01.dvi @ Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 1 - 12 Merotopies associated withquasi-uniformities�Akos Cs�asz�ar�Abstract. To an arbitrary quasi-uniformity on the set X,a merotopy on X is assigned. There are results concerning thequestion whether this merotopy is compatible with the topologyinduced by the quasi-uniformity and whether the closure opera-tion induced by the merotopy, admits a compatible uniformity.More precise results are obtained in the case of transitive quasi-uniformities.2000 AMS Classi�cation: 54E15, 54E17Keywords: Quasi-uniformity, merotopy, semi-symmetric, transitive1. IntroductionThe purpose of the present paper is to establish a relation between two well-known kinds of topological structures, namely quasi-uniformities and mero-topies.Notation and terminology concerning quasi-uniformities will be used accord-ing to [4]. The concept of a merotopy has been introduced in [8], but we shalluse according to [3] a more advantageous description of them due to [7]. Thusa merotopy C on a set X will mean a non-empty collection of covers of X (wedenote by �(X) the collection of all covers of X) with the properties:(1.1) If c 2 C, c0 2 �(X) and c re�nes c0 then c0 2 C,(1.2) c1;c2 2 C implies c1(\)c2 2 Cwhere we say that c re�nes c0 (in symbol c < c0) i� C 2 c implies the existenceof C0 2 c0 satisfying C � C0, andc1(\)c2 = fC1 \ C2 : Ci 2 cig;�Research supported by Hungarian Foundation for Scienti�c Research, grant No. T032042. 2 �Akos Cs�asz�ar(\) is obviously an associative operation. Equivalently, (1.2) may be replacedby(1.3) c1;c2 2 C implies the existence of c 2 C satisfying c < ci (i = 1;2).The topological category Qunif is composed of the objects of quasi-uniformspaces (X;U) where U is a quasi-uniformity on X, and of the morphisms ofquasi-uniformly continuous maps [4]. The category Mer contains the objectsof merotopic spaces (X;C) where C is a merotopy on X and of the morphisms ofmerotopically continuous maps, where f : X ! X0 is said to be merotopicallycontinuous or (C;C0)-continuous, C and C0 being merotopies on X and X0 respec-tively, i� c0 2 C0 implies f�1(c0) 2 C (of course, f�1(c0) = ff�1(C0) : C0 2 c0g).We know ([4]) that each quasi-uniformity U on X induces a topology �(U)on X for which the neighbourhood �lter of x 2 X is given by fU(x) : U 2 Ug.Similarly, each merotopy C on X induces a closure operation on X (i.e. a mapc : expX ! expX such that c(?) = ?, A � c(A), c(A [ B) = c(A) [ c(B)where exp X is the power set of X) and c = c(C) is de�ned byx 2 c(A) , A 2 sec vc(x)(for b � �(X), where �(X) is the collection of all non-empty subsets of thepower set expX, we writeA 2 sec b , A � XA \ B 6= ? for each B 2 b)and the c-neighborhood �lter vc(x) of x 2 X is generated by the �lter basefst(x;c) : c 2 Cg. Also each topology � on X may be considered as a closurec = c� = cl� special in the sense that c(c(A)) = c(A) for every A � X.2. Merotopies associated with quasi-uniformitiesLet U be an entourage [4] on X. De�ne cU = fU(x) : x 2 Xg. Then cU isa cover on X and, both U and U 0 being entourages on X with U � U 0, clearlyU(x) � U 0(x) for x 2 X so that cU < cU0. Therefore, if U is a quasi-uniformityon X, then B = fcU : U 2 Ug is a base [3] for a merotopy CU. More generally,if B is a base for U and we set B = fcU : U 2 Bg then B is still a base forCU. Moreover, if (X0;U0) is another quasi-uniform space and f : X ! X0 isquasi-uniformly continuous then f is (CU;CU0)-continuous as well: if U 2 U,U 0 2 U0 and (x;y) 2 U implies (f(x);f(y)) 2 U 0 then f(U(x)) � U 0(f(x)) sothat cU < f�1(cU0).Hence we can state:Theorem 2.1. If we associate with each quasi-uniformity U on the set X themerotopy CU with base(2.4) B = fcU : U 2 Ugwhere(2.5) cU = fU(x) : x 2 Xg;then �((X;U)) = (X;CU), �(f) = f for f : X ! X0 de�ne a (covariant)functor � : Qunif ! Mer. Merotopies associated with quasi-uniformities 3It is an interesting question which merotopies can be represented in the formCU with some quasi-uniformity U, or which covers have the form cU for someentourage U. The collection of all covers of the form cU clearly does not coincidewith �(X): if c = cU then there is a surjection f : X ! c such that x 2 f(x) foreach x 2 X, consequently there is a bijection g : X0 ! c for some X0 � X suchthat x 2 g(x) for x 2 X0, or equivalently there is an injection g�1 = h : c ! Xsuch that h(C) 2 C for C 2 c, i.e., in the terminology of [8], there is a transversalfor c. Now clearly, if t 2 �(X) and h is a transversal for t, then necessarily thefollowing condition must hold:(2.6) t0 � t implies t0j 5 j [ t0jbecause h(t0) � [t0. Consequently, if c = cU for some entourage U then (2.6)has to be ful�lled for t = c.According to [6], the condition (2.6) is su�cient for the existence of a trans-versal for t in the case when t and each T 2 t are �nite, or even, according to[5], in the case when t is in�nite but each T 2 t is �nite. However, probablythere are no further results on the su�ciency of (2.6) in the general case (ifsome T 2 t can be in�nite then (2.6) certainly does not guarantee the existenceof a transversal, cf. [9]). So we can formulate:Problem 2.2. Look for necessary and/or su�cient conditions for a cover c ofX for the existence of an entourage U satisfying c = cU.Problem 2.3. Look for necessary and/or su�cient conditions for a merotopyC on X for the existence of a quasi-uniformity U satisfying C = CU.If U is a quasi-uniformity on X and we look for the closure c = c(CU) thenit is easy to see:Lemma 2.4. c = c(CU) is coarser than c�(U), i.e.c�(U)(A) � c(A) (A � X):Proof. Clearly vc(x) is generated by the �lter base composed of all sets st(x;cU)where U 2 U, and(2.7) st(x;cU) = [fU(y) : y 2 U(x)g = U(U�1(x)):Obviously U(x) � U(U�1(x)). �In general, c 6= c�(U); e.g. if X = R and U is the Sorgenfrey quasi-uniformitygenerated by the base fU" : " > 0g where U"(x) = [x;x+") then U"(U�1" (x)) =(x � ";x + ") so that c(CU) is the Euclidean topology on R. It is even possiblethat the closure c(CU) it not a topology:Example 2.5. Let X = fa;b;cg and U be an entourage on X such that U(a) =fag; U(b) = fa;bg; U(c) = fa;cg. Clearly U2 = U so that fUg is a base for aquasi-uniformity U on X and fcUg is a base for the merotopy CU. For c = c(CU),we have c(fbg) = fa;bg and c(fa;bg) = X. 4 �Akos Cs�asz�arHowever, it is not di�cult to characterize those quasi-uniformities U forwhich c(CU) = c�(U). Recall ([4]) that a quasi-uniformity U on X is said tobe point-symmetric i�, for each x 2 X and U 2 U, there is V 2 U such thatV �1(x) � U(x) or, equivalently, i� �(U) is coarser than �(U�1).Theorem 2.6. The equality c(CU) = c�(U) holds i� U is point-symmetric.Proof. By Lemma 2.4, we need, for x 2 