Studia Scientiarium Mathematicarum Hungarica 27. (1992)

1-2. szám - Makai Jr., E.: The full embeddings of the categories of uniform spaces, proximity spaces and related categories into themselves and each other. II

Studio Scientiarum Mathematicarum Hungarica 27 (1992), 1 -24 THE FULL EMBEDDINGS OF THE CATEGORIES OF UNIFORM SPACES, PROXIMITY SPACES AND RELATED CATEGORIES INTO THEMSELVES AND EACH OTHER. II* 1 E. MAKAI, Jr.2 We use the same notations as in Part I. For the definitions we refer to Part I. Here we just recall some special notations and definitions (others are rather standard). Sff denotes the category with objects all pairs (X,X), X a set, {0,X}cA C 2^, and morphisms /: (Xi, Aj) —» (X2, Xgf) characterized by /: X\ —> X2, /-1(A2) C X\. Coz, the category of cozero-spaces, is the full subcategory of S(f determined by ObCoz = {(X,X) | 3 uniformity on X, X — { cozero-sets of all uniformly continuous real functions w.r.t. this uniformity}}. A uniform (etc.) space is called special if for any uniform (etc., resp.) space Y with UY = UX (= underlying set of X) U(Y,Y) = U(X,X) ( = {uniformly continuous functions X —* A}) implies Y = X. For a concrete category C the underlying set functor is denoted by Uc (or U; this will be sometimes omitted). J: Prox—> Unif is the concrete functor associating to a proximity the compatible precompact uniformity. Subcategories will always be assumed to be full. § 5. Embeddings of subcategories of Prox (Unif) in S0 For the case of full embeddings Prox —» Sq we have a negative result. For analogous negative results, concerning full embeddings of categories connect­ed with closure spaces into iS^ cf. [3], [2]. The proof of the following proposition is related to the proof of [27], Proposition 13. Proposition 2. 1) Let C c Unif and let ObC contain a uniform space Co which does not have a base composed of all partitions of cardinality less 1980 Mathematics Subject Classifications (1985 Revision). Primary 54E15; Second­ary 54E05, 54B30. Key words and phrases. Categories of uniform (proximity, cozero, Tgi ) spaces, cat­egory of set systems, full embeddings, special spaces, inductive (projective) generation, strongly rigid class of objects, spaces Yjr, ultraproximities, free ultraspaces. 1 Part I of this paper has appeared in the same journal, 25 (1990), 199-208. 2 Research (partially) supported by Hungarian National Foundation for Scientific Re­search Grant no. 1807. Akadémiai Kiadó, Budapest MAGYAR TUDOMÁNYOS AKADÉMIA KÖNYVTÁRA

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