Studia Scientiarium Mathematicarum Hungarica 19. (1984)
1. szám - Makai Jr. E.: Uniformities uniquely determined by their uniformly continuous self-maps
UNIFORMITIES In the second case let {ya}= {convergent sequences with any of their limits}, {^p}— {filter generated by the tails of the respective convergent sequence}. Choose for x£X an h£C(X,N*) with /j-i(oo)= {*}. Let «=1, A'1 = A'\{a:}. If Ya = = {y*}U {y'}, where yk-*y, and Z satisfies the hypothesis of Proposition 1, then Z contains an infinite subsequence {yi(/i)} of {yk}. Let now (p: N*-»Y, (p(k)=ytw, <p(°°)=y- Then f=(ph satisfies /(fjjcZ, f(x)=y. The remark in the brackets follows from considering an y£.4\T (Ac Y), then choosing y„€A, yB—y, and choosing arcs joining y to y„, in elements of a neighbourhood base of y. Proposition 2. Let X, Y be uniform spaces. Let Y have a family of subspaces {Lj such that ZxbZ2o3oc (ZxDYJbiZzilYJ. Let ®={{Aß,Bß}} be a subbasis of uniform coverings of the discrete proximity on X (Aß, BßAX). Let V {Aß, Bß}£ati Va VZX, Z2cYa with ZxbZ23n,m£N, 3 Ax, ..., A„, Bx, ..., Bm, X\Aß=\JAt, 1 m X\Bß=(JBj, V/ (1 S/Sn), V/ (lS/S/n) 3f€U(X, Y) f(Ai)czZ1, f(BficZ2. Let X', Y' be other uniformities on the underlying sets of X resp. Y, with U(X, Y)c cU(X', Y'). Then either pX' is a discrete proximity or pY' is coarser than pY. Proof. We have to show pY' is coarser than pY, i.e. ZxbYZ2=s-ZxbY, Z2. By the hypothesis on {ya} 3a (ZxnY0,)SY(Z2r\Y!X). Hence for proving the above implication Zx, Z2 can be replaced by Zx = Zxi]Ycl, Z2 =Z2C\YX. Suppose pX' is not a discrete proximity. Since <%={{Aß, Bß}} is a subbasis of the discrete proximity, some {Aß,Bß} is not a uniform cover of pX', i.e. (X\Aß)bx, (X\Bß). For this n m Aß,Bß we have by hypothesis At, X\Bß = \J Bj. Therefore 3/, j with AjbX' Bj. By hypothesis 3f€U(X, Y)cU(X\ Y'), f(At)cZi, f(Bj)cZ't. Thus by Atbx, Bj we have Z[bY, Z2, which was to be shown. Corollary 2. Let X, Y be uniform spaces. Let VAcX, 0AA AX 3g: X-~ — {1/k} (c[0, 1] — where k<£N), g(^)flg(A'\y4) = 0. (This class of spaces contains each countable X with zX discrete and is closed under taking subspaces, finer uniformities and sums.) Let for the completion ypY ofpY hold: Zx, Z2c Y, ZxbZ2=> 3y((ypY, 3yXJ£Zx, 3y2J<zZ2 (l£N), yXj-*y, y2,i-~y- Then the hypothesis and statement of Proposition 2 hold. Proof. Let Zx,Z2cY, ZxbZ2. Then 3ylfi€Zlf y2>)€Z2, y2>/ — y. Thus choose {Yx}—{{y„}cY\3y£ypY, y„-~y}, and also ^—{two-element partitions of X}. Let A c X (fbjíAjí X). Thus {A, Z\^4} = {Aß, Bß}£ °U. By hypothesis 3g: X--{l/k} with the desired properties. Let m=n— 1. We can suppose g(X\Aß)c c{l/(2l-l)},g(X\Bfi)c{l/2l). Let h(l/(2l-l))=ylt„ h(l/2l)=y2,>. Then f=hg satisfies f(X\Aß)cZx, f(X\Bß)cZ2. The remark in the brackets follows from the fact that, for zX discrete, X is finer than the proximity on X corresponding to the one-point compactification. Remark 1. The condition of Corollary 2 implies zX is discrete. It would be interesting to determine the class of spaces X satisfying the above condition. A prop l*