X and U 2 U, the existence of W 2 Usuch that W(W�1(x)) � U(x). Now this condition clearly implies the point-symmetry of U. On the other hand, if, for U 2 U, we choose U0 2 U satisfyingU20 � U, then, given x 2 X, V 2 U such that V �1(x) � U0(x), �nally we setW = V \ U0 2 U, obviously W(W�1)(x) � U0(V �1(x)) � U20 (x) � U(x). �It is easy to �nd examples of point-symmetric quasi-uniformities. In fact,recall (cf. [1]) that a topology c (i.e. a closure c = c� for a topology �) issaid to be S1 i� x 2 G implies c(fxg) � G whenever G is c-open. Also recall([4]) that the Pervin quasi-uniformity P associated with the topology c (andinducing c) is de�ned by the quasi-uniform subbase fUG : G is c-openg whereUG(x) = G if x 2 G and UG(x) = X if x 2 X � G. More generally, if B is abase for the topology c then the entourages UB (B 2 B) constitute a subbasefor a transitive quasi-uniformity U(B) compatible with c (see e.g. [2]). If thetopology c is S1, we can also consider the entourages Ux;B = UB \ UX�c(fxg)where x 2 B 2 B to obtain a subbase for a transitive quasi-uniformity U1(B)�ner than U(B) and coarser than P, hence still compatible with c.Now we can state:Proposition 2.7. If c is an S1 topology admitting a base B then every quasi-uniformity U �ner than U1(B) and compatible with c is point-symmetric.Proof. Given x 2 X and U 2 U, there is a B 2 B such that x 2 B � U(x). ByS1, we have c(fxg) � B. Let H denote the c-open set H = X � c(fxg). Then,for V = UB \ UH 2 U1(B) � U, we have V �1(x) = c(fxg) � B � U(x). �The condition for a quasi-uniformity U of being point-symmetric has anotherimportant consequence for the merotopy CU. Recall ([3]) that a merotopy C issaid to be Lodato i� c 2 C implies int c 2 C where int c = fint C : C 2 cg andintC = X � c(X � C), c = c(C). Now we can state:Theorem 2.8. If U is point-symmetric then CU is a Lodato merotopy.Proof. For c 2 C, choose U 2 U such that cU < c and U0 2 U such thatU20 � U. Then, by U0(x) � intU(x), cU0 < int cU < intc and cU0 2 C impliesintc 2 C. �3. Semi-symmetric quasi-uniformitiesRecall ([3]) that a semi-uniformity U on a set X is a �lter on X � X havinga base composed of symmetric entourages; it induces a closure c(U) such that,if c = c(U) and x 2 X, then vc(x) = fU(x) : U 2 Ug is the neighborhood �lterof x for c. Merotopies associated with quasi-uniformities 5Now if U is an arbitrary entourage on X then clearly UU�1 (we write ABfor A � B if A;B � X � X) is a symmetric entourage on X so that, wheneverU is a quasi-uniformity on X, fUU�1 : U 2 Ug is a base for a semi-uniformityU�; by Lemma 2.4(3.8) c(U�) = c(CU):We look for those quasi-uniformities U which admit a corresponding semi-uniformity U� that is a uniformity. For this purpose, let us say that U issemi-symmetric i�, given U 2 U, there is V 2 U satisfying V �1V � UU�1;the pair (U;V ) is said to be semi-symmetric in this case and, in particular, theentourage U is said to be semi-symmetric i� (U;U) is semi-symmetric. Now itis easy to prove:Theorem 3.1. For a quasi-uniformity U, the semi-uniformity U� is a unifor-mity i� U is semi-symmetric.Proof. If U� is a uniformity then, for U 2 U, there is V 2 U such thatV V �1V V �1 � UU�1 whence clearly V �1V � UU�1. Conversely, if the con-dition in the statement is ful�lled, let U 2 U and U0 2 U be chosen such thatU20 � U, then let V 2 U satisfy V �1V � U0U�10 . Now we can suppose V � U0 asV can be replaced by V \U0. Then V (V �1V )V �1 � U0U0U�10 U�10 � UU�1. �Of course, each uniformity is an example of a semi-symmetric quasi-uniform-ity. But it is easy to �nd non-symmetric examples, too. E.g. if U is the Sorgen-frey quasi-uniformity on X = R whose base is composed of the entourages U" =f(x;y) : x 5 y < x + "g (" > 0) then U"U�1" = U�1" U" = f(x;y) : jx � yj < "g.Similarly if U is the Michael quasi-uniformity on X = R, i.e. the base is com-posed of fU" : " > 0g where U"(x) = (x � ";x + ") if x 2 Q and U"(x) = fxg ifx 2 R�Q , then U"U�1" (x) = (x�2";x+2"), while clearly U"(x) � (x�";x+")and U�1" (x) � (x � ";x + ") so that U�1" (U"(x)) � U"(U�1" (x)). On theother hand, e.g. Example 2.5 is not semi-symmetric: U(U�1(b)) = fa;bg andU�1(U(b)) = X.Corollary 3.2. If a quasi-uniformity U is both semi-symmetric and point-symmetric then the topology �(U) is completely regular.Proof. By Theorem 2.6 c�(U) = c(CU), by (3.8) and Theorem 3.1 the latter is atopology induced by a uniformity. �It is easy to see that point-symmetry and semi-symmetry are properties ofa quasi-uniformity independent of each other. In fact, the Sorgenfrey quasi-uniformity is semi-symmetric without being point-symmetric, while if c is anS1 topology that is not completely regular then its Pervin quasi-uniformity ispoint-symmetric by Proposition 2.7 but not semi-symmetric by Corollary 3.2.Semi-symmetric quasi-uniformities have rather good invariance properties.Recall that, if f : X ! Y , then the inverse image f�1(U) of a quasi-uniformityU on Y is generated by the entourages f̂�1(U) for U 2 U where f̂(x;y) =(f(x);f(y)). 6 �Akos Cs�asz�arLemma 3.3. If f : X ! Y is surjective and U is a semi-symmetric quasi-uniformity on Y then f�1(U) is semi-symmetric.Proof. If U;V 2 U and V �1V � UU�1, further (f(x);f(y)) 2 V , (f(y);f(z)) 2V �1 then (f(x);f(z)) 2 V �1V � UU�1 so that there is some w 2 Y satisfying(f(x);w) 2 U�1, (w;f(z)) 2 U, and choosing u 2 X such that w = f(u),we get (f(x);f(u)) 2 U�1, (f(u);f(z)) 2 U, i.e. (x;u) 2 f̂�1(U�1), (u;z) 2f̂�1(U). �The condition of surjectivity cannot be dropped as semi-symmetry is nothereditary:Example 3.4. Let X = fa;b;c;dg, U(a) = fag, U(b) = fa;bg , U(c) = fa;cg,U(d) = X. Then U2 = U, so that fUg is a base for a quasi-uniformity U onX. The semi-symmetry of U is easily checked using the formulas for U(x) andthose U�1(a) = X; U�1(b) = fb;dg; U�1(c) = fc;dg; U�1(d) = fdg. De�neX0 = fa;b;cg; U0 = U \ (X0 � X0). Then UjX0 coincides with the quasi-uniformity in Example 2.5 which fails to be semi-symmetric.Lemma 3.5. If Ui is a semi-symmetric quasi-uniformity on Xi (i 2 I) andX = QfXi : i 2 Ig then U = QUi is semi-symmetric on X.Proof. Let U 2 U be given. We can suppose U = QUi where Ui 2 Ui fori 2 F and a �nite F � I, Ui = Xi � Xi otherwise. Choose Vi 2 Ui such thatV �1i Vi � UiU�1i for i 2 F and Vi = Xi � Xi otherwise. For V = QVi, we haveV �1V � UU�1. �Some partial results concerning heredity may be obtained by introducingthe following de�nition: let us say that U is strongly semi-symmetric i�, givenU 2 U, there is V 2 U such that V �1V � U[U�1; in this case (U;V ) is stronglysemi-symmetric and, in particular, U 2 U is strongly semi-symmetric i� so is(U;U).Lemma 3.6. A strongly semi-symmetric quasi-uniformity is semi-symmetricas well.Proof. If