PERVIN NEARNESS SPACES

Transcription

1 Voume 9, 1984 Pages 7 30 htt://tooogy.auburn.edu/t/ PERVIN NEARNESS SPACES by John W. Carson Tooogy Proceedings Web: htt://tooogy.auburn.edu/t/ Mai: Tooogy Proceedings Deartment of Mathematics & Statistics Auburn University, Aabama 36849, USA E-mai: tooog@auburn.edu ISSN: COPYRIGHT c by Tooogy Proceedings. A rights reserved.

2 TOPOLOGY PROCEEDINGS Voume PERVIN NEARNESS SPACES John W. Carson Introduction The coection of a finite oen covers of a tooogica sace generates the Pervin quasi-uniform structure for that sace. A covers refined by some finite oen cover forms a nearness structure for a symmetric tooogica sace. This nearness structure wi be caed the Pervin nearness structure. The Pervin nearness structure ays an interesting roe in the famiy of a comatibe nearness structures on a sace. It is the smaest tot:ay bounded structure, the smaest contigua structure, and the argest utrafiter generated structure. Herrich, in [10], has shown that the cometion of the contigua refection of a T tooogica nearness sace is the Waman comactification. But the contigua refection of a tooogica nearness structure is the Pervin nearness structure. Hence, for T saces, the Pervin nearness structure is induced by the Waman comactification. 1 A rime extension is one for which each trace fiter is a rime oen fiter. Since t~he Pervin nearness structure is utrafiter generated it foows that the Waman comactification or the Stone-Cech comactification, if the sace is norma, are rime extensions. For a T sace it is shown that the trace fiters of the Waman comactification are the minima rime oen

3 8 Carson fiters. Moreover, we can construct the Waman comactification for a T sace, or the Stone-Cech comactification for a norma sace, using the strict extension tooogy on the famiy of a minima rime oen fiters. For T saces, concrete nearness structures ~ on X are induced by a strict extension Y. There is a one-to-one corresondence between the oints of Y and the ~-custers. Baanced near coections are introduced and it is shown that there is a one-to-one corresondence between the nonemty cosed subsets of Y and the baanced near coections on X. This corresondence aied to the Pervin nearness structure yieds a corresondence between the nonemty cosed subsets of the Waman comactification and the baanced cosed fiters on X. Using this resut, certain cosed sets in SN are characterized in terms of certain fiters on N. 1. Preiminaries We wi assume that the reader is basicay famiiar with the concet of a nearness sace as defined by Herrich in [9] and [10]. Definition 1.1. Let X be a set and ~ a coection of covers of X, caed uniform covers. Then (X,~) is a nearness sace rovided: (N) A E: ~ and A refines B imies B E: ~. (N2) {X} E: ~ and ~ ~ ~. (N3) If A E: ~ and B E: ~ then A A B {A n B: A E: A and B E: B} E: ~.

4 TOPOLOGY PROCEEDINGS Voume (N4) A E ~ imies {int (A): A E A} E 11. (int (A) {x: {X-{x},A} E ~}). For a given nearness sace (X,~) the coection of sets that are "near" is given by ~ = {A c 'P(X): {X-A: A E A} 11}. The micromeric coections are given by A E Y if and ony if {B c X: A n B ~ $ for each A E A} E ~. The cosure oerator generated by a nearness sace is given by c~a = {x: {{x},a} E ~}. If we are rimariy using these "near" coections we wi denote the nearness sace by (X,~). The underying tooogy of a nearness sace is aways symmetric; that is, x E TYT imies y E TXT. Definition 1.2. Let (X,~) be a nearness sace. The nearness sace is caed: (1) tooogica rovided A E ~ imies na ~ $. (2) comete rovided each ~;-custer is fixed; that is, na ~ ~ for each maxima eement A in ~. (3) concrete rovided each near coection is contained in some ~-custer. (4) contigua rovided A ~; imies there exists a finite B c A such that B ~. (S) totay bounded rovided A ~ ~ imies there exists a finite B c A such that nb = ~. 2. Cosed Fiters The foowing notation wi be used in this aer. Definition 2.1. Let] be a cosed fiter in a tooogica sace (X,t). (1 ) ~ (]) = {A ex: A E ]}.

