Studia Scientiarium Mathematicarum Hungarica 19. (1984)
1. szám - Makai Jr. E.: Uniformities uniquely determined by their uniformly continuous self-maps
E. MAKAI JR. nalities <a}+K„) ([7], p. 134). A topology is saturated if arbitrary intersections of open sets are open, y denotes completion. A concrete functor is a functor between concrete categories commuting with the respective underlying set functors. A map betwen uniform (etc.) spaces is always meant as a morphism of the category in question. §1 The following proposition follows the lines of [17]. Proposition 1. Let X, Y be topological spaces. Let Y have a family of subspaces {7J, and for each a a family offilters {J^} on Yx such that BczY is closed iff [\/a, ß, Br\Y<zf&rIZß-*y£Ycl implies Bf\Yf$y]. Let further V xfX V aV ZczYx with (3 ß,y, Ze&'ß-ytY', Z$y) 3y* £YXC)(Z\Z), 3n€N, 3Xlt...,X„, Z\{*}=Ü X» V/(Is zki = n) 3f£C(X, Y), f(Xi)czZ,f(x) = y*. Let X', Y' be other topologies on the underlying sets of X resp. Y, with C(X, Y)czC(X', Y'). Then either X' is a discrete space or Y' is coarser than Y. Proof. Let BczY be closed in Y'. We show it is closed in Y, too, i.e. Va, ß Bf)Y^^ß-'-y^Y7 implies BClY^y. Denote Z=BC\Y0l, and suppose this implication is false for Z. Suppose X' is not discrete. Then 3x{X, X\{x}xfx. Thus for one of the Xrs (1 ^i^n), assured by hypothesis, we have Xffx. Since 3feC(X,Y)aC(X',Y'), f(Xi)czZ, f(x)=y\ therefore Br'z>Zr'z>f(XTlf(x)= =y*. Thus y* £BY'\B, contradicting our assumption BY'—B. Corollary 1. Let X be a T3i space in which every point is a G0-set. Let Y be a space with unique limits of such sequences which are contained in Peano continua cz Y, and let Y have the weak topology w. r. t. its subspaces which are Peano continua (e.g. Y is T2, first countable and locally arcwise connected). Then the hypothesis and statement of Proposition 1 hold. The same is valid if X is a zero-dimensional space in which every point is a Gs-set, and Y is sequential. Proof. In the first case let {7^}= {subspaces of 7 homeomorphic to N*(— = one point compactification of a countable discrete space)}, Va {cofinite filter on Yfi, which evidently satisfy the property in Proposition 1. Choose for x£X an h£C(X, [0, 1]) with /i_1(0)={x}. Let n== 2, X1 = h-Y( U (2~2j+\ 2~2'+2]), X2 = h-Ul) (2~2J, 2_2j+1]). i=1 3=1 Let Zc7, satisfy the condition on Z from Proposition 1. Then Z={yk}, in Yx 3 limyk=y*, ZY«={yk}[){y*}. Since {yk} is not closed in 7, for some Peano continuum g([0, l])c;7 (where g: [0, 1] —7) Z= {yt}Dg([0, 1]) is not closed in g([0, 1]). Denote this infinite subsequence once more by {yk}. Choose ukfg~'i(yk). We may suppose 3 lim uk = u (otherwise choose a subsequence). Thus by condition g(u)= y*. Let Ijr. [0, l]—[0, 1] map (2~2k+1, 2~2k+2] to uk; thus i//(0)=u. So Ü (2_2J+1, 2~2J+2])cz {yk},gf(0)=y*. Then f=gifh: 7-7 satisfies f{Xf)czZ, j=i f(x)=y*. In case of X2 we proceed analogously.