V �1V � U [ U�1 and (x;y) 2 V �1V then either (x;y) 2 U or(x;y) 2 U�1. In the �rst case, let (x;x) 2 U�1, in the second one let (y;y) 2 U.In both cases, (x;y) 2 UU�1. �E.g. the Sorgenfrey quasi-uniformity is strongly semi-symmetric becausef(x;y) : jx � yj < "g = f(x;y) : x 5 y < x + "g [ f(x;y) : x � " < y 5 xg. Thesame holds for the Michael quasi-uniformity: U"(x) [ U�1" (x) = (x � ";x + ")if x 2 Q and = fxg [ ((x � ";x + ") \ Q) if x 2 R � Q, while U�1� (U�(x)) �(x � 2�;x + 2�) if x 2 Q and = fxg [ ((x � �;x + �) \ Q) if x 2 R � Q.In Example 3.4, we �nd a semi-symmetric but not strongly semi-symmetricquasi-uniformity; in fact strong semi-symmetry is hereditary:Lemma 3.7. If f : X ! Y and U is strongly semi-symmetric on Y thenf�1(U) is strongly semi-symmetric on X. Merotopies associated with quasi-uniformities 7Proof. Assume U;V 2 U and V �1V � U [ U�1. If (x;y) 2 f̂�1(V �1) f̂�1(V )then (f(x);f(y)) 2 V �1V � U [ U�1, so (x;y) 2 f̂�1(U) [ f̂�1(U�1). �However, the analogue of Lemma 3.5 is not valid for strongly semi-symmetricquasi-uniformities:Example 3.8. Let X = R2, U be the Sorgenfrey quasi-uniformity, and considerU � U. We know that both factors are strongly semi-symmetric. For U =U1 �U1, no V� = U� �U� is suitable: (0; 34�) 2 U�, (34�; 12�) 2 U�1� , (0; 14�) 2 U�,(14�;�12�) 2 U�1� , so ((0;0);(12�;�12�)) 2 V �1� V� but ((0;0);(12�;�12�)) =2 U [U�1 = (U1 � U1) [ (U�11 � U�11 ) because (0; 12�) =2 U�11 and (0;�12�) =2 U1.4. The transitive caseProblems 2.2 and 2.3 have partial solution in the case of transitive entouragesand quasi-uniformities, respectively. In order to see this, consider a systemt 2 �(X) and de�ne an operation � : �(X) ! �(X) by(4.9) �(t) = fT(x) : x 2 Xgwhere(4.10) T(x) = \fT 2 t : x 2 Tgand we de�ne \? = X. Clearly x 2 T(x), hence �(t) is always a cover of X sothat � : �(X) ! �(X).Lemma 4.1. The operation � is idempotent.Proof. Let t 2 �(X) and t0 = �(t). For x;y 2 X and x 2 T(y) we havefT 2 t : y 2 Tg � fT 2 t : x 2 Tg, consequently T(x) � T(y), so thatTfT 0 2 t0 : x 2 T 0g = TfT(y) 2 t0 : x 2 T(y)g � T(x) while obviouslyT(x) 2 t0; x 2 T(x) imply TfT 0 2 t0 : x 2 T 0g � T(x). By this, TfT 0 2 t0 : x 2T 0g = T(x) and �(t0) = �(�(t)) = �(t). �Let us say that a system t 2 �(X) is point-true i� �(t) = t; hence a point-truesystem is always a cover of X. In other words,Lemma 4.2. A system t is point-true i� a) TfT 2 t : x 2 Tg 2 t if x 2 X andb) if T 2 t, there is x 2 T such that x 2 T 0 2 t implies T � T 0.Now let U be a transitive (i.e. such that U2 = U) entourage on X. Asx 2 U(y) implies U(x) � U(y) (because (x;z) 2 U and (y;x) 2 U imply(y;z) 2 U), we have U(x) = TfU(y) : x 2 U(y)g, so that:Lemma 4.3. If U is a transitive entourage on X then the cover cU is point-true.Conversely:Lemma 4.4. If c is a point-true cover of X then there is a transitive entourageU on X such that c = cU. 8 �Akos Cs�asz�arProof. De�ne (x;y) 2 U � X �X i� x 2 C 2 c implies y 2 C. Then (x;x) 2 Ufor x 2 X and (x;y) 2 U, (y;z) 2 U imply (x;z) 2 U so that U is a transitiveentourage on X. By de�nition, U(x) = TfC 2 c : x 2 Cg 2 c by Lemma 4.2a), and, if C 2 c, there is by Lemma 4.2 b) an x 2 C such that C = U(x).Consequently c = fU(x) : x 2 Xg. �Lemma 4.5. The transitive entourage U in the above lemma is uniquely de-termined by c.Proof. Let U1 and U2 be transitive entourages on X such that