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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 99 - 114

Preservation of completeness undermappings in asymmetric topologyHans-Peter A. K�unzi�Abstract. The preservation of various completeness proper-ties in the quasi-metric (and quasi-uniform) setting under open,closed and uniformly open mappings is investigated. In partic-ular, it is noted that between quasi-uniform spaces the propertythat each costable �lter has a cluster point is preserved underuniformly open continuous surjections. Furthermore in the realmof quasi-uniform spaces conditions under which almost uniformlyopen mappings are uniformly open are given which generalize cor-responding classical results for uniform spaces. As a by-productit is shown that a quasi-metrizable Moore space admits a left K-complete quasi-metric if and only if it is a complete Aronszajnspace.2000 AMS Classi�cation: 54C10, 54E15, 54E35, 54E50Keywords: uniformly open mapping, almost uniformly open mapping, openmapping theorem, quasi-metrizable, left K-complete, open mapping, closedmapping, supercomplete, Aronszajn space1. IntroductionIn Section 2 the preservation of topological completeness properties related toleft K-completeness under open continuous mappings between quasi-metrizablespaces is studied. Section 3 contains similar investigations for uniformly openmappings between quasi-metric and quasi-uniform spaces. Furthermore Section4 deals with the classical problem of determining conditions under which almostuniformly open mappings are uniformly open, again for the case of mappingsbetween quasi-metric and quasi-uniform spaces. Those two sections show thatin order to obtain satisfactory results, for our purposes it is appropriate to�This paper was written while the author was supported by the Swiss National ScienceFoundation under grant 2000-056811.99. He also acknowledges support during a visit to theUniversity of Oxford by the Stiftung zur F�orderung der wissenschaftlichen Forschung an derUniversit�at Bern.



100 Hans-Peter A. K�unziwork under some conditions of supercompleteness. Section 5 �nally recordsseveral results on the preservation of these completeness properties under closedcontinuous mappings between quasi-metrizable spaces.2. Preservation of completeness properties under open mappingsLooking for an adequate version of completeness for the present investiga-tions, we found that conditions from the area of left K-completeness were espe-cially useful. Therefore in the following we shall concentrate on such propertiesand notions. To �x our notation and terminology let us recall the following ba-sic concepts and conventions. By N we shall denote the positive integers. LetX be a set. As usual, a function d : X � X ! [0;1) that satis�es d(x;y) = 0if and only if x = y; and d(x;z) � d(x;y) + d(y;z) whenever x;y;z 2 X iscalled a quasi-metric on X: The induced topology �(d) is the topology gener-ated by the base consisting of the balls B2�n(x) = fy 2 X : d(x;y) < 2�ngwhere x 2 X and n 2 N: A sequence (xn)n2N in a quasi-metric space (X;d)is called left K-Cauchy provided that for each k 2 N there is nk 2 N suchthat d(xn;xm) < 2�k whenever n;m 2 N and nk � n � m: A quasi-metricspace (X;d) is left K-complete provided that each left K-Cauchy sequence con-verges in (X;�(d)) (compare [32]). For further concepts from the theory ofquasi-uniform spaces we refer the reader to [9]. (Note however that we shalluse Us instead of U� to denote the coarsest uniformity �ner than some givenquasi-uniformity U:)It is well known that Hausdor� showed that each open continuous imageof a completely metrizable space is completely metrizable provided that it ismetrizable. A modern proof of this fact is now usually based on the result thata paracompact open continuous (Hausdor�) image of a �Cech complete spaceis �Cech complete (compare [4, pp. 114{116]). On the other hand we do notknow whether each open continuous quasi-metrizable image of a left K-completequasi-metric space admits a left K-complete quasi-metric. (Recall that Kofner[20] showed that quasi-metrizability need not be preserved under open compactcontinuous mappings.) For the further discussion of that problem it is usefulto be aware of the characterization of R0-spaces possessing a �-base given byWicke and Worrell in [38, Theorems 3.2 and 3.3]: A topological R0-space Xhas a �-base if and only if there exists a sequence (Gn)n2N of bases for X suchthat every decreasing representative (Gn)n2N of (Gn)n2N with nonempty termsconverges to some x 2 X and also to every element of \n2NGn: (In the followingwe shall call such a sequence of bases a �-base sequence.) It is straightforward toverify that a quasi-metric space (X;d) is left K-complete if and only if (Gn)n2N,where Gn = fB2�k(x) : x 2 X;k � n;k 2 Ng whenever n 2 N; is a �-basesequence (compare [33, Theorem 1]). Since each open continuous R0-image ofa space possessing a �-base has a �-base [37, Theorem 1], the question ariseswhether each quasi-metrizable space with a �-base admits a left K-completequasi-metric. We observe that it was shown in [24, Propositions 10 and 11]that a (Tychono�) �Cech complete or scattered quasi-metrizable space admits aleft K-complete quasi-metric. We also note that for regular spaces, because a



