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
E. MAKAI, JR. than some cardinal. Let F: C —*■ S0 be a full embedding. Then VC E ObC for FC = (X, X) (where for simplicity we assume Uc — Us-F) we have X D D {Ac X \ A, X \ A are far in C}. 2) Let VC £ Ob C VAcUC, A,(UC)\ A proximal in C 3D £ ObC (dB, B' C UD, B, (CO) \ O near, B\ {UD) \ B' near Bg £ U(D, O), ^(B) = B') 3f £ U(D,C), 3f?i, B2 near subsets of D, f(Bi) C A, f(B2)c(UC)\A. (A space D has this property e.g. if, denoting by N* the proximity on a countable discrete topological space corresponding to its one-point compactification, we have B CU D, B,(U D) \ B near in D => 3 embedded copy of N* in D, which has infinite intersections both with B and (U D)\B and which is a retract of D by a retraction mapping B into B and (UD)\B into (UD)\B (e.g. D has a discrete topology and its completion is a zero-dimensional compact metric space). An above / £ U(D,C) exists for such a D with non-discrete proximity if we have: A C UC, A, (UC) \ A near in C =>* 3g: N* —»C, 3 infinite disjoint subsets Ni, N2 of N* such that g(N\) C A, g(N2) C (UC)\ A (e.g. the completion of the To-reflection of C is compact Fréchet-Urysohn).) Let F: C -* S(( be a concrete full embedding satisfying the conclusion of statement 1), C <f_ {indiscrete spaces}. Then VC E ObC for FC = (X, X) we have X = {A C X I A,X \ A are far in C}. 3) If 3Ci,C2 € ObC such that for their reflections rCi,rC2 in (C € € Ob Unif|C has a basis consisting of finite partitions} there holds U{C\ ,C2) ^ ^ U(rC\, rC2) (where for simplicity we assume the universal map Ci —* rCi has underlying function luCi) then there exists no concrete full embedding F: C —y S({ satisfying the conclusion of statement 2). In particular {C € Ob Prox| 8dC = 0,the To-reflection of C has a metric completion} has a unique full embedding into Sq — up to natural isomorphy — namely the one given under 2), but no subcategory of {C € Ob Prox | the To-reflection of C has a metric completion} strictly containing the above subcategory admits one. Proof. 1) By [22], Corollary to Lemma 2 Uc and U<--F are naturally °0 isomorphic, thus we may assume Uc = Us-F. We have by [23], Remark 2 U(C0, C0) # X*°, where X0 = UC0. Hence for FC0 = (X0X0) {0,XO} # X0 ± ^ 2X°. Let 0 ^ A0 9 X0, A0 G X0 and let C G Ob C, X = UC. If A, X \ A are far in C, define / € Í7(X,X0) = hom((X, X), (X0, X0)) by f(A) C {zo}> /(X \ A) C {r/o}, where x0 G A0, yo G X0 \ A0. Then A = f~1(A0) e X. 2) For the second statement we first show for FD = (UD,V) that T> = = {B C UD I B, (UD) \ B are far in D}. By the conclusion of statement 1) V D {B C UD I B, (UD) \ B are far in D}. If, however, 3B £ V, B,(UD) \ B are near in D then \UD\ > 1 and for any other set B' C UD with B',(UD)\B' near 3g £ U(D, D) = hom((UD,V),(UD, V)) such that B' = g~x(B), hence B' £ V. Therefore V = 2UD, hence U(D,D) = hom((UD,V),(UD,V)) = = (UD)^ud\ hence by [23], Remark 2 D has for basis all partitions of UD