cU1 = cU2. Givenx 2 X, there is y 2 X satisfying U1(x) = U2(y). Then x 2 U1(x) impliesx 2 U2(y), hence U2(x) � U2(y) = U1(x) and U2(x) � U1(x). ThereforeU2 � U1. Similarly U1 � U2. �Theorem 4.6. There is a bijection from the set of all transitive entourages onX to the set of all point-true covers of X given by the formulas(4.11) U 7! cU;(4.12) c 7! Uc;Uc(x) = \fC 2 c : x 2 Cg(x 2 X):Concerning the behaviour of transitive quasi-uniformities, let us �rst remark:Lemma 4.7. Let Ui be transitive entourages on X for i = 1; :::;n and U =Tn1 Ui. Then cU = �((T)n1 cUi).Proof. Let us denote cUi = ci, cU = c. Then, for x 2 X, we have by (4.12), forthe element of c corresponding to x, U(x) = Tn1 Ui(x) = Tni=1 TfCi 2 ci : x 2Cig = TfCi 2 ci : x 2 Ci; i = 1; :::;ng = TfC 2 (T)n1ci : x 2 Cg; the latter Tis the element of �((T)n1 ci) corresponding to x. �Observe that � cannot be omitted because c1(\)c2 may fail to be point-truefor point-true covers ci (i = 1;2).Example 4.8. Let X = R, c1 = f(2n;2n+2) : n 2 Zg[f(2n�2;2n+2) : n 2 Zgand c2 = f(2n � 1;2n + 1) : n 2 Zg [ f(2n � 1;2n + 3) : n 2 Zg. It is easyto check using Lemma 4.2 that both c1 and c2 are point-true covers. Nowc1(\)c2 = f(n;n +1) : n 2 Zg[f(n;n +2) : n 2 Zg[f(n;n+3) : n 2 Zg[f?gis not point-true since neither (n;n + 3) nor f?g does ful�l Lemma 4.2 b).Now we can prove:Theorem 4.9. If U is a transitive quasi-uniformity then the merotopy C = CUful�ls(4.13) C has a base B composed of point-true coverssuch that(4.14) if ci 2 B for i = 1; : : : ;n then �((T)n1ci) 2 B.Conversely if C is a merotopy satisfying (4.13) and (4.14) then there exists atransitive quasi-uniformity U such that C = CU. Merotopies associated with quasi-uniformities 9Proof. (4.13) is obvious if B = fcU : U 2 U is transitiveg. If ci 2 B (i = 1; :::;n)then there are transitive entourages Ui 2 U such that ci = cUi. By Lemma 4.7,�((T)n1cUi) = cU 2 B for U = Tn1 Ui 2 U and B ful�ls (4.14).Conversely, if the merotopy C satis�es (4.13) and (4.14), let B denote thebase for C occurring in (4.13). By Lemma 4.4, there are transitive entouragesU such that c = cU for each c 2 B. Denote by B the set of all these U. ByLemma 4.7 and (4.14), B is a �lter base on X �X and by U2 = U, it is a basefor a transitive quasi-uniformity U. Clearly CU = C. �In contrast to Lemma 4.5, there is no uniqueness in the above theorem:Example 4.10. Let X = R, c = f[2n;2n + 2) : n 2 Zg and c1 = c [ f[0;1)g,c2 = c [ f[1;2)g. Each of the point-true covers c and ci (i = 1;2) de�nemerotopic bases fcg, fcig for the same merotopy C (observe ci < c < ci).However, if we choose transitive entourages Ui such that ci = cUi (cf. Lemma4.4) then fUig is a base for a quasi-uniformity Ui and CUi = C while U2 " U1(e.g. 1 2 U2(0) � U1(0)), so U1 6= U2.Observe that this Example shows: if Ui (i = 1;2) are transitive entouragesand cU1 < cU2 < cU1 then U1 = U2 need not hold. Also fc1;c2g is a base forC but fU1;U2g is not a quasi-uniform base at all as U1 " U2 " U1. Certainly,it is a quasi-uniform subbase; however, if U = U1 \ U2, then fUg is a basefor a quasi-uniformity U but, since by Lemma 4.7 cU = f[2n;2n + 2) : n 2Z� f0gg [ f[0;1); [1;2)g, we have C 6= CU as cU < c and c � cU.Example 4.10 contains a merotopy and quasi-uniformities inducing very badtopologies. However, it is possible the �nd a better example:Example 4.11. Let X = R�Z = Sn2ZIn where In = (n;n+1). Let � denotethe subspace topology on X of the Euclidean one on R. Denote by B thebase for � composed of all (�)-open sets B contained in some In. Consider the(point-true) covers of X cx;B = ffxg;B � fxg;X � fxgg; clearly cx;B = cUx;B.Denote also c0 = fXg[fI2k�1 : k 2 Zg, c00 = fXg[fI2k : k 2 Zg. Clearly bothc0 and c00 are point-�nite, point-true covers of X. We write c0 = cU0, c00 = cU00with transitive entourages U 0, U 00. Let U0 be the transitive quasi-uniformityde�ned by the subbase fUx;B : x 2 B 2 Bg [ fU 0g, and similarly de�ne U00with the help of the subbase fUx;B : x 2 B 2 Bg [fU 00g.We have U0 6= U00. In fact, assume the contrary; then U 0 � U = Tn1 Uxi;Bi\U 00for suitable xi 2 Bi, 1 5 i 5 n. There is a k 2 Z such that I2k�1 is disjoint fromall sets B1; :::;Bn so that U 0(x) = I2k�1 for x 2 I2k�1 while U(x) is co�nite asUxi;Bi(x) = X � fxig and U 00(x) = X.Let us write C0 = CU0, C00 = CU00. For an arbitrary cover c 2 C0, we can �nd,according to Lemma 4.7, xi 2 Bi 2 B such that �((T)n1cxi;Bi(\)U 0) < c. Weclaim �((\)n1cxi;Bi) < �((\)n1cxi;Bi(\)U 0):In fact, if x 2 Bi for some i then the member containing x of the left handside is contained either in Bi \ I2k�1 = Bi for some k or in Bi \ X = Bi; bothsets belong to the right hand side. If x =2 Bi for each i = 1; :::;n, then there 10 �Akos Cs�asz�aris a k such that I2k is disjoint from all sets Bi occurring on the left hand sideand then the member of the left hand side containing some y 2 I2k is the sameas the one containing x; therefore this member is the one containing y of theright hand side. Thus the left hand side, belonging to C00, re�nes c and c 2 C00,C0 � C00. A similar argument furnishes C00 � C0 so that �nally C0 = C00 = C.Clearly both U0 and U00 induce the (very good) topology �. According toProposition 2.7, they are point-symmetric, so that the merotopy C induces �as well (see Theorem 2.6).Example 3.4 shows that the invariance properties of semi-symmetry are es-sentially the same in the transitive case as in the general one. However, we canestablish useful criteria guaranteeing the symmetry of a transitive entourage orthe semi-symmetry of a transitive quasi-uniformity.Lemma 4.12. If c is a point-true cover of X, U = Uc is the correspondingtransitive entourage, then cU�1 = �(cc) where cc = fX � C : C 2 cg.Proof. Let V = U�1, x 2 X. Now y 2 V (x) i� x 2 U(y) = TfC 2 c : y 2 Cg i�y 2 C 2 c ) x 2 C i� x =2 C 2 c ) y =2 C i� x 2 X �C; C 2 c ) y 2 X �C i�y 2 TfX�C : C 2 c; x 2 X�Cg and the latter T is the element correspondingto x of �(cc). �Observe that � cannot be dropped: let X = [0;1] � R, c = f[0;x] : 0 5 x <1g [ f1g; now cc = f(x;1] : 0 5 x < 1g [ [0;1) is not point-true.Theorem 4.13. Let c be a point-true cover of X and U = Uc. U is symmetrici� c is a partition of X.Proof. Necessity: Suppose U(x) \ U(y) 6= ?, say, z 2 U(x) \ U(y). ThenU(z) � U(x)\U(y) by the transitivity, x 2 U(z) and y 2 U(z) by the symmetry,and U(x)[U(y) � U(z) by the transitivity again. Hence U(x) = U(z) = U(y).Su�ciency: If U(x) = C0 then U�1(x) = TfX � C : C 2 c; x =2 Cg byLemma 4.12, hence U�1(x) = C0 provided c is a partition. �Theorem 4.14. Let C = CU for a transitive quasi-uniformity U. The latter issemi-symmetric i� there is a base B for C composed of covers cU with transitiveU 2 U and such that these U