —2/) Denote the interval <—1, 1> x <—1,1> a s / . Put/(o: , ?/) =• fi(x, y), where 1 ^ i ^ 8 . / is unambiguously defined on I by means of the functions ft. I t is easy to check that / is not somewhat continuous on I while the sections fz and fy are somewhat continuous for every x e <—1, 1>, y e <—-1, 1>, respectively. REFERENCES [1] GENTRY, K. R.—HOYLE, H . B . : Somewhat continuous functions. Czechosl. Math. J . 21 96, 1971, 5 - 1 2 . [2] KEMPISTY, S.: Sur les Functions quasicontinues. Fund. Math. 19, 1932, 184—197. [3] MARTIN, N. F . G.: Quasicontinuous functions on product spaces. Duke Math. J . 28 1961, 3 9 - 4 4 . Received September 24, 1974 Katedra matematickej analyzy Prirodovedeckej fakulty UK Mlynskd dolina 816 31 Bratislava 9 9 

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A generalized continuity and product spaces

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A generalized continuity and product spaces

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Institute of Mathematics AS CR, v. v. i.

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