5 10 Carson (2) O(J) = {O E t: there exists F E J with F co}. (3) sec(j) = {A c X: A is cosed and A n F ~ ~ for each F E J}. 2 (4) sec (J) = sec(sec(j». (5) If A has the finite intersection roerty then J(A) wi denote the cosed fiter generated by A. (6) J is caed baanced rovided J = n{m: mis a cosed utrafiter containing J}. (7) J is caed rigid rovided J n{m: mis a fixed cosed utrafiter containing J}. (8) J is caed nontrivia rovided J ~ {X}. (9) J is caed a sarce cosed fiter if for each nontrivia cosed fiter H there exists a cosed utrafiter Ywith H Y and J y. (10) A coection of cosed fiters {J : a E I} is caed a uniformy sarce if for each nontrivia cosed fiter H there exists a cosed utrafiter y such that H Yand J. Y for each a I. a Lemma 2.1. Let J and J be cosed fiters on a too 2 ogica sace. Then J c J if and ony if y(j ) c y(j ). 2 2 Lemma 2.2. Let J be a cosed fiter on a tooogica sace. (1) J = n{m: mis a cosed utrafiter containing J} if and ony if y(j) = n{y(m): mis a cosed utrafiter containing J}. (2) sec (J) u{m: mis a cosed utrafiter containing J}.

6 TOPOLOGY PROCEEDINGS Voume (3) sec 2 (]) n{m: mis a cosed utrafiter containing ]}. (4) sec 2 (]) is a cosed fiter containing ]. (5) ] is baanced if and ony if ] = sec 2 (]). (6) ] is rigid if and ony if there exists a nonemty cosed set A such that] = ]({A}). Lemma 2.3. Let (X,t) be a tooogica sace. (1) If X is T and x E X then Ox' the oen neighborhood fiter of x, is a minima rime oen fiter. (2) If] is a cosed utraf1:ter then 0 U) {o E t: x-o ~ ]}. (3) 0 is a minima rime oen fiter if and ony if there exists a cosed utrafiter] such that 0 = 0 U). Proof. To show (3), et 0 be a minima rime oen fiter and set] = {F: F is cosed in X and X-F ~ OJ. Then ] is a cosed utrafiter and 0 = O(]). The remainder of the roof foows in a natura way. Lemma 2.4. Let (X,t) be a T tooogica sace and ] a cosed fiter on X. The fozowing statements are equivazent. (1) ] is a sarce cosed fizter. (2) For each nonemty oen set 0 there exists FE] such that 0 F. 3. Pervin Nearne88 Structure For a tooogica sace (X,t) the coection {S(O): o E t} where S(O) = (0 x 0) U «X-O) x X), is a subbase

7 12 Carson for a comatibe quasi-uniform structure on X caed the Pervin quasi-uniform structure. The Pervin quasi-uniform structure can aso be generated as a covering quasi-uniform structure using the famiy of a finite oen covers [7]. It is shown in [5] that the coection of a covers refined by a finite oen cover is a comatibe nearness structure. It is natura to ca this the Pervin nearness structure. Definition 3.1. Let (X,t) be a symmetric tooogica sace. Let ~ be the famiy of a covers of X that are refined by a finite oen cover. ~ is caed the Pervin nearness structure on X. For the uroses of this aer it is more convenient to work with ~, the corresonding famiy of near coec tions. Theorem 3.1. Let (X,t) be a symmetric toozogicaz sace. Then: ~ = {A c ~(X): A has the finite intersection roerty}. From [3], we have the foowing resut. Theorem 3.2. Let (X,t) be a symmetric toozogicaz sace. Then: comzete. (1) X is comact if and onzy if t; is uztrafizter (2) X is H-cZosed if and onzy if X is Hausdorff and t; is oen uztrafizter comzete.