Preservation of completeness under mappings in asymmetric topology 101regular T0-space is a complete Aronszajn space if and only if it has a �-base (see[37, p. 256]), our problem was already formulated by Romaguera (Question3 of [33]) when he asked whether each complete Aronszajn quasi-metrizablespace admits a left K-complete quasi-metric. While the latter question remainsopen, in this section we shall show that Romaguera's problem has a positiveanswer in the class of quasi-metrizable Moore spaces. Our method of proofmay be of independent interest, since as a by-product of our slightly moregeneral argument we obtain a new proof of Kofner's classical result [19] thateach 
-space with an ortho-pair-base is quasi-metrizable which seems easierto comprehend than the original one. Our proof will make use of some ideascontained in Junnila's thesis [11] (see also [12]) and in [14]. In particular, thefollowing concept will be used: A neighbornet U of a topological space X iscalled unsymmetric provided that x;y 2 X;x 2 U(y) and y 2 U(x) imply thatU(x) = U(y):The de�nition of an ortho-pair-base (for T1-spaces) is due to Kofner [19]. Acollection B of pairs (G;G0) (with G � G0) of open sets in a topological spaceX is called a pair-base for X provided that whenever H is open and x 2 H thenthere is (G;G0) 2 B such that x 2 G � G0 � H: The concept of a local pair-baseat some x 2 X will now be self-explanatory. A pair-base P of a topologicalspace X is called an ortho-pair-base provided that for each subcollection P0 ofP and each x 2 \fG : (G;G0) 2 P0g such that x 62 int \ fG0 : (G;G0) 2 P0g;the collection P0 is a local pair-base at x.Proposition 2.1. Let X be a topological space that possesses an ortho-pair-base. Then for every unsymmetric neighbornet S of X there exists a neighbornetV of X such that V 2 � S.Proof. Suppose that G is an ortho-pair-base for X that is well-ordered by �.For each x 2 X choose the �rst element (G;G0) 2 G with respect to � suchthat x 2 G � G0 � S(x) and call it (Gx;G0x); furthermore set V (x) = TfG0y :x 2 Gyg \ Gx: First we want to show that V (x) is a neighborhood at x :Otherwise x cannot have a smallest neighborhood and fG0y : x 2 Gyg is aneighborhood base at x; since G is an ortho-pair-base. Therefore there is Gysuch that x 2 Gy � G0y � Gx: Then fx;yg � Gx � G0x � S(x) and fx;yg �Gy � G0y � S(y): Thus S(x) = S(y) by unsymmetry of S: Furthermore we have(Gx;G0x) < (Gy;G0y) or (Gx;G0x) > (Gy;G0y) which contradicts the de�nition of(Gy;G0y) resp. (Gx;G0x): We conclude that V is a neighbornet of X: Supposethat x 2 X and y 2 V (x): Then y 2 Gx: Thus V (y) � G0x � S(x): We haveshown that V 2 � S. �Recall that for a neighbornet V of a topological space X the neighbornet V +is de�ned as follows: V +(x) = TfV (G) : G is a neighborhood at xg wheneverx 2 X (see [19]). Note that V + � V 2:Proposition 2.2. Let X be a topological T1-space with an ortho-pair-base.Then for each neighbornet U of X, U+ contains an unsymmetric neighbornetS of X.



102 Hans-Peter A. K�unziProof. Suppose that G0 = f(G�;G0�) : � < �g [ f(fxg;fxg) : x is isolated inXg is an ortho-pair-base for X, where we can assume that each G0� is not asingleton. Set H0 = X and de�ne inductively, given an ordinal �; H�+1 =intfx 2 H� : SfG0 : x 2 G;(G;G0) 2 G�g * U(x)g and G�+1 = f(G;G0) 2 G� :G0 � H�+1g n f(G;G0) 2 G� : G0� � G0g where the second set of that de�nitionis assumed to be empty if G0� is unde�ned; furthermore for a limit ordinal �set G� = T�<� G� and H� = T�<� H�:Clearly, for each �; H�+1 is open and the trans�nite sequence (H�)� is de-creasing. Note that certainly H�+1 = ?: So we assume that the induction stopsat the �rst ordinal � such that H� = ?: Observe also that SfG0 : (G;G0) 2G�g � H� for each limit ordinal �: We want to show next by induction on� that for each x 2 H� we have that G� contains a neighborhood pair-baseat x : If this condition is satis�ed for some �; then clearly it is also ful�lledfor � + 1; since no G0� is a singleton. For a limit ordinal � we argue as fol-lows: Suppose that x 2 T�<� H�: Since x 2 H�+1 whenever � < �; we haveSfG0 : x 2 G;(G;G0) 2 G�g * U(x) whenever � < �: Consequently for each� < � there exists (E�;E0�) 2 G� such that x 2 E� and E0� * U(x): Thusx 2 intT�<� E0� by de�nition of the characteristic property of the ortho-pair-base G0: Let (L;L0) 2 G0 be such that x 2 L and L0 � intT�<� E0�: Supposethat (L;L0) 62 G�: Then there is some minimal � < � such that (L;L0) 62 G�:Note that � necessarily is a successor ordinal and so (L;L0) 2 G��1: ThenL0 * H� or G0��1 � L0: Therefore E0� * H� or G0��1 � E0�: Thus (E�;E0�) 62 G�|a contradiction. We conclude that f(L;L0) : x 2 L;(L;L0) 2 G�g is a neigh-borhood pair-base at x: In particular, since (L;L0) 2 \�<�G�; we deduce thatx 2 L0 � \�<�H� = H�: So H� is also open if � is a limit ordinal.Let x 2 X and let �x be the �rst ordinal � such that x 62 H�: Note that�x necessarily is a successor ordinal and so x 2 H�x�1: Let (Gx;G0x) be the�rst element of G�x�1 with respect to the well-ordering of G0 such that x 2 Gx:Set S(x) = Gx: Then the neighbornet S = Sx2X(fxg � S(x)) is unsymmetric:If x;y 2 X and fx;yg � S(x) \ S(y); then �x = �y: Otherwise suppose forinstance that �x < �y: Then x 62 H�y�1; but x 2 Gy where (Gy;G0y) 2 G�y�1and hence x 2 G0y � H�y�1 |a contradiction. Therefore we conclude that�x = �y and so S(x) = S(y):Let x 2 X and let G be an arbitrary neighborhood of x: By de�nition ofH�x; there is y 2 S(x)\G\H�x�1 such that SfG0 : y 2 G;(G;G0) 2 G�x�1g �U(y) � U(G): Thus S(x) = Gx � G0x � U(G): Hence we have shown thatS � U+: �Remark 2.3. Combining the two preceding propositions we obtain the resultdue to Kofner that in a T1-space X with an ortho-pair-base for each neighbornetU of X there is a neighbornet V of X such that V 2 � U+. As Kofner observedin [19, Proposition 3], the latter result implies that for each neighbornet U of X,U+ is a normal neighbornet (compare [19, Theorem 1]) so that in particular each
-space with an ortho-pair-base is quasi-metrizable [19, Theorem 2]. (Recallthat a T1-space X is a 
-space provided that it possesses a sequence (Vn)n2N of



