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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number 2, September 1975 T-REGULAR-CLOSED CONVERGENCE SPACES D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK ABSTRACT. It is known compactification space. is almost It is shown T-regular-closed that that a convergence identical a less extensions restricitive with space to a completely the class universal which has tegular a regular topological of convergence property of the spaces have Stone-Cech compactification. 1. Introduction. space has tinuous a Hausdorff functions a convergence space space. In this versal property sure paper spaces. by relinquishing Instead, space the filters iff the space having the requirement highly that topological which has a uni- of con- nonidempotent the that is regular regular compactification" spaces than, for con- of [6] and u] tot a class we require but more general convergence property as a completely a "regular includes Hausdorff In [7] two of us showed compactification convergence we obtain (which A convergence between a regular that each with an extension regular like the compactifications be compact. resembling, has ultrafilter spaces operators) space compactification into compact and has the same vergence In [6] one of us showed clo- "compactification" that it be T-regular-closed, a concept compactness. (X, —>) consists of a set on X and the elements X and a relation of X, subject "_," to the following conditions: (1) x'-. x, all x eX; ra<f . .. (2) J —> x and , cr J < (ú g implies , , cr . (3) J* —»x implies For x e X, x' empty subset expression x". denotes g —» x; S n x —»x. the fixed of X, then "A —» x" (o cr ultrafilter A* will denote should be read generated the filter "the filter by ixi; if A is a non- of all oversets .f converges We will use the abbreviation "u.f." Throughout will mean convergence Presented the paper, to the Society, space for "ultrafilter". June 21, 1975; received AMSiMOS)subject classifications 54D35. Key words and phrases. Convergence regular-closed spaces, T-regular-closed of A. The to the point space. by the editots We will May 25, 1974. (1970). Primary 54A05, 54A20, 54D10, 54D25, spaces, extensions. T-regular convergence spaces, T- Copyright © 1975, American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 461 462 D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK usually refer to a space dorff if each filter to be Hausdorff Fischer unless "cl,," definition "X" rather is the closure operator standard. cL v J —* x; XX, the topological of the set alent X equipped structure space regular, if X is a topological tion of important is productive space set of any T-regular space in which which than the iff A = civ- A. A T- are equiv- in the next regularity, sec- T-regularity spaces, it can be embedded. [4, Example spaces that space T-regua T- a continuous is closed. in T-regular-closed regular sub- of T-regular-closed things, of spaces compact A compact 3.10] describes A study other classes locally if X is a closed spaces. are shown spaces, func§4 is to be T-reg- and lattices with convergence. X be a space, X with the coarsest relative f(x), to which C (X) the set convergence the natural is continuous. c-embedded (R denotes if the evaluation of continuous structure map (called real-valued continuous line with its usual on convergence) cú: C (X) x X —* R, defined the real functions by &)(/, x) = topology.) X is map /: X —» C C (X), defined by /(x)(/) = fix), all / £ C(X), is an embedding. spaces Feldman [2] has proved that all c-embedded are T-regular. A convergence contains Proof. space a compact Proposition hence coarser are given onto a T-regular T-regular The following c-embedded Let filter on X. it J—► of regularity T-regular-closed among space with embedding 2. Examples. order J of X, is the space topology Like is not compact. in §3; we show, tion from a T-regular-closed ular: spaces. T-regular-closed; space is given concerned modification Some examples will be called is obviously regular-closed spaces J" is a filter A is ÀX-closed of T-regular J —» x whenever and hereditary. A T-regular lar space is Haus- are assumed X to be T-regular and the two versions space. classes for X and with the finest of X. A subset is clearly A space All spaces if cl^ We define consisting convergence —>)". X to be regular x implies regular "(X, one point. indicated. a space has become than to at most otherwise [3] defined —• x, where This as converges 2.1. Let closed. A locally J —» x. Then As a subspace lows from [7, Lemma clx J = cL^ X is said J. Since to be locally compact if each convergent set. l] that compact J regular contains a set space A which of X, A is a compact cl^ and cl^x X is regular, cl,y X is T-regular. coincide J —* x. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use regular is compact space, for subsets and and it fol- of A; thus 463 T-REGULAR-CLOSED CONVERGENCE SPACES Pervin gence and Biesterfeldt is regular. T-regular. [5] have We will give We use the filter ix 6 X, there exists dually. 3 a lattice with order showing that such of order convergence conver- a space is given by ordered set, J" a filter on X. Let L(J") = F e 3 such that x < y fot all y e FÎ, and let 1/(3") be order converges to x if x = inf Uij) well known that order convergence plete that proof characterizations Ward [9]. Let X be a partially defined shown a shorter = sup need not be topological, L(j ). It is even in a com- lattice. Proposition 2.2. Proof. J Let Order convergence order converge in any lattice to x. Let X is T-regular. § be the filter generated by all sets of the form la, b] = {y: a < y < b\, fot a e L(5") and b £ i/(3). Since L(§) = L(j) converges to x. Also, (see and t/(§) sets = (7(3), of the form [9J), which is known > y, and so cl^ investigated on this topic closed spaces) topological see [lj. space spaces Regular-closed § order topology convergence. on X Thus 3.1. in various Equivalently, that to that space, ways spaces for a summary spaces the concept cl,y J (but not T-regular- regular-closed obtained for topological identical to that have of results of a regular-closed of a topological and the results from those 3.1 is almost for conver- spaces. of Theorem 2.10 of be omitted. A T-regular J topological convergence in [4]. Note of Theorem ter 3 072 X, cl.v filter that in the interval order Regular-closed convergence differ Theorem than of mathematicians; is not equivalent [4] and will therefore closed spaces. are studied The proof la, b] ate closed to be coarser by a number (or T-regular-closed) gence immediately J —> x. 3. T-regular-closed been it follows space X is T-regular-closed has an adherent a T-regular iff, for each fil- point. space X is T-regular-closed iff each maximal space is T-reg- on X converges. Corollary 3.2. A closed subspace of a T-regular-closed ular-closed. Theorem space 3.3. X onto a T-regular Proof. Let 3 on ¡A such /_1(3) Let f be a continuous VA', space A be a closed that where 3 —< y in Y. Then subset Y. Let A* is the filter function from a T-regular-closed f is a closed of X, y e cly, /A. map. Then there § be an u.f. on X which of all oversets of A. Since License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use is an u.f. is finer X is than 464 D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK T-regular-closed, is closed, there A e cl.v- is an adherent a, We next there set 3.4. are of the real Let and only infinite structure such subset that The spaces However, infinite 3A and no free u.f. can converge, Theorem 3.5. set, and §^ which // X: the first and X x £ Xy maps —»x in Xj. Lemma Proof. such map P. there is a filter Hence, K = (P^ the second ent point Then PjH sub- A such that B of X. Let Xj be A ¿ —> x in Xj subset A. by Theorem in the product then that 3.1. space is a closed g E2^l // X. finer clxx Conversely, J has an adherent map, than P.cl^y. P.cl^y, §) Vcl,y projection Xj is T- map J let 3 = cl^y J such is a filter P2 is closed, J point. P$* that tj —> x, on Y, and cl^y P2J\ = such that A —>y, fot some y £ X,. - ^> ana- so ^2^1 —' ^ *n *2' is compact X, x X. 3.3. cl^y K = Kj is a filter on Y. Also, P,H, > clxx Similarly, 3.6. infinite are closed. §, and so PjK Hence> K. "~► to cl xy 3. regular and X2 is T-regular-closed, then is T-regular-closed. It is easy to see that the second projection Let J be a filter on Y = Xj x X2« By hypothesis, y. Hence g converges contain that infinite by Theorem (x, y) in Y, and so (x, y) is adherent the product to each T-regular-closed show PAli. Let K be an u.f. finer than P Ji Then KVPJ in X. X with the finest convergence are T-regular-closed, is necessary projection cl^y. K = H. Since point of X (c the cardinality which the diagonal Y = Xj x X2; we must by hypothesis that and so Xj x X2 is not T-regular-closed. The condition on for some contains A of cl ^y, 3, The example subsets be the set X2 ate clearly point x a fixed subset structure X iff both projection Proof. be a filter convergence A. Let since is not productive. of X, we can assign C¡A —» x in X2 fot each X. regular-closed and infinite on each c subsets free u.f.'s, X with the finest for each Since spaces. T-regular-closed X be a countable 2C free u.f.'s line) However, is an adherent of T-regular-closed of being [j. y e/A. B lt B 4 A and 3^ 4 §B for all infinite the set x X. products A of X two distinct ^A ^ 3 —> y. Thus that the property Example Since cl.y, consider shows x of cl^. and so x e A. But /(x) and so fix) = y, since follows point there is a filter in X2 to y. Let —♦ x, fot some adherent point of cl.y g finer that clxx clxx H be an u.f. on V finer x e Xj, and map P, P'23 = E2clXy than License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ^> and P~ !