8 (4) ~ = n{~: ~ is a comatibe totay bounded near ness structure on X}. TOPOLOGY PROCEEDINGS Voume 9 ' (3) X is countaby comact if and ony if J has the countabe intersection roerty for each fiter J in ~. (4) X is Lindeof if and onz.y if every fiter in ~ with the countabe intersection roerty custers. Every contigua nearness structure is totay bounded; and it is easy to see that ~ is contigua. In fact, the smaest comatibe totay bounded nearness structure on a symmetric tooogica sace is ~ and thus equas the smaest comatibe contigua nearness structure on the sace. Theorem 3.3. Let (X,t) be a symmetric tooogica sace. Then: (1) ~ is contigua. (2) ~ is the smaest comatibe contigua nearness structure on X. (3) ~ = n{~: ~ is a comatibe contigua nearness structure on X}.,(5) ~ is the smaest comatib~e totay bounded nearness structure on X. (6) The Pervin nearness structure is contained in each comatibe totay bounded nearness structure on X. Definition 3.2. Let (X,t) be a symmetric tooogica sace and 5 a coection of free utrafiters on X. Set ~ (5) = {A c if (X): nif :I < or there: exists J E 5 such that ~ c ]}. A nearness structure ~ is caed utrafiter

9 14 Carson generated rovided there exists S such that ~ ~ (S). It is shown, in [8], that ~(S) is a comatibe nearness structure on (X,t). Theorem 3.4. Let (X,t) be a symmetric tooogica sace. Then the Pervin nearness structure is the argest comatibe utrafiter generated nearness structure on (X,t). Thus, for a given symmetric tooogica sace, the Pervin nearness structure is the argest utrafiter generated nearness structure and the smaest totay "bounded nearness structure comatibe with (X,t). Symboicay: ~ = Totay Bounded n Utrafiter Generated Coroary 3.5. Let (X,t) be a symmetric tooogica sace. Let ~ be any comatibe totay bounded nearness structure and ~2 any comatibe utrafiter generated nearness structure. Then ~2 c ~. That is~ each comatibe totay bounded nearness struature aontains every aomatibe utrafiter generated nearness structure. Since every contigua nearness structure is concrete we have the foowing coroary. Coroary 3.6. Every Pervin nearness structure is concrete. Theorem 3.7. Let (X,t) be a symmetric tooogica sace. Then:

10 TOPOLOGY PROCEEDINGS Voume (1) ~ {A c P(X): A c (J) for some cosed utrafiter ]}. (2) A is a ~ -custer if and ony if there exists a cosed utrafiter J such that A' = ~(]). (3) y = {A c P(X): there exists a minima rime oen fiter 0 that corefines A}. (Note: 0 is said to corefine A if for each 0 E 0 there exists an A E A such that A co.) 4. Extensions An extension Y of a sace X is a sace in which X is densey embedded. Uness otherwise noted, we wi assume for notationa convenience that X c Y. It is we known that for any extension Y of X there exists an equivaent extension Y' with X c Y'. If Y is an extension of X then ~ = {A c P(X): nciya., <t>} is caed the nearness structure on X induced by Y. Let (Y,t) be a tooogica sace and X = Y. For each Y E Y, set 0 y = {O n X: yeo E t}. Then {O y : y E Y} is caed the fiter trace of Y on X. Y is caed a rime extension of X if 0 y is a rime oen fiter for each y E Y. Note: Ox is aways a rime oen "fiter for x E X. Let t(strict) be the tooogy on Y generated by the base {O*: 0 E t(x)}, where 0* = {y E Y: 0 EO}. Let y t(sime) be the tooogy on Y generated by the base {O U {y}: 0 E 0 y,y E Y}. Then t(strict) and t(sime) are such that Y with either of these~ tooogies is an extension of (X,t(x), caed a strict extension, or sime extension

