Alas, O. T.

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

Toposym 3Ofelia Teresa AlasUniform continuity in paracompact spacesIn: Josef Novák (ed.): General Topology and its Relations to Modern Analysis and Algebra,Proceedings of the Third Prague Topological Symposium, 1971. Academia Publishing House of theCzechoslovak Academy of Sciences, Praha, 1972. pp. 19--22.Persistent URL: http://dml.cz/dmlcz/700711Terms of use:© Institute of Mathematics AS CR, 1972Institute 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.cz19 UNIFORM CONTINUITY IN PARACOMPACT SPACES O. T. ALAS Sao Paulo Let Ex and E2 be two nondiscrete paracompact Hausdorff spaces, VLX and U2 uniformities in Ex and £ 2 , respectively, and Ux ® U2 the product uniformity in Et x £ 2 . On [0, 1] we consider the usual metric topology (thus there is a unique uniformity U' in [0, 1]). We shall study the following question: Under what conditions every continuous function of Et x £ 2 (with the product topology) into [0, 1] is a uniformly continuous map of (Et x £ 2 , Ut ® U2) into ([0, 1], U')? 1. Preliminaries Let £ be a nondiscrete completely regular Hausdorff space. Definition. The index of £ is the least cardinal number for which there is a family (with this cardinality) of open subsets of £, whose intersection is not an open set. (Let us denote by m the index of £.) Let £ t and £ 2 be two nondiscrete completely regular Hausdorff spaces. For i = 1, 2 let U,- be a uniformity in Et (i.e. compatible with the topology for £;); Hi ® U2 denotes the product uniformity in Et x £2 , i.e. the uniformity in £ t x £ 2 for which the set {Ut ® U2 \ U1 e U l 5 1/ 2 e U2}, where Ut ® U2 = {((xt, x2), 0'i> yi)) | (*b yd e Ui9 i = 1, 2}, is a basis. Proposition. Suppose m > X0 and every locally finite open covering of E has cardinality less than m. We have: 1) if E is a normal space, then every subset of E of cardinality m has an accumulation point (in £) and every open covering of E of cardinality m has a subcovering of cardinality less than m; 2) if E is a normal space, then every closed subset of E which is the inter-section of at most m open subsets of E has a fundamental system 93 of open neigh-borhoods, whose cardinality |93| is less than or equal to m; 3) if m is the pseudoweight at some point x e £, then m is the weight at the point x; 4) if E is a topological group and m is the pseudoweight at the neutral element of E, then E is paracompact. 20 O. T. ALAS Proof. By the definition of index, m is a regular cardinal number. Since m > N0 and E is completely regular, every point of E has a fundamental system of neigh-borhoods, which are open-closed in E. (The closed G -̂subsets of E are open.) The union of less than m closed subsets of E is closed in E. Assertion 1) follows easily from the above consideration. Assertions 2) and 3) are proved by using the same technique. We shall prove 3). Let us denote by fi the first ordinal number of cardinality m and by M the set of all ordinals less than p. Since m is the pseudoweight at the point x e E9 there is a family (ViiieM of neighborhoods of x, whose intersection is {x}. (We can and will suppose that the Vt are open-closed in E.) Put W0 = V0 and Wt = f] V% for each i e M - { 0 } . J<i The family (W^)ieM is a fundamental system of neighborhoods of x. Indeed, let U be an open-closed neighborhood of x (x has a fundamental system of neighborhoods of this type). Consider the set {Wt - (Wr vU)\ieM - {0}}, where V is the ordinal successor of i. It is a discrete collection of open-closed subsets of E. So there is p e M such that Wi - (Wr u U) = 0 for every i e M, i > p. Thus Wp. c U. (If it were t e Wp. and t $ U there would be a minimal k e M such that t $ Vk. By the construction of Wp,9 k is greater than p; and t belongs to Wk and does not belong to Wr\ but Wk is contained in Ww u U9 which is a contradiction.) Proof of 4). By virtue of assertion 3) the neutral element of E has a fundamental system 93 of neighborhoods, with |93| = m. Since m is greater than K0 we can choose elements of 93 such that W= V1 = V for every Ve 93. For each Ve 93, {Vx \ x e E} is a discrete open covering of E. The paracompactness of E follows from the fact that U {Vx | x e E} is an open basis of the topology on E. 2. Main results Let Et and E2 be two nondiscrete completely regular Hausdorff spaces and mt and m2 their