(§) VclAy, P2K —i y in X2, so that 3- is closed. P23 has an adher- (x, y) is an 3. 465 T-REGULAR-CLOSED CONVERGENCE SPACES Theorem closed, 3.7. then // both X. and Proof. It is sufficient, jection maps in Y = Xj x X , and let which P. B, and so are known and T-regular- P, point is a closed map. topological to be false, The question and Lemma of whether regular-closed a product is an unsolved rem 3.5 is valid P. Then and contains to prove that the pro- is closed, there let is an u.f. a compact set B be J on A. By Lemma Also, B. = B Ci (Ax X A is a closed an adherent For regular-closed that P.S. 3) V B*. is such that to [j, g has P.cly, compact theorem, To show x 6 clx to x in X, dxx preceding are closed. is T-regular-closed. § = (P~ belongs P2 converges 3.6, A x X Hence, and by the closed P.ß X2 are locally X. x X2 is T-regular-closed. spaces, § = §, and since Hence x £ P.cly Corollary 3.2 and Theorem 3.6 and Theorem 3.7 are known topological 3.3 to be true. spaces We do not know whether topological A x X2 B. C of regular-closed problem. for regular-closed clxy (x, b). set. is or not Theo- spaces. A. Embedding theorems. Theorem closed 4.1. space Proof. Each T-regular space X can be embedded in a T-regular- X.. Assume that X is not T-regular-closed. Let y be a point X, and let X, = X U iyl; let Xj be the set X U iyi, equipped convergence structure satisfying x 4 y, iff J contains X and the restriction 3 —» y iff 3 > y" A §, where has no adherent point that X is a subspace adherent an adherent ent to cl^x 3. assumption that We shall in which erty. each Let cluding compactification X in X, then by Theorem to be the set ing the following cly, 3 —» x in X A —» x in X., X. for to X in X; X such that that cl^x is T-regular, has the problem of AX; let Y, equipped whenever space such space. guarantees of finding that § and Let with the finest that a class extension on X y is adher- Finally, X is dense the in Xj. spaces with universal map from convergence does of T-regular V = ßXX denote (1) If 3 is a filter 3j of X., AX is a completely (f> be the embedding <p to X. If cl^x is T-regular-closed. a T-regular-closed topological conditions: J» by the construction 3.1, X. X is not T-regular-closed X be a T-regular Hausdorff) containing to verify 3 has a restriction now consider member conditions: of 3 to X converges of Xj. Let 3 be a filter on X., If 3 =y , then y is point Thus, § is a filter in X. It is easy to y. Otherwise, not have the following not in with the finest prop- regular (in- the Stone-Cech AX into structure containing Y. Define satisfy- (fix, then 3 —» tp~ x in X; (2) If § is an u.f. on X License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 466 D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK which in contains X cbX and to the same Theorem then X 4.2. Z, then to which to AX-converge, it converges // X is a T-regular space and the ¡unction a unique First tion of X The first on X that X that X condition is continuous; is a filter continuous note it is clear injection. l,[j> fails is a dense f : X is T-regular. The function of X that (pX and regular space —> Z. Y; from this guarantees regular, embedding. map into a compact in the definition cly, J, where (f>X£J § converges Y. is finer than contains cl„ and XX is completely extension the second which in then c/j: X —> X if f: X —' Z is a continuous f has Proof. —> X point is T-regular Furthermore, (p~ § fact and the construc- (f> is certainly guarantees that (fiX is dense an (b: X in X . If Cj g —» x in X , then, by Condition and (f>~ J —' (f>~ x in X. Since X is T-regu- lar, clxx <p- l<3f— <p~lx in X. But 4>~*g > <p~!clAx. 3 = clxx (p—!3, and so (f>~ § —• (p~ x in X. Thus dense (f>~ is continuous, and <p: X —» X isa embedding. Finally, Proposition consider /: X -* Z. Then l], AZ is a compact extension f : Y --> AZ. Let /: AX —► AZ is continuous. Hausdorff topological § —» x in X . Assume space. first By [7, Thus that / has y > cly an J, where tp"1? — (f>~Xxin X. Then /*(3) = fich~ *3) -» fi<p~ lx) = f*x in Z. Also, cl„ uous, and so / (cly / 3 —' f x in Z, since Z. Next, assume 4>X such that that g > cly (p~ 3 fails It follows 3, where f : X —» Z is continuous. that on X / g —* f x in which J —» x in AZ have By regularity /§> AclY 3)> clz /*3, which implies that Then l], Z and and so / 3 —»/ x in Z. But / : Y —» AZ is continimmediately 3 is an u.f. to AX-converge. —►/ x in AZ. By [7, Theorem vergence, Z is regular. 