11 16 Carson of X, resectivey. Note that t(strict) < t < t(sime). Moreover, a tooogy s on Y with the same fiter trace as t forms an extension of (X,t(X)) if and ony if it satisfies the above inequaity. (See Banaschewski[].) Herrich's cometion of a nearness sace was resented in [9]. A brief descrition of it aears in [2] which we rovide here for the convenience of the reader. Let (X,~) be a nearness sace and et Y be the set of a ~-custers Awith emty adherence. Set X* = X U Y. For each A c X, define c(a) {y E Y: A E y} U c~a. A nearness structure ~* is defined on X* as foows: B E ~* rovided A {A c X: there exists B E B with B c c(a)} E ~. (X*,~*) is a comete nearness sace with c~*x = X*. Aso, for A c X, c~*a = c (A). The foowing imortant theorem is due to Herrich and Benty [2]. Theorem A. For any T nearness sace (X,~) the foow ing aonditions are equivaent. extension. (1) ~ is a nearness structure induced on X by a strict (2) The cometion (x*,~*) of (X,~) is tooogica. (3) (X,~) is concrete. It is shown in [4] that a concrete nearness structure is utrafiter generated if and ony if it is induced by a rime strict extension. The foowing theorem aso aears in [4].

12 TOPOLOGY PROCEEDINGS Voume Theorem B. For any T nearness sace the foowing conditions are equivaent. (1) ~ is induced on X by a rime strict extension. (2) The cometion (x*,~*) of (X,~) is tooogica and X* is a rime extension of x. (3) ~ is concrete and utrafiter generated. Since the Pervin nearness structure is utrafiter generated and concrete it foows that ~ is induced by a rime strict extension, rovided the underying tooogy is T. We now show that for T saces the Pervin nearness structure is induced by the Waman comactification. Let (X,~) be a nearness sace and set ~c = {A c P(X) : each finite B c A beongs to ~}. The foowing theorem is due to Herrich [10]. Theorem C. If (X,~) is T and tooogica then (x*,~~) is the Waman comactificaiion. If ~ is tooogica then ~c = {A c P(X): A has f.i..} = ~. Thus the foowing theorm is an immediate consequence of Herrich's resut. Theorem D. Let (X,t) be a T tooogica sace. Let 1 ~ be the Pervin nearness structure. Then (X*,~*) is the Waman comactification of x. Since the Waman comactification of a norma sace v is the Stone-Cech comactification we have the foowing coroary.

13 18 Carson Coroary 4.1. Let (X,t) be a norma tooogica sace and ~ the Pervin nearness structure. Then (X*,~*) P is SX, the Stone-Cech comactification of x. Coroary 4.2. Let (X,t) be a T tooogica sace. Then the Waman comactification is a rime extension of X. If X is norma then SX is a rime extension of x. Coroary 4.3. Let (X,t) be a T tooogica sace. Then the Pervin nearness structure is the nearness structure on X induced by the Waman comactification of x. Simiary, if X is norma then the Pervin nearness structure is the nearness structure induced by the Stone-Cech comactification of x. By coroary 4.3 and the resuts obtained for searated and reguar nearness saces obtained in [2], we have the foowing theorem. Theorem 4.4. Let (X,t) be a T tooogica sace. The following statements are equivalent. (1) X is norma. (2) ~ is sear~ted. (3) ~ is reguar. Theorem 4.5. Let (X,~) be a T nearness sace and (x*,~*) its cometion. Then the trace fiters on X are given by: {o E t(~): x E o} for x E X, and {o E t(~): x-o ~ A} for A E X*-X.