indices. Theorem 1. Let Ux and tl2 be uniformities in Et and El9 respectively. If every continuous function of Ex x E2 (with the product topology) into [0,1] is a uniformly continuous map of (Et x JE2, Ut <g) U2) into ([0,1], IT), then every locally finite open covering of E% has cardinality less than m^ (hj = 1, 2). Proof. We shall prove, for instance, that every locally finite open covering of Ex has cardinality less than m2. It is sufficient to prove that every discrete family of nonempty open subsets of.fi! has cardinality less than m2. On the contrary, let us suppose that there exists a discrete family of nonempty open subsets of El9 (Wt)t(sT9 whose cardinality \T\ is equal to m2. There is a point d G £2 and a family of open neighborhoods of d9 (Vt)teT9 such that f| V% is not a neigh-teT O. T. ALAS 21 borhood of d. For each te T we fix a point a, 6 W, and two continuous functions / , : E{ -+ [0, 1] and gt : E2 -+ [0, l ] satisfying the conditions: 1) f,(at) = 1 , / , (£ , - Wt,) = {0} ; 2) 0,(d) = 1 , g,(E2 - V,) = {0} . The function g defined below is continuous (because the family (Wt x Vt)teT is discrete in Et x E2): g:Exx E2^ [0, 1] (x, y) i-> 0 if (x, v) does not belong to \J Wt x Vt (x, y) -*/,(x) gt(y) if (x, y)eWtxV19 te T. By the hypothesis, there are Ux e Ux and U2 e U2, such that (x, y) e Ux and (u, V) e U2 imply |g(x, u) — g(y, v)\ < | . But this is not possible, because then U2\d\ would be contained in fl Vv ((av at) e U\ and (u, d) e U2 imply g(av u) ^ | , so u e Vv) The proof is completed; notice that m% = m2. Remark 1. Suppose mt = m2 = p. If Ex and E2 are paracompact and every locally finite open covering of Et (i = 1, 2) has cardinality less than p, then Ex x E2 is paracompact and, further, every locally finite open covering of Ex x E2 has cardin-ality less than p. So if Xlx and U2 are the universal uniformities in Ex and 2?2, Ux ® U2 is the universal uniformity in Ex x E2. The next theorem follows easily. Theorem 2. Let E be a nondiscrete paracompact space and m the index of E. The following conditions are equivalent: 1) there is a uniformity U in E such that every continuous function ofExE (with the product topology) into [0, 1] is a uniformly continuous map of (E x £, 11 ® 11) into ([0, 1], IV); 2) every locally finite open covering of E has cardinality less than m; 3) there is a uniformity XI in E such that U ® ... ® 11 (n times) is the universal uniformity in the product topological space £", n = 2, 3 , . . . ; 4) there is a uniformity XI in E such that XI ® XI is the universal uniformity in the product topological space E x E. Hint. If E satisfies the condition 2), then En is a paracompact space for each n = 2, 3, . . . On the other hand, it is well-known that if X is a paracompact space, the set { U Y xY\ a is a locally finite open covering of X} is a basis of the universal Yem uniformity in X. Remark . The implication 2) => 4) is a particular case of Theorem 35 ([7], p. 137). 22 O. T. ALAS For topological groups we have the following theorem ([2]): Theorem 3. Suppose E is a paracompact topological group. E satisfies the condition 2 of Theorem 2 if and only if the right uniformity in E is the universal uniformity in E. For other results in the same area see, for instance, [3], [5], [7] and [8]. Professor L. Nachbin also investigated a similar question for metric spaces. References II] O. T. Alas: Paracompact topological groups and uniform continuity. (To appear.) [2] O. T. Alas: Normal topological groups and universal uniformities. An. Acad. Brasil. Ci. 42 (1970), 411-413. [3] M. Atsufi: Uniform continuity of continuous functions on uniform spaces. Canad. J. Math. 13 (1961), 654-663. [4] 1V. Bourbaki: Elements de Mathematique. Topologie Gen6rale. Livre 3, chapitres 1 et 2. Hermann, Paris, 1965. [5] W. W. Comfort and K. A. Ross: Pseudocompactness and uniform continuity in topological groups. Pacific J. Math. 16 (1966), 483-496. (6] R. Engelking: Outline of General Topology. North-Holland Publishing Co, Amsterdam, 1968. [7] / . R. Isbell: Uniform spaces. Math. Surveys 12. Amer. Math. Soc, 1964. 18] N Onuchic: On the Nachbin uniform structure. Proc. Amer. Math. Soc. 11 (1960), 177—179. 

Uniform continuity in paracompact spaces

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Uniform continuity in paracompact spaces

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