Preservation of completeness under mappings in asymmetric topology 103neighbornets such that fV 2n (x) : n 2 Ng is a neighborhood base at x wheneverx 2 X.) Obviously it also follows from these results that in a topological spacewith an ortho-pair-base for each unsymmetric neighbornet U there exists anunsymmetric neighbornet V such that V 2 � U. Kofner noted in [19, p. 1440]that each developable 
-space possesses an ortho-pair-base. Next we want toapply the preceding results to our discussion of the �-base property.Proposition 2.4. Let X be a T1-space possessing a �-base and having theproperty that for each unsymmetric neighbornet U there exists an unsymmetricneighbornet V such that V 2 � U. Then X admits a left K-complete quasi-metric.Proof. Since X has a base of countable order [37] and thus a primitive base [39,Theorem 4.1], X possesses a sequence (Hn)n2N of unsymmetric neighbornetssuch that fHn(x) : n 2 Ng is a neighborhood base at x whenever x 2 X (see[8, p. 147]). Let (Bn)n2N be a �-base sequence of X: We can suppose thateach base Bn is well-ordered by �n. Inductively we shall de�ne unsymmetricneighbornets Vn and Bn such that V 2n+1 � Hn \ Bn and Bn � Vn whenevern 2 N: Set V1(x) = X whenever x 2 X: Suppose now that, for some n 2 N;the unsymmetric neighbornet Vn is de�ned. Then for each x 2 X we �nd the�rst element B 2 Bn such that x 2 B � Vn(x) and set Bn(x) = B: Similarlyas above, note �rst that the neighbornet Bn is unsymmetric: If x;y 2 X andx;y 2 Bn(x) \ Bn(y); then Vn(x) = Vn(y) by unsymmetry of Vn: By de�nitionof Bn it follows that Bn(x) = Bn(y):By our assumption on unsymmetric neighbornets of X we can �nd an un-symmetric neighbornet Vn+1 of X such that V 2n+1 � Hn \ Bn; since Hn \ Bnis unsymmetric. The induction having carried out, we note that B2n+1 � Bnand Bn+1 � Hn whenever n 2 N: Then fBn : n 2 !g is a base for a com-patible quasi-metrizable quasi-uniformity V on X: Let d be a quasi-metric onX inducing V and let (xn)n2N be a left K-Cauchy sequence in (X;d): Thereis a strictly increasing sequence (nk)k2N in N such that for each k 2 N;(xn;xm) 2 Bk whenever nk � n � m and n;m 2 N: For each k 2 N n f1g wehave xnk+1 2 Bk(xnk) and thus Bk(xnk+1) � Bk�1(xnk): Since Bk(xnk+1) 2 Bkwhenever k 2 N we conclude by the �-base property that fBk(xnk+1) : k 2 Ngand thus (xn)n2N converges to some x 2 X (compare [34, Lemma 1]). We haveshown that (X;d) is left K-complete. �Corollary 2.5. Each T1-space with an ortho-pair-base that also possesses a�-base admits a left K-complete quasi-metric.In particular we conclude that a Moore space admits a left K-complete quasi-metric if and only if it is a complete Aronszajn quasi-metrizable space (compare[33, Theorem 1]). Moore spaces that are complete Aronszajn spaces are alsocalled semicomplete Moore spaces in the literature [30]. The Tychono� exampledue to [4, Example 2.9] shows that a quasi-metrizable semicomplete Moorespace need not be �Cech complete. Quasi-metrizability of this space is clear, sinceit is a metacompact Moore space (see [9, Theorem 7.26]). Moreover it has a �-base, because it is locally completely metrizable (compare [4, Proposition 2.2]).



104 Hans-Peter A. K�unziObserve that this example answers negatively another question of Romaguera[33, Question 2], since each sequentially complete quasi-metric Tychono� spaceis �Cech complete [22, Proposition 4]. Let us �nally state explicitly the twoquestions discussed in this section.Problem 2.6. Does each quasi-metrizable space with a �-base admit a left K-complete quasi-metric?Problem 2.7. Suppose that X admits a left K-complete quasi-metric and f :X ! Y is an open continuous surjection onto a quasi-metric space Y: Does Yadmit a left K-complete quasi-metric? (As mentioned above, these conditionsimply that Y possesses a �-base [37, Theorem 1]. Observe also that Theorem 8of [37] asserts that a regular T0-space has a �-base if and only if it is an opencontinuous image of a completely metrizable space.)3. Preservation of completeness properties under uniformly openmappingsIn this section we shall show that as in the classical, symmetric case betterresults than in Section 2 can be achieved if we assume that the mappings areuniformly open with respect to some given (quasi-)uniform structures on thespaces under consideration. Let (X;U) and (Y;V) be quasi-uniform spaces. A(multi-valued) mapping F : X ! Y is called uniformly open provided that foreach U 2 U there is V 2 V such that V (F(x)) � F(U(x)) whenever x 2 X(compare [5]). It is known that in the area of uniform (Hausdor�) spaceseach open continuous mapping with compact domain is uniformly open [7,Proposition 2.2]. In fact the following more general result holds.Proposition 3.1. Let (X;U) be a compact uniform space and let the mappingf : (X;U) ! (Y;V) be open and continuous where (Y;V) is a quasi-uniformspace. Then f is uniformly open.Proof. Let U 2 U: There is P 2 U such that P2 � U: Since f is open, foreach a 2 X we �nd Wa 2 V such that W2a (f(a)) � f(P(a)): By continuity off and since U is a uniformity, we can consider the open cover fint(P�1(a) \f�1Wa(f(a))) : a 2 Xg of X: Since X is compact, there is a �nite subset Fof X such that Sa2F int(P�1(a) \ f�1Wa(f(a))) = X: Set W = Ta2F Wa andnote that W 2 V: Consider x 2 X: There is b 2 F such that x 2 P�1(b) \f�1Wb(f(b)): Therefore f(x) 2 Wb(f(b)) and W(f(x)) � W2b (f(b)) � fP(b) �fP2(x) � fU(x): We have shown that f is uniformly open. �Applying the preceding result to the identity mapping on a compact Haus-dor� space, we draw the following conclusion.Corollary 3.2. [9, Proposition 1.47] The uniformity is the coarsest quasi-uniformity on a compact Hausdor� space.The identity mapping on a topological space X admitting two quasi-uniform-ities U and V such that U is not contained in V also shows that the conclusionof Proposition 3.1 can only hold under some strong conditions.