14 TOPOLOGY PROCEEDINGS Voume Proof (1) foows immediatey since X is a dense subsace of X*. (2). Let A be a ;-custer and set 5 = {O E t(;): X-O ~ A}. Let 0 E 5. Then X-O f A. Now c(x-o) is cosed in X* and A f c(x-o). Thus Q* X* c(x-o) is oen in X* and 0 = Q* n X and A E Q*. Thus o E 0A and thus 5 c 0A. Let 0 E 0A. Then there exists Q*, oen in X*, with A E Q* such that 0 = X n Q*. Now c(x-o) is cosed in X* but A E Q* and Q* n (X-O) =~. Hence A E c(x-o) and thus X-O E A. Therefore, 0 E 5 and 0A c 5. Theorem 4.6. Let (X,t) be a T tooogica sace and ; the Pervin nearness structure. Then the trace fiters generated by the cometion (X*,;*) are recisey the mini ma rime oen fiters on X. Proof. By theorem 4.5, the trace fiters for the cometion are of the form: (1) 0 {O E t: x EO}, for x E X; or x (2) 0A {O E t: x-o f A} for A a ; -custer. By emma 2.3, each 0 x If A is a ~-custer is a minima rime oen fiter. then by theorem 3.7 there exists a cosed utrafiter] such that 14 = ~ (J). Thus 0A = {O E t: X-O ~ ~(])} = {O E t: X-O f ]}. Thus, by emma 2.3, 0A is a minima rime oen fiter. Thus each Ox and 0A is a minima rime oen fiter; a that remains to be shown is that if is a minima = 0A for some ;-custer A. rime oen fiter then either 0 o for some x E X or x

15 20 Carson Let 0 be a minima rime oen fiter. Then, by emma 2.3, there exists a cosed utrafiter J such that 0 = O{J). If {x} E J then 0 c 0 and, since 0 is a minima rime x oen fiter, it foows that 0 = Ox. If nj = 4> then ~(J) E X*-X. Then O~{J) {O E t: x-o f ~(J)} = {O E t: x-o f J} = O{J) = O. Coroary 4.7. The famiy of minima rime oen fiters is the fiter trace of the Waman comactification for a T sace; and corresondingy, the fiter trace of the Stone-Cech comactification for a norma sace. Let (X,t) be a tooogica sace and Y' a coection of oen fiters on X containing the oen neighborhood fiters. For 0 E t, et 0* = {O E Y': 0 EO}. Let t' be the tooogy on Y' generated by the base {O*: 0 E t}. Then, by Banaschewski [1], (Y',t') is a strict extension of X where the embedding ma e: X ~ Y' is given by e{x) = Ox' the oen neighborhood fiter of x. Many we known extensions can be constructed in this manner. The foowing exames, among others, are given in [1] (I) If (X,t) is cometey reguar and Y' is the famiy of a maxima cometey reguar oen fiters then (Y',t') v is the Stone-Cech comactification. (2) If (X,t) is Hausdorff and Y' is the famiy of a oen utrafiters then (Y',t') is the Fomin H-cosed extension of X. (3) If (X,t) is a ocay comact, non-comact Hausdorff sace and Y' is the coection of a oen neighborhood

16 TOPOLOGY PROCEEDINGS Voume fiters together with the fiter of oen sets with comact comements then (Y',t') is the I-oint Aexandroff comactification of X. By theorems D and 4.6, we have that the Waman comactification and the Stone-Cech comactification for norma saces can be constructed in this manner, using the famiy of a minima rime oen fiters. Theorem 4.8. Let (X,t) be a T tooogica sace and et Y' be the famiy of a minima rime oen fiters on X. Then: (1) (Y',t') is the Waman oomactification of x. (2) If (X,t) is norma then (Y',t') is the stone-cech comactification of x. In [6], nearness saces whose cometions were second category or Baire tooogica saces were characterized. In order to reate these resuts to the Pervin nearness structure we state a definition and two theorems from.[6]. Definition 4.1. Let (X,~) be a nearness sace. Let 5 c (X). Then 5 is caed a sarce near coection if 5 E ~ and for each B E ~, such that B is not contained in each ~-custer, then there exists a ~-custer A such that B. A and 5. A. Let {S : a E I} c~. a Then {S : a E I} is caed a uniformy sarce famiy if for each B E ~, a such that B is not contained in each ~-custer, then there exists a ~-custer A such that B. A and 5. A for each a E I. a