3) > cl2 / 3. contains Y, and so f J the same ultrafilter of Z, clzf con- 3—> f x, and /*§ -*• /*x. Thus we have established The uniqueness of the extension is obvious, and so the proof is complete. Unfortunately, the conditions enough to insure that dition, we obtain the desired Theorem regular 4.3. and locally X Let compact. continuous has a continuous closed Proof. By adding T-regular-closed Then function extension In view of Theorems Since on X in Theorem is T-regular-closed. X be a T-regular of X, and each is T-regular-closed. imposed space 4.2 are not an additional such that XX is completely (X , t/j) is a T-regular-closed from con- extension. X into a compact regular extension space Z to X . 3.3 and 4.2, it remains AX is locally compact, only to show that AX is an open subspace License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use X T-REGULAR-CLOSED CONVERGENCE SPACES of Y = /SAX. Thus no u.f. on If g is an u.f. on X z in Y - (f>X. Thus g > cly suits Y - cßX can converge - tpX, then there //. From the definition lead us to conclude relative to u.f.'s X contain and Y to a point relative K on (f>X such of X , Condition that which g converges is an u.f. in 467 to that Y to some point K —» z in at the conclusion and so AX is compact. the same Y — r/jX. Furthermore, To show that maximal closed the maximal filters X closed (relative filters to Theorem then if g —> x in described the same it is sufficient X ) converges. relative to X // X has coincides u.f. convergence, to show that By the preceding are the same are fixed, a regular since every paragraph, as the maximal Y is compact. compactification with the regular From [7, Theorem and Stone-Cech l], X is T-regular In the proof of the preceding theorem, the same convergence for u.f.'s account the respective sired have facts, closed Thus X XX is locally compactification com- of X in ll]. Proof. 1 ar. AX Y and these and the proof is complete. 4.4. X to Y, and the latter is T-regular-closed, pact, Y and re- u.f. convergence AX = Y. is T-regular-closed, filter relative that Thus, Y, and 2, g —» z in X . These Y have 4>X £ g, then (ß~ § —* x in AX, and so g —* x in AX . Combining we arrive in (pX. and comparing and AX is completely it is shown containing regu- X and Y have Y - rpX. By taking this fact into constructions, that one is led to the de- conclusion. In [8, V3], a space pact, c-embedded, However AX and have is not locally T-regular-closed. X satisfying the closure operator number. blence which a completely regular compact, hand, a simple in contrast of a regular to any topological alteration iterations, in Theorem com- modification. that of this X. is not example 4.3 with the property where to the stringent compactification, as described to be locally topological of Theorem for X has 72-distinct extension is shown and it can be shown the conditions Therefore, [7] for the existence ular-closed is constructed On the other to a space natural X that 72 is an arbitrary conditions a space leads required can have 4.3 and yet bear little in a T-reg- resem- space. REFERENCES 1. M. P. Berri, J. R. Porter and R. M. Stephenson, Jr., A Survey of minimal topological spaces, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968), Academia, Prague, 1971, pp. 93-114. MR 43 #3985. 2. W. A. Feldman, Axioms of countability in C(X), Pacific 81-89. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use J. Math. 47 (1973), 468 D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK 3. H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269-303. 4. D. C. Kent and G. D. Richardson, Math. Soc» 160 (1971), 487-499. 5. W. J. Pervin Minimal convergence MR 22 #225. spaces, Trans. Amer. and poséis, Proc. MR 44 #3279. and H. J. Biesterfeldt, Jr., Regular topologies Amer. Math. Soc. 17 (1966), 903-905. MR 34 #8368. 6. G. D. Richardson, A Stone-Cech compactification for limit spaces, Proc. Amer. Math. Soc. 25 (1970), 403-404. MR 41 #992. 7. G. D. Richatdson and D. C. Kent, Regular compactifications of convergence spaces, Proc. Amer. Math. Soc. 31 (1972), 571-573. MR 44 #3290. 8. -, A note on C (X), Proc. Amer. Math. Soc. 49 (1975), 441-445. 9. A. J. Ward, On relations between certain intrinsic topologies in partially or- dered sets, Proc. Cambridge Philos. Soc. 51 (1955), 254-261. MR 17, 67. DEPARTMENT OF MATHEMATICS, WASHINGTON STATE UNIVERSITY, PULLMAN, WASHINGTON99163 DEPARTMENT OF MATHEMATICS, EAST CAROLINA UNIVERSITY, GREENVILLE, NORTH CAROLINA 27834 DEPARTMENT OF MATHEMATICS, ARKANSAS STATE UNIVERSITY, STATE UNIVERSITY, ARKANSAS72467 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use