Preservation of completeness under mappings in asymmetric topology 105The following classical result from Kelley's book [15, p. 203] is well known:Let f be a uniformly open continuous mapping from a complete pseudo-metriz-able space into a uniform Hausdor� space. Then the range of the mapping fis complete. On the other hand, it is known that if G is a topological groupwhose left uniformity is complete and N is a closed normal subgroup, thenthe left uniformity of the quotient group G=N need not be complete, althoughthe quotient mapping is continuous and uniformly open (compare [31, p. 195]and [27]). Such examples show that completeness of the domain space is notsu�cient to generalize the afore-mentioned result from Kelley's book to uniformspaces.In order to extend our investigations on quasi-metric spaces from Section 2to general quasi-uniform spaces, we recall that a �lter F on a quasi-uniformspace (X;U) is called left K-Cauchy provided that for each U 2 U there isF 2 F such that U(x) 2 F whenever x 2 F . A quasi-uniform space (X;U)is called left K-complete provided that each left K-Cauchy �lter converges(compare [34]). The negative uniform result mentioned above however suggeststhat in an arbitrary quasi-uniform space (X;U) we should consider a propertystronger than left K-completeness, for instance, that each costable �lter has a�(U)-cluster point, where a �lter F on a quasi-uniform space (X;U) is calledcostable provided that for each U 2 U we have TF2F U�1(F) 2 F: Costable�lters characterize hereditary precompactness in the sense that a quasi-uniformspace (X;U) is hereditarily precompact if and only if each �lter on (X;U) iscostable (see e.g. [13, Proposition 2.5]). An ultra�lter on a quasi-uniformspace is costable if and only if it is a left K-Cauchy �lter [34, Proposition 1].Costable �lters were called Cs�asz�ar �lters by P�erez-Pe~nalver and Romaguerain [29]. They said that a quasi-uniform space (X;U) is Cs�asz�ar complete pro-vided that each costable �lter of (X;U) has a �(Us)-cluster point. The latterconditions strengthens the well-known property of Smyth completeness, whichmeans that each left K-Cauchy �lter has a �(Us)-cluster point (equivalently, a�(Us)-limit point). P�erez-Pe~nalver and Romaguera also remarked that for anytopological space X the well-monotone quasi-uniformity WX has the propertythat each costable �lter on (X;WX) has a cluster point [29, Proposition 2].It was noted (compare [26, p. 169], [32]) that for a quasi-pseudometric space(X;d), each costable �lter of the induced quasi-uniform space (X;Ud) clustersif and only if each left K-Cauchy sequence (resp. each left K-Cauchy �lter)converges. So for quasi-pseudometric spaces the property considered in thefollowing is indeed equivalent to left K-completeness. For uniform spaces theproperty under consideration is equivalent to supercompleteness. A uniformspace X is called supercomplete if each stable �lter has a cluster point [2, 10].For instance that condition is satis�ed by a complete bilateral uniformity ofa topological group of pointwise countable type [35]. It is well known thatsupercompleteness characterizes completeness of the Hausdor� uniformity onthe hyperspace of nonempty closed subsets (equivalently, nonempty subsets)of a uniform space. On the other hand, for a quasi-uniform space (X;U) thecondition that each costable �lter clusters in (X;U) is only necessary, but not



106 Hans-Peter A. K�unzisu�cient that its Hausdor� quasi-uniformity (on the collection of nonemptysets) is left K-complete (see [25]).Proposition 3.3. Let f be a uniformly open continuous mapping from a quasi-uniform space (X;U) in which each costable �lter F has a cluster point ontoa quasi-uniform space (Y;V): Then each costable �lter on (Y;V) has a clusterpoint.Proof. Let F be a costable �lter on (Y;V) and �x U 2 U: Since f is uniformlyopen, there is V 2 V such that V (f(x)) � f(U(x)) whenever x 2 X: Because the�lter F is costable in (Y;V); there is F0 2 F such that F0 � TF2F V �1(F): Wewant to show that f�1(F0) � TF2F U�1(f�1(F)) : Let F 2 F and a 2 f�1(F0):Hence f(a) = f0 for some f0 2 F0: Thus f0 2 V �1(e) for some e 2 F: Thene 2 V (f0) � f(U(a)): Therefore e = f(c) for some c 2 U(a): It follows thata 2 U�1(c) and c 2 f�1(F): We have shown that a 2 U�1(f�1(F)): Weconclude that f�1(F0) � TF2F U�1(f�1(F)) and f�1F = �lff�1(F) : F 2Fg is costable in (X;U): Suppose now that x is a cluster point of f�1F: Bycontinuity of f; f(x) is a cluster point of F: �Corollary 3.4. A uniform space that is the image of a supercomplete uniformspace under a uniformly open continuous mapping is supercomplete.Since in a quasi-uniform space each left K-Cauchy �lter is costable and con-verges to its cluster points (see [34]), the next result is a consequence of the pre-ceding proposition and the observation about quasi-pseudometric spaces men-tioned above.Corollary 3.5. Let (X;U) and (Y;V) be quasi-uniform spaces and f : (X;U) !(Y;V) be a uniformly open continuous surjection. If U is quasi-pseudometrizableand left K-complete, then V is left K-complete.Because uniformly continuous mappings between quasi-uniform spaces arecontinuous with respect to the associated supremum uniformities, the followingcorollary is also readily veri�ed.Corollary 3.6. Let (X;U) and (Y;V) be quasi-uniform spaces and f : (X;U) !(Y;V) a uniformly open uniformly continuous surjection. If U is Cs�asz�ar com-plete, then V is Cs�asz�ar complete.A �lter F on a quasi-uniform space (X;U) is called a weakly Cauchy �lteror Corson �lter provided that TF2F U�1(F) 6= ? whenever U 2 U (see e.g.[29]). Obviously, each costable �lter is weakly Cauchy. The property (compare[9, Proposition 5.32]) that each weakly Cauchy �lter has a cluster point is oftencalled co�nal completeness and even in metric spaces is strictly stronger thancompleteness (see e.g. [2, Example 1]). It is known that each uniformly locallycompact and each paracompact �ne uniform space is co�nally complete (e.g.[2, Corollaries 4 and 5]). In [36] it is shown that a (Tychono�) topologicalgroup is locally compact if and only if it is of pointwise countable type and itsleft uniformity is co�nally complete. The following strengthening of Cs�asz�arcompleteness was considered in [29]. A quasi-uniform space (X;U) is called