17 22 Carson Easiy each member of a uniformy sarce famiy is itsef a sarce near coection. Theorem 4.9 (Carson [6]). Let (X,~) be a T nearness sace. The foowing statements are equivaent. category T (1) ~ is a nearness structure induced on X by a second strict extension. (2) The cometion (x*,~*) of (X,~) is tooogica and second category. (3) (X,~) is concrete and for each countabe coection {5.: i 1 E N} of sarce near coections there exists a ~-custer A such that 5. U A ~ for each i E N. 1 sace. Theorem 4.10 (Carson [6]). Let (X,~) be a T nearness 1 The foowing statements are equivaent. (1) ~ is a nearness structure induced on X by a strict T Baire extension. (2) The cometion (x*,~*) of (X,~) is tooogica and a Baire sace. (3) (X,E:) is concrete and each countabe famiy of sarce near coections is uniformy sarce. Theorem Let (X,t) be a T tooogica sace and ~ the Pervin nearness structure on X. Let A E ~. (1) A is not contained in each -custer if and ony ~ if J = J (A) is not the trivia cosed fiter. J (2) A is a sarce near coection if and ony if J(A) is a sarce cosed fiter. Proof (1) foows from the fact that A is contained in each ~-custer if and ony if each A E A is dense in X.

18 TOPOLOGY PROCEEDINGS Voume (2) Suose A is a sarce near coection. Then J = J(A) is a cosed fiter. Let H be a nontrivia cosed fiter. Then H E ~ and H is not contained in each ~ -custer. Since A is a sarce near coection there exists a ~ -custer B such that A Band H B. Now there exists a cosed utrafiter such that B = {B c X: B E }. Easiy J and H. Therefore, J is a sarce cosed fiter. Now suose J = J(A) is a sarce cosed fiter. Let H E ~ and H not contained in each ~ -custer. Then m= J(H) is a nontrivia cosed fiter. Since J is a sarce cosed fiter there exists a cosed utrafiter such that J and m. Let B = {B c X: B E }. Then B is a ~ -custer and m B and A B. Hence A is a sarce near coection. Theorem Let (X,t) be a T tooogica sace. The foowing statements are equivaent. category. (1) The Waman comactification of X is second (2) For each countabe coection {J i : i E N} of sarce cosed fiters there exists a cosed utrafiter ~ such that J. ~ for each i E N. 1 on X. Proof. Let ~ denote the Pervin nearness structure Then (X*,s*) is the Waman comactification of X and the resut hods by theorems 4.9 and Theorem Let (X,t) be a T tooogica sace. The foowing statements are equj~vaent.

19 24 Carson sace. (1) The Waman comactification of X is a Baire (2) Each countabe famiy of sarce cosed fiters is uniformy sarce. Proof. Let ~ denote the Pervin nearness structure on X. Then (x*,~*) is the Waman comactification of X and the resut hods by theorems 4.10 and Coroary Let (X,t) be a norma tooogica sace. Then each countabe famiy of sarce cosed fiters is uniformy sarce. Proof. This foows immediatey from the fact that each comact Hausdorff sace is a Baire sace. 5. Baanced Near Coections Definition 5.1. Let (X,~) be a nearness sace. Let A c ~(X) and et c(a) denote the set of a ~-custers that contain A. Let (A) denote the set of a the fixed ~-custers that contain A. Set b(a) nc(a). A is said to be a baanced near coection rovided A E s and A nc(a) and A is caed a rigid near coection rovided A E ~ A = n (A) and and A E ~. Theorem 5.1. Let (X,~) be a concrete nearness sace (1) A c nc(a). (2) b(a) is the smaest baanced near coection containing A. (3) Each ~-custer is a baanced near coection.