constitute a base for U, further, if c 2 B, there isa c0 2 B such that, whenever C0i 2 c0 and C01 \C02 6= ?, there is C 2 c satisfyingC01 [ C02 � C.Proof. Necessity: Let B = fcU : U 2 U is transitive g. Given c = cU 2 B,U 2 U transitive, choose a transitive V0 2 U such that V �10 V0 � UU�1 andset V = V0 \ U 2 U. Finally let c0 = cV . Now if C01 = V (x), C02 = V (y) andC01 \ C02 6= ?, we have some z such that z 2 V (x) \ V (y), hence y 2 V �1(z) �V �1(V (x)) � U(U�1(x)). Consequently there is some u satisfying u 2 U�1(x),y 2 U(u), i.e. x;y 2 U(u), therefore C01 [ C02 = V (x) [ V (y) � U(x) [ U(y) �U(u) by the transitivity of U. For C = U(u) 2 c we obtain C01 [ C02 � C.Su�ciency: Given U 2 U, choose a transitive U0 2 U such that U0 � Uand cU0 belongs to the base B in the hypothesis. Set c = cU0, then choosec0 2 B satisfying C01 [ C02 � C 2 c whenever C0i 2 c0 and C01 \ C02 6= ?, and Merotopies associated with quasi-uniformities 11let c0 = cV for some transitive V 2 U. If x 2 X and y 2 V �1(V (x)), thenV (x); V (y) 2 c0 and V (x)\V (y) 6= ? so that V (x)[V (y) � C = U0(z) � U(z)for a suitable z 2 X. Then x;y 2 U(z), hence z 2 U�1(x) and y 2 U(U�1(x)).From V �1(V (x)) � U(U�1(x)) we obtain V �1V � UU�1. �A similar (but simpler) argument furnishes:Corollary 4.15. Let c = cU for a transitive entourage U. The latter is semi-symmetric i�, whenever Ci 2 c and C1 \ C2 6= ?, there exists C 2 c satisfyingC1 [ C2 � C.Semi-symmetry and point-symmetry are independent concepts also for tran-sitive quasi-uniformities. In fact, the example given above for a point-symmetricbut not semi-symmetric quasi-uniformity was a Pervin quasi-uniformity, hencetransitive. For a semi-symmetric but not point-symmetric, transitive quasi-uniformity, consider:Example 4.16. Let X = fa;bg, c be the closure associated with the Sierpi�nskitopology f?;fag;Xg, U the (transitive) Pervin quasi-uniformity of c generatedby the base fUg where U = Ufag and cU = ffag;Xg. Then U(a) = fag,U(b) = X, U�1(a) = X, U�1(b) = fbg. Clearly U�1(U(a)) = U�1(U(b)) =U(U�1(a)) = U(U�1(b)) = X so that U is semi-symmetric, but it is not point-symmetric because U�1(a) " U(a).Acknowledgements. The author thanks Professor Vera T. S�os for helpfulsuggestions. References[1] �A. Cs�asz�ar, General Topology, Akad�emiai Kiad�o, Budapest, 1978.[2] �A. Cs�asz�ar, D-completions of Pervin-type quasi-uniformities, Acta Sci. Math. (Szeged)57 (1993), 329{335.[3] �A. Cs�asz�ar and J. De�ak, Simultaneous extensions of proximities, semi-uniformities, con-tiguities and merotopies I, Math. Pannon. 1/2 (1990), 67{90.[4] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, Inc., New Yorkand Basel, 1982.[5] M. Hall, Jr., Distinct representatives of subsets, Bull. Amer. Math. Soc. 54 (1948), 922{926.[6] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26{30.[7] H. Herrlich, Topological structures, Topological Structures (Proc. Symp. in honour of J.de Groot, Amsterdam, 1973) Math. Centre Tracts 52 (1974), 59{122.[8] M. Kat�etov, On continuity structures and spaces of mappings, Comment. Math. Univ.Carolinae 6 (1965), 257{278.[9] L. Mirsky, Transversal Theory, Academic Press, New York and London, 1971.Received January 2000 12 �Akos Cs�asz�ar�Akos Cs�asz�arDepartment of AnalysisE�otv�os Lor�and UniversityM�uzeum krt. 6{81088 BudapestHungaryE-mail address: csaszar@ludens.elte.hu