Preservation of completeness under mappings in asymmetric topology 107Corson complete provided that each weakly Cauchy �lter has a �(Us)-clusterpoint. As we show next, these two completeness properties are preserved underuniformly open uniformly continuous surjections, too.Proposition 3.7. Let (X;U) and (Y;V) be quasi-uniform spaces and f : (X;U)! (Y;V) a uniformly open continuous surjection. If X is co�nally complete,then Y is co�nally complete.Proof. It will su�ce to show that f�1F is a weakly Cauchy �lter on (X;U)provided that f is uniformly open and F is a weakly Cauchy �lter on (Y;V):So suppose that F is a weakly Cauchy �lter on (Y;V): Let U 2 U: By uniformopenness of f, there is V 2 V such that V (f(x)) � f(U(x)) whenever x 2 X:By our assumption, there is y 2 Y such that V (y) \ F 6= ? whenever F 2 F:Let x 2 X be such that y = f(x): Then fU(x)\F 6= ? whenever F 2 F: ThusU(x) \ f�1F 6= ? whenever F 2 F: Therefore f�1F = �lff�1F : F 2 Fg is aweakly Cauchy �lter. �Corollary 3.8. Let f : (X;U) ! (Y;V) be a uniformly open uniformly con-tinuous mapping from a Corson complete quasi-uniform space (X;U) onto aquasi-uniform space (Y;V): Then (Y;V) is Corson complete.Our �nal proposition in this section applies Corollary 3.5 to the questionconsidered in Section 2.Proposition 3.9. Suppose that f : X ! Y is an open continuous mapping froma topological space X onto a T1-space Y: If X admits a left K-complete quasi-metric d such that all �bers of f are precompact in (X;d�1) then Y admits a leftK-complete quasi-metric. In particular, a T1-image of a completely metrizablespace under an open compact mapping admits a left K-complete quasi-metric.Proof. We shall work with the quasi-metric quasi-uniformity Ud = �lfB2�n :n 2 Ng on X: For each y 2 Y and n 2 N set Vn(y) = Tx2f�1fyg f(B2�n(x))whenever y 2 Y: Then fVn : n 2 Ng is a base for a quasi-metrizable quasi-uniformity V on Y; because V 2n+1 � Vn whenever n 2 N and Tn2N Vn = f(y;y) :y 2 Y g: Since the �bers are precompact in (X;d�1); we see that V is compatible:By our assumption for each y 2 Y and U 2 Ud there exists a �nite subsetF � f�1fyg such that f�1fyg � Sx2F U�1(x) and thus for each x0 2 f�1fygthere is x 2 F such that x0 2 U�1(x) and so f(U(x)) � f(U2(x0)): ThereforeTx2F f(U(x)) � Tx02f�1fyg f(U2(x0)): Since Tx2F f(U(x)) is a neighborhoodof y and f is continuous, we deduce that V is compatible. Since f : (X;Ud) !(Y;V) is uniformly open by de�nition of V, we conclude that V is left K-completeby Corollary 3.5. �4. Almost uniformly open mappingsIn this article a (multi-valued) mapping F : X ! Y between quasi-uniformspaces (X;U) and (Y;V) is called almost uniformly open provided that for eachU 2 U there is V 2 V such that V (F(x)) � cl�(V�1)F(U(x)): Note that thisde�nition yields the usual concept of almost uniform openness for mappings



108 Hans-Peter A. K�unzibetween uniform and metric spaces. Extending classical work on metric spaces(see [15, p. 202]) Dektjarev [6] proved the following result for supercompleteuniform spaces: Let F be an almost uniformly open multi-valued mapping withclosed graph from the supercomplete uniform space X into an arbitrary uniformspace Y: Then, for any entourages U and V in X and any point x0 2 X; theinclusion FU(x0) � FV U(x0) is valid.In this section we want to address the problem under which conditions an al-most uniformly open mapping between quasi-uniform spaces is uniformly open.To this end we �rst recall that a quasi-uniform space (X;U�1) is called rightK-complete provided that each left K-Cauchy �lter on (X;U) converges with re-spect to the topology �(U�1) (compare [34]). In the following we shall considera stronger condition and further variant of the uniform property of supercom-pleteness, namely the condition that each costable �lter on the quasi-uniformspace (X;U) has a �(U�1)-cluster point. The latter condition was already stud-ied to some extent by K�unzi and Ryser [26, Proposition 6] where it was shownto be equivalent to right K-completeness of the Hausdor� quasi-uniformitytransmitted by U�1 onto the collection of nonempty subsets of X: We also re-call that a quasi-metric space (X;d�1) is called right K-sequentially completeprovided that each left K-Cauchy sequence of (X;d) converges in (X;�(d�1)):It is known that right K-sequential completeness (for non-R1-spaces) can bestrictly weaker than right K-completeness of the induced quasi-uniformity inquasi-metric spaces [1, Remark 2]. This complication suggests that we should�rst establish a version of Dektjarev's result for quasi-metric spaces and only af-terwards consider the more general quasi-uniform case. We remark that Khanhhas already obtained a quantitative version of our next proposition in [16, The-orem 2]. On the other hand, Cao and Reilly [3, Lemma 5.3] gave some bitopo-logical version of that result. Related to Khanh's studies further investigationsin quasi-uniform spaces were conducted by Chou and Penot [5].Proposition 4.1. (compare [16]) Each almost uniformly open mapping f :X ! Y from a quasi-metric space (X;d) into a quasi-metric space (Y;d0)such that the graph of f is �(d�1) � �((d0)�1)-closed and (X;d�1) is right K-sequentially complete is uniformly open.Proof. Let U resp. V be the quasi-metric quasi-uniformities on (X;d) resp.(Y;d0) generated by the standard bases fU� : � > 0g resp. fV� : � > 0g: Byour assumption on f for each U 2 U there is V 2 V such that V (f(x)) �cl�(V�1)fU(x) whenever x 2 X: Hence it su�ces to show that cl�(V�1)fU�(x) �fU�+�(x) whenever �;� > 0 and x 2 X:Fix �;� > 0: For each n 2 N; set �n = �2n and choose �n � 1n suchthat V�n(f(x)) � cl�(V�1)fU�n(x) whenever x 2 X: Fix x 2 X and let y 2cl�(V�1)fU�(x): Find x1 2 U�(x) such that (f(x1);y) 2 V�1: Inductively we de�nea sequence (xn)n2N in X such that (f(xn);y) 2 V�n and (xn;xn+1) 2 U�n when-ever n 2 N : Suppose that xn was chosen for some n 2 N such that the inductionhypothesis is satis�ed. Therefore we have y 2 V�n(f(xn)) � cl�(V�1)f(U�n(xn)):Hence we �nd xn+1 2 U�n(xn) such that f(xn+1) 2 V �1�n+1(y): This completes the