20 TOPOLOGY PROCEEDINGS Voume (4) If n is a nonemty coection of ~-custes then nn is a baanced nea coection. (5) If A is a igid nea coection then A is a baanced nea coection. Let Y be a strict T extension of X and ~ the nearness structure on X induced by Y. For y E Y, et A y = {A c X: Y E cya}. Then {y} ncya and each ~-custer is of the y form ~ for some y E Y. For <P ~ F c y, et S(F) = {A c X: F c cya}. We sha see that the baanced near coections are recisey of this form. Since the extension is strict it foows that Theoem 5.2. Let Y be a T stict extension of x. Set ~ = {A c 7' (X): nc ya ~ <}. (1) If ~ # FeY then c(s(f» 5(F)is a baanced nea coection. (2) If A is a baanced nea coection then thee exists a cosed set F in y such that A = 5(F). (3) If <P ~ FeY then 5(F) = 5(c y F). (4) If F and E ae nonemty subsets of y then 5(F U E) = 5(F) n 5(E). (5) Thee exists a one-to-one coesondence between the nonemty cosed subsets of Y and the baanced nea coections. Theoem 5.3. Let Y be a stict T extension of X and ~ the neaness stuctue on X induc~d by Y. Let A c 7'(X).

21 26 Carson (1) If T {x: Ax E (A)} then T = ncxa. (2) A is a rigid near coection if and ony if there exists a nonemty cosed set T in X such that A = {A c X: T c cxa}. (3) A is a rigid near coection if and ony if (A) A is a baanced near coection, and (B) cy(ncxa) = ncya. (4) Let F be a nonemty cosed subset of Y. The foowing statements are equivaent. (A) F = cy(f n X) (B) There exists a unique cosed subset of X, say G, such that F = cyg. (C) 5(F) is a rigid near coection. We now characterize the baanced near coections in a Pervin nearness sace. Theorem 5.4. Let (X,t) be a T tooogica sace and ~ the Pervin nearness structure on x. iet A c (X). P (1) A is a baanced near coection in ~ if and ony if there exists a baanced cosed fiter J such that A = (J). (2) There exists a one-to-one corresondence between the nonemty cosed sets in (x*,~*) and the baanced cosed fiters in x. Moreover; each nonemty cosed set F* c x* is of the form F* = ncx*j = { (m): mis a cosed utrafiter containing J}, where J is a baanced cosed fiter on x. (3) A is a rigid near coection in ~ if and ony if there exists a rigid cosed fiter J such that A = G).

22 TOPOLOGY PROCEEDINGS Voume Coroary 5.5. Let (X,t) be a T (Norma) tooogica sace and wx its Waman comactification (ax its Stone Cech comactification). Then there exists a one-to-one corresondence between the nonemty cosed sets in wx (ax) and the baanced cosed fiters in X. Secificay, if F is a nonemty cosed set in wx (ax) then there exists a unique baanced cosed fiter J such that F = ncwxj (F = ncaxj). 6.,f3 N Let N denote the natura numbers with the discrete tooogy and et ~ denote the Pervin nearness structure on N. Since N is norma, (N*,~~) ~ an. Now the oints of an are the ~-custers; that is, they are of the form <m), where mis an utrafiter. Since the tooogy is discrete it foows that (m) = m. (Throughout this section, m wi denote an utrafiter on N.) Thus, the oints of an are simy the utrafiters on N. Since N* is a strict extension of N it foows that the sets of the form cn*e, for E c sets of an. N, form a base for the cosed Now, cn*e = {m: E Em}, which we wi ca V(E). This agrees with a standard construction of ani see for exame, M. E. Rudin [11]. It is aso known that the oencosed subsets of an are of the form VeE) for some E c N. Moreover; {V(E): E c N} is a base for the oen sets and a base for the cosed sets in an. Lemma 6.1. Every fiter on N is a baanced cosed fiter.

