Studia Scientiarium Mathematicarum Hungarica 19. (1984)
1. szám - Makai Jr. E.: Uniformities uniquely determined by their uniformly continuous self-maps
Studia Scientiarum Mathematicarum Hungarica 19 (1984). 1—12. UNIFORMITIES UNIQUELY DETERMINED BY THEIR UNIFORMLY CONTINUOUS SELF-MAPS E. MAKAI, JR. Abstract We prove for a class of uniform (proximity, resp. topological) spaces that any space of this class is uniquely determined (among all uniform, proximity, resp. topological spaces) by its uniformly (proximally) continuous (resp. continuous) self-maps. This class contains e.g. all Peano continua and the long line ([2], exercises 6J, 15R, 16H) (with the unique uniformity), resp. all countable precompact uniform spaces with discrete topology and zero-dimensional metric completion. Our results largely parallel analogous results for topological spaces (cf. [17]). In fact most of our auxiliary results treat the topological case — usually considering instead of C(X, X) the more general case C(X, Y). As applications we determine the coarsest concrete functors between some subcategories of uniform spaces. §0 Uniform and proximity spaces are not assumed to be separated. The category of uniform, resp. proximity spaces is denoted by Unif resp. Prox. Definition. A uniform (or proximity) space A" is called special if for any uniform (or proximity) space Y on the same underlying set horn (Y, É) = hom (X, X)=>Y= X. Remark. [17] defines similarly special topologies. Restricting our attention to non-empty spaces by [11], p. 197, this property is equivalent to the following: the existence of a semigroup-isomorphism /': horn (A, A)—horn (Y, Y), for any space Y, implies the existence of an isomorphism j: X-*Y, with i(f)(y)=j[f(j~1(y))']. First we recall some concepts. A topological space is Fréchet—Urysohn if x£Xz)A, x£Ä=>3xn£A, (n£N), x„ — x, and it is sequential if sequentially closed sets are closed. A topological space is S1 (S2, S3i) if its T,, "inflection >s Tx (T2, th). A Peano continuum is a connected, locally connected compact metric space. Equivalently (if it is not empty), it is a T2 continuous image of [0, 1] ([10], § 45, II. 2, p. 185, [1], § 2.10, Prop. 17). Peano continua are arcwise connected ([10], § 45, II. 1, p. 184 and I. 2, p. 182). Pseudocompact is meant to imply S3i. Pseudocompact spaces with the fine uniformity are just the precompact fine spaces ([7], p. 135). For a uniform space X the generated topology is denoted by iX and the precompact reflection by pX. Proximity spaces will be identified with precompact uniform spaces. The covering character (cov char) of a uniform space X is min {a | X has a discrete subspace of cardinality ß =>•/?< a}+ (=min {a | X has a basis of coverings of cardi- 1980 Mathematics Subject Classification. Primary 54E15; Secondary 54H15. Key words and phrases. Semigroups of uniformly (proximally) continuous, resp. continuous mappings, special uniform (proximity, topological) spaces, Peano continua, fine uniform spaces, concrete functors.