Preservation of completeness under mappings in asymmetric topology 109induction. It follows that (xn)n2N is a left K-Cauchy sequence in (X;d): By ourassumption on completeness of X there is x0 2 X such that (xn)n2N convergesto x0 in (X;�(U�1)): We conclude that d(x;x0) < � + �, because d(xn;x0) ! 0and thus d(x1;x0) � � by the triangle inequality. Consequently x0 2 U�+�(x):Since the graph of f is �(d�1) � �((d0)�1)-closed and d0(f(xn);y) ! 0; we seethat y = f(x0): Thus cl�(V�1)fU�(x) � fU�+�(x): We have shown that f isuniformly open. �We shall now give a version of Dektjarev's argument for quasi-uniform spaces.Proposition 4.2. Let (X;U) be a quasi-uniform space such that each costable�lter on (X;U) has a �(U�1)-cluster point and let F be an almost uniformlyopen multi-valued mapping from (X;U) into an arbitrary quasi-uniform space(Y;V). Suppose that the graph of F is �(U�1) � �(V�1)-closed. Then for anyentourages U und W in U and any point x0 2 X; we have cl�(V�1) FU(x0) �FWU(x0), In particular F is uniformly open.Proof. Suppose that fUi : i 2 Ig is a base for U and fVi : i 2 Ig is a base for V.With every entourage P of U, we associate a sequence of entourages (Pn)n2Nsuch that P21 � P and P2n+1 � Pn whenever n 2 N: By our assumption onF , we can suppose that for each U 2 U there is UF 2 V such that UFF(z) �cl�(V�1)FU(z) whenever z 2 X: Fix now U;W 2 U: Without loss of generalitywe assume that (Ui)n � Wn+1 whenever i 2 I and n 2 N: Let x0 2 X andy 2 cl�(V�1)FU(x0): Furthermore let D be the collection of nonempty �nitesubsets of I partially ordered by inclusion and for any � 2 D denote the numberof elements of � by j�j: We shall construct for each � 2 D a nonempty setB� � W1U(x0) such that B� � (Ti2�(Ui)j�j)�1(B�) whenever � 2 D and� � �:The sets B� are constructed by induction on j�j: For each i 2 I, set Bi =fx 2 U(x0) : V �1i (y) \ (((Ui)1)F )�1(y) \ F(x) 6= ?g. Furthermore for each� 2 D with j�j � 2 set B� = fx 2 S���(Ti2�(Ui)j�j)(B�) : (\i2�V �1i (y)) \((Ti2�(Ui)j�j)F)�1(y) \ F(x) 6= ?g. We shall verify next that the sets B�(� 2 D) satisfy the stated conditions: Since y 2 cl�(V�1) FU(x0), there is anet (z�)�2E in FU(x0) converging to y in (Y;V�1). For each � 2 E chooseu� 2 U(x0) such that (u�;z�) 2 F . We conclude that for each i 2 I; u� 2Bi eventually, and, thus, also, for each � 2 D, we have u� 2 B� eventually.Hence each B� (� 2 D) is nonempty. For all i 2 I; the inclusion Bi � U(x0)holds by de�nition. Let j�j = k � 2. Inductively we can assume that for all� for which j�j < k, the inclusion B� � Wj�jWj�j�1 : : :W2U(x0) is satis�ed.(In particular, we have B� � U(x0) for j�j = 1:) Then, by de�nition, B� �S���(Ti2�(Ui)j�j)(B�) � S��� Wj�j+1(B�) � S��� Wj�j+1Wj�j : : :W2U(x0) =Wj�j : : : W2U(x0). Hence B� � W1U(x0) whenever � 2 D. Consider now�;� 2 D such that � � � and x 2 B�. From ((\i2�(Ui)j�j)F )�1(y) \F(x) 6= ?,that is y 2 (Ti2�(Ui)j�j)F (F(x)) � cl�(V�1)F(\i2�(Ui)j�j)(x), we see that thereexists x0 2 (\i2�(Ui)j�j)(x) such that (\i2�V �1i (y)) \ ((Ti2�(Ui)j�j)F )�1(y) \F(x0) 6= ?: Therefore x0 2 B� by de�nition of B�. Furthermore we deduce that



110 Hans-Peter A. K�unzix 2 (\i2�(Ui)j�j)�1(x0), that is B� � (Ti2�(Ui)j�j)�1(B�). This concludes theveri�cation of the stated conditions.For each � 2 D set C� = S��� B�: Clearly fC� : � 2 Dg is a �lterbase onX: We shall show that the generated �lter F is costable in (X;U): Let H 2 Uand � 2 D: There is i 2 I such that Ui � H: Consider x 2 Cfig: Consequentlyx 2 B� for some � 2 D such that i 2 �: Note that � = � [ � 2 D: Thenx 2 B� � (Tj2�(Uj)j�j)�1(B�) � U�1i (B�) � H�1(B�) � H�1(C�): Hence wehave shown that Cfig � T�2D H�1(C�): Thus F is costable in (X;U). Observenext that the set of cluster points of F in (X;U�1) belongs to WU(x0); sinceeach C� � W1U(x0) (� 2 D):By our assumption there exists a �(U�1)-cluster point x of F: Considerarbitrary i;k 2 I: Choose � 2 D such that fi;kg � � and U�1i (x) \ B� 6= ?:Find x0 2 U�1i (x) \ B�: Then V �1k (y) \ F(x0) 6= ? by de�nition of B�: Weconclude that (U�1i (x) � V �1k (y)) \ F 6= ?: Thus (x;y) 2 F by closedness of Fwith respect to the topology �(U�1)��(V�1): We have shown that y 2 F(x) �FWU(x0): It follows that F is uniformly open. �Corollary 4.3. (compare [28]) An almost uniformly open mapping with a closedgraph from a supercomplete uniform space into an arbitrary uniform space isuniformly open. In particular, an almost uniformly open continuous mappingfrom a supercomplete uniform space into a uniform Hausdor� space is uniformlyopen.Corollary 4.4. Let (X;U) be a Cs�asz�ar complete quasi-uniform space andf : (X;U) ! (Y;V) an almost uniformly open uniformly continuous mappingonto a quasi-uniform T1-space (Y;V): Then f is uniformly open and (Y;V) isCs�asz�ar complete.Proof. Only the �nal paragraph of the proof of Proposition 4.2 has to be modi-�ed. This time F has a �(Us)-cluster point x in X: Let k 2 I: By continuity off there is i 2 I such that f(Ui(x)) � Vk(f(x)): Find � 2 D such that fi;kg � �and there is x0 2 Ui(x) \ U�1i (x) \ B�: Thus f(x0) 2 Vk(f(x)); furthermoref(x0) 2 V �1k (y) by de�nition of B�: Consequently (f(x);y) 2 \V and thusy = f(x) 2 fWU(x0): We conclude that f is uniformly open. The secondassertion is a consequence of Corollary 3.6. �5. Preservation of completeness under closed mappingsWe �nish this article with three results on closed continuous mappings be-tween quasi-metrizable spaces. Let us recall that Kofner [18] has shown thateach �rst-countable closed continuous image of a quasi-metrizable space is quasi-metrizable. His techniques can be modi�ed to yield the following two results.Proposition 5.1. The image of a left K-complete quasi-metric space under aperfect continuous mapping admits a left K-complete quasi-metric.Proof. Let f : X ! Y be a perfect continuous mapping from a left K-completequasi-metric space (X;d) onto a topological space Y: For each y 2 Y and n 2 N