23 28 Carson By coroary 5.6, there exists a one-to-one corresondence between the nonemty cosed subsets of (N*,~;) and the baanced cosed fiters on(n,~). In ight of emma 6.1, we have that there exists a one-to-one corresondence between the nonemty cosed subsets of an and the fiters on N. The foowing theorem rovides a characterization of the nonemty cosed subsets of an. Theorem 6.2. Let ~ #F can. F is azosed if und onzy if there exists a fizter J on N suah that F = {m E an: J em}. Proof. If F is cosed and nonemty in an = N* then F = nc *5(F). N By theorem 5.2, 5(F) is a baanced near co ection and by theorem 5.4 and emma 6.1 there exists a fiter J on N such that 5(F) (J) = J. Now, for F c N, / cn*f = {m: F E m}. Hence F nc *5(F) = nc J = N N* {m: Jc m}. On the other hand, if J is a fiter on N, then F ncn*j is a cosed set in N* = SN. Caim. F = {m: J em}. Suose me F = ncn*j. Then A E mfor each A E J. That is, J c m. Conversey, if J c mthen me cn*a for each A E J. Thus me ncn*j F. o ~ o Coroary 6.3. If 0 is an oen subset of an and an, then there -exists a fiter J on N suah that {m: J m}. Coroary 6.4. If A is a nontrivia oen-aosed subset of an then there exists fizters J and H on an suah that A = {m: J em} = {m: H A}.

24 TOPOLOGY PROCEEDINGS Voume Thus, if A is a nontrivia oen-cosed subset of SN, there must exist a air of fiters J and Hon N such that each utrafiter on N contains one and ony one of these fiters. It is easy to show that the ony ossibiity is for J {E c N: SeE} and H = {E c N: N-S c E} for some ~ ~ S ~ N. Thus, the nontrivia oen-cosed subsets of SN are simy the V(S) = {m: S E m} as stated in [11]. Theorem 6.5. There exists a natura one-to-one corresondence between the foowing casses of fiters on N and the resective secia nonemty cosed subsets of an. N SN (1) utrafiters oints (2) fiters cosed sets (3) J(A) fiters oen-cosed sets (4) fiters with countabe. base cosed Go sets (5) fiters with countabe base zero sets Proof. The corresondence is as discussed; namey, F = {m: J c m} where F is a cosed set in SN and J is a fiter on N. The corresondences for (1), (2) and (3) have aready been noted. Proof of (4) and (5). In a norma sace, a subset F is a cosed Go set if and ony if it is a zero set. Let F be a nonemty zero set in SN. From (6E) in [8], we have that every ~ero set in ax is a countabe intersection of sets of the form csxz where Z is a zero set in X. Since every subset of N is a zero set in N it foows that F = n{csnf : F c Nand i EN}. Since F is nonemty, i i it foows that there exists an utrafiter containing

25 30 Carson {F i : i EN}. Hence the F have the f.i.. and we et J i denote the fiter generated by these sets. Then J has a countabe base and moreover F = {PJ: J c PJ}. Now et J be a fiter with a countabe base, say {G i : i EN}, and F = {PJ: J c PJ}. Then F = n{c1 G i : i E N} SN and hence F is a zero set in SN. References 1. B. Banaschewski, Extensions of toozogiaaz saaes, Can. Math. Bu. 7 (1964), H. L. Bentey and H. Herr1ich, Extensions of toozogiaaz saaes, Tooogica Proc., Memhis State Univ. Conference, Marce1~Dekker, New York (1976), J. W. Carson, Too ZogiaaZ roerties in nearness saaes, Gen. To. A1. 8 (1978), , Prime extensions and nearness struatures, Cornmentationes Math. Univ. Caro1inae 21 (1980), , Nearness and quasi-uniform struatures, To. A1. 15 (1983), , Baire saae extensions, To. A1. 12 (1981), P. Fetcher and W. F. Lindgren, Quasi-uniformities with a transitive base, Pacific J. Math. 43 (1972), L. Giman and M. Jerison, Rings of aontinuous funations, Van Nostrand Reinhod Press, Princeton, New Jersey, H. Herr1ich, A aonaet of nearness, Gen. To. A1. 5 (1974), , TooZogiaaZ struatures, Mathematica Centre Tracts 52, Amsterdam, M. E. Rudin, Leatures on sit theoretia toozogy, CBMS Regiona conference series-no. 23, Amer. Math. Soc., Providence, Emoria State University Emoria, Kansas 66801