Preservation of completeness under mappings in asymmetric topology 111set Vn(y) = fy0 2 Y : f�1fy0g � B2�n(f�1fyg)g: Then clearly, by the assump-tion made on f; fVn : n 2 Ng is a base of a compatible quasi-metrizable quasi-uniformity V on Y (see [17, Theorem 2]). Let e be a quasi-metric on Y inducingV: Furthermore let (yn)n2N be a left K-Cauchy sequence in (Y;e): There is astrictly increasing sequence (nk)k2N in N such that (ynk;yp) 2 Vk wheneverp 2 N and p � nk: Hence f�1fynk+1g � B2�k(f�1fynkg) whenever k 2 N:By compactness of the �bers of f; we �nd �nite subsets Fnk of f�1fynkg suchthat f�1fynkg � B2�k(Fnk) and therefore Fnk+1 � f�1fynk+1g � B2�(k�1)(Fnk)whenever k 2 N: By K�onig's Lemma [21] applied to the sequence of �nite sets(Fnk)k2N we see that there exists a sequence (y0nk)k2N such that y0nk 2 Fnkand d(y0nk;y0nk+1) < 2�(k�1) whenever k 2 N: Thus by left K-completenessof (X;d) we can �nd x 2 X such that the left K-Cauchy sequence (y0nk)k2Nconverges to x: Therefore by continuity of f; the sequence (ynk)k2N and henceby [34, Lemma 1] the sequence (yn)n2N converges to f(x): Hence (Y;e) is leftK-complete. We conclude that the topological property of admitting a leftK-complete quasi-metric is preserved under perfect continuous surjections. �Proposition 5.2. A �rst-countable image Y of a right K-sequentially completequasi-metric space (X;d) under a closed continuous mapping f admits a rightK-sequentially complete quasi-metric.Proof. For any y 2 Y , let fVn(y) : n 2 Ng be a decreasing basic sequence ofopen neighborhoods at y: Set Wn(y) = fz 2 Y : f�1fzg � B2�n(f�1fyg) \f�1Vn(y)g whenever y 2 Y and n 2 N: Furthermore set cWn = SfWkp � : : : �Wk1 : 2�k1 + : : : + 2�kp � 2�n and k1; : : : ;kp;p 2 Ng whenever n 2 N: Notethat cW2n+1 � cWn whenever n 2 N: Kofner's argument [18, p. 334] showsthat the quasi-metrizable quasi-uniformity W generated by fcWn : n 2 Ng iscompatible on Y: Note that if a;b 2 Y , s 2 N and a 2 Ws(b) we can �nd for anya0 2 f�1fag some b0 2 f�1fbg such that a0 2 B2�s(b0): Let e be a quasi-metricon Y inducing W: It su�ces to show that e is right K-sequentially complete.Let (yn)n2N be a left K-Cauchy sequence in (Y;e�1): For each k 2 N there isa strictly increasing sequence (nk)k2N in N such that (yl;ynk) 2 cWk wheneverl 2 N and l � nk: In particular (ynk+1;ynk) 2 cWk whenever k 2 N: It followsthat for each k 2 N there are p 2 N, s1; : : : ;sp 2 N and a1; : : : ;ap�1 2 Y suchthat 2�s1 + : : : + 2�sp � 2�k and (ynk+1;a1) 2 Ws1; : : : ;(ap�1;ynk) 2 Wsp: (Inparticular, (ynk+1;ynk) 2 Ws1 if p = 1:) Choose y0n1 2 X such that yn1 = f(y0n1):Inductively over k 2 N we can �nd points a0p�1; : : : ;a01 and y0nk+1 2 X such thatf(y0nk+1) = ynk+1; for each i = 1; : : : ;p � 1 we have f(a0i) = ai and (y0nk+1;a01) 2B2�s1 ; : : : ;(a0p�1;y0nk) 2 B2�sp: Thus for each k 2 N; (y0nk+1;y0nk) 2 B2�(k�1): Weconclude that (y0nk)k2N is a left K-Cauchy sequence in (X;d�1) and convergesto some x in (X;d): Then the sequence (ynk)k2N converges to f(x) by continuityof f: Since (yn)n2N is a left K-Cauchy sequence in (Y;e�1); it also convergesto f(x) in (Y;e) (see [34, Lemma 1]). We deduce that Y admits a right K-sequentially complete quasi-metric. �



112 Hans-Peter A. K�unziProblem 5.3. Does a �rst-countable image of a left K-complete quasi-metricspace under a closed continuous mapping admit a left K-complete quasi-metric?Finally we would like to mention that it is well known that under appro-priate hypotheses preimages of quasi-uniform spaces which possess some kindof completeness property also satisfy that type of completeness condition (seee.g. [26, Proposition 7]). We �nish this article with another such result. (Itis known on the other hand that the property of quasi-metrizability behavesrather badly under preimages (compare [23]).)Proposition 5.4. Let f : X ! Y be a closed continuous mapping from aquasi-metric space (X;d) such that all �bers are left K-complete onto a leftK-complete quasi-metric space (Y;d0): Then X admits a left K-complete quasi-metric. (The analogous result also holds for right K-sequential completenessinstead of left K-completeness.)Proof. For each n 2 N set Vn = f(x;y) 2 X � X : d0(f(x);f(y)) < 2�nand d(x;y) < 2�ng: Let e be a quasi-metric on X inducing the (compati-ble) quasi-uniformity generated by fVn : n 2 Ng: Furthermore let (xn)n2Nbe a left K-Cauchy sequence in (X;e): Note �rst that the left K-Cauchy se-quence (f(xn))n2N converges to some y 2 Y: By our assumption on the �bers,(xn)n2N has a cluster point and thus, by [34, Lemma 1], converges providedthat (f(xn))n2N has a constant subsequence. So let us assume that this is notthe case. In particular we can suppose that f(xn) 6= y for n larger than somen0 2 N: By closedness of f, we deduce that y 2 f(cl�(d)fxn : n > n0;n 2 Ng):Choose x 2 cl�(d)fxn : n > n0;n 2 Ng such that f(x) = y: Then evidently x isa cluster point and thus by [34, Lemma 1] a limit point of the sequence (xn)n2N:We conclude that (X;e) is left K-complete. A similar argument establishes thesecond assertion. �References[1] E. Alemany and S. Romaguera, On right K-sequentially complete quasi-metric spaces, ActaMath. Hung. 75 (1997), 267{278.[2] B.S. Burdick, A note on completeness of hyperspaces, General Topology and Applications: FifthNortheast Conference at The College of Staten Island, 1989, ed. by S.J. Andima et al., Dekker,New York, 1991, 19{24.[3] J. Cao and I.L. Reilly, On pairwise almost continuous multifunctions and closed graphs, IndianJ. Math. 38 (1996), 1{17.[4] J. Chaber, M.M. �Coban and K. Nagami, On monotonic generalizations of Moore spaces, �Cechcomplete spaces and p-spaces, Fund. Math. 84 (1974), 107{119.[5] C.C. Chou and J.-P. Penot, In�nite products of relations, set-valued series and uniform opennessof multifunctions, Set-Valued Anal. 3 (1995), 11{21.[6] I.M. Dektjarev, A closed graph theorem for ultracomplete spaces, Sov. Math. Dokl. 5 (1964),1005{1007.[7] V.V. Fedorchuk and H.-P.A. K�unzi, Uniformly open mappings and uniform embeddings of func-tion spaces, Topology Appl. 61 (1995), 61{84.[8] P. Fletcher and W.F. Lindgren, �-spaces, Gen. Top. Appl. 9 (1978), 139{153.[9] P. Fletcher and W.F. Lindgren, Quasi-uniform Spaces, Dekker, New York, 1982.[10] J.R. Isbell, Supercomplete spaces, Paci�c J. Math. 12 (1962), 287{290.



Preservation of completeness under mappings in asymmetric topology 113[11] H.J.K. Junnila, Covering properties and quasi-uniformities of topological spaces, Ph. D. Thesis,Virginia Polytechnic Institute and State University, Blacksburg, Va., 1978.[12] H.J.K. Junnila, Neighbornets, Paci�c J. Math. 76 (1978), 83{108.[13] H.J.K. Junnila and H.-P.A. K�unzi, Stability in quasi-uniform spaces and the inverse problem,Topology Appl. 49 (1993), 175{189.[14] H.J.K. Junnila and H.-P.A. K�unzi, Ortho-bases and monotonic properties, Proc. Amer. Math.Soc. 119 (1993), 1335{1345.[15] J.L. Kelley, General Topology, D. van Nostrand, New York, 1955.[16] P.Q. Khanh, An induction theorem and general open mapping theorems, J. Math. Anal. Appl.118 (1986), 519{534.[17] J. Kofner, Quasi-metrizable spaces, Paci�c J. Math. 88 (1980), 81{89.[18] J. Kofner, Closed mappings and quasi-metrics, Proc. Amer. Math. Soc. 80 (1980), 333{336.[19] J. Kofner, Transitivity and ortho-bases, Can. J. Math. 33 (1981), 1439{1447.[20] J. Kofner, Open compact mappings, Moore spaces and orthocompactness, Rocky Mountain J.Math. 12 (1982), 107{112.[21] D. K�onig, Sur les correspondances multivoques des ensembles, Fund. Math. 8 (1926), 114{134.[22] H.-P.A. K�unzi, Complete quasi-pseudo-metric spaces, Acta Math. Hung. 59 (1992), 121{146.[23] H.-P.A. K�unzi, Perfect preimages having G�-diagonals of quasi-metrizable spaces, Indian J.Math. 41 (1999), 33{37.[24] H.-P.A. K�unzi and S. Romaguera, Some remarks on Doitchinov completeness, Topology Appl.74 (1996), 61{72.[25] H.-P.A. K�unzi and S. Romaguera, Left K-completeness of the Hausdor� quasi-uniformity, Ros-tock. Math. Kolloq. 51 (1997), 167{176.[26] H.-P.A. K�unzi and C. Ryser, The Bourbaki quasi-uniformity, Topology Proc. 20 (1995), 161{183.[27] M. Leischner, An incomplete quotient of a complete topological group, Arch. Math. (Basel) 56(1991), 497{500.[28] V.L. Levin, Open mapping theorem for uniform spaces, Am. Math. Soc., Transl., II. Ser. 71(1968), 61{66; translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2(45) (1965), 86{90.[29] M.J. P�erez-Pe~nalver and S. Romaguera, Weakly Cauchy �lters and quasi-uniform completeness,Acta Math. Hung. 82 (1999), 217{228.[30] T.M. Phillips, Completeness in Aronszajn spaces, Stud. Topol., Proc. Conf. Charlotte, N.C.,1974, 457{465 (1975).[31] W. Roelcke and S. Dierolf, Uniform Structures on Topological Groups and Their Quotients,McGraw-Hill, New York, 1981.[32] S. Romaguera, Left K-completeness in quasi-metric spaces, Math. Nachr. 157 (1992), 15{23.[33] S. Romaguera, On complete Aronszajn quasi-metric spaces and subcompactness, in Papers onGeneral Topology and Applications, Eighth Summer Conf. at Queens College, 1992, Ann. NewYork Acad. Sci. 728 (1994), 114{121.[34] S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, ActaMath. Hung. 73 (1996), 159{178.[35] S. Romaguera and M. Sanchis, Completeness of hyperspaces on topological groups, J. Pure Appl.Algebra (to appear).[36] S. Romaguera and M. Sanchis, Locally compact topological groups and co�nal completeness, J.London Math. Soc. (to appear).[37] H.H. Wicke and J.M. Worrell, Jr., Open continuous mappings of spaces having bases of countableorder, Duke Math. J. 34 (1967), 255{271; errata 813{814.[38] H.H. Wicke and J.M. Worrell, Jr., Topological completeness of �rst countable Hausdor� spacesI, Fund. Math. 75 (1972), 209{222.[39] H.H. Wicke and J.M. Worrell, Jr., A characterization of spaces having bases of countable orderin terms of primitive bases, Can. J. Math. 27 (1975), 1100{1109.Received March 2000



114 Hans-Peter A. K�unziHans-Peter A. K�unziDepartment of Mathematics and Applied MathematicsUniversity of Cape TownRondebosch 7701South Africa
Current address:Institut de math�ematiquesUniversit�e de FribourgChemin du Mus�ee 23CH-1700 FribourgSuisseandDepartment of MathematicsUniversity of BerneSidlerstr. 5CH-3012 BerneSwitzerlandE-mail address: hans-peter.kuenzi@math-stat.unibe.ch