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CHARACTERIZATIONS OF REGULAR-CLOSED SPACES Larry L. Herrington* . (received 11 March, 1975; revised 21 April, 1976) 1. Introduction Characterizations of regular-closed spaces are given in terms of open regular filterbases [l, p.104]. Open filterbases, of course, determine nets but not every net determines an open filterbase. In this paper we give characterizations of regular-closed spaces in terms of nets and arbitrary filterbases. These characterizations are obtained mainly through the introduction of a type of convergence for filterbases and nets that we call s-convergence. Throughout, cl04) will denote the closure of a set A will denote the set of accumulation points of a filterbase will assume that the condition of regularity includes the and 4(F) F. We T^ separa tion axiom. 2. Preliminary definitions and theorems Let A be a non-empty subset of a topological space S(A) = {0^ : A c 0^,, a € A} Then S(A) X and let be a family of open subsets containing A. is called a shrinkable family of open sets containing if for each 0 a € S(A) , there exists an On € S(A) 8 A such that cl(O a) c 0g . Definition 2.1. A filterbase X s-oonverges to a point • a ^ A} p € X (denoted by F in a topological space p) if for each *This research was supported by a grant from the National Science Foundation, Grant No. SER76-08651, Research Initiation for Minority Institution Improvement. Math. Chronicle 5(1977) 168-178. 168 shrinkable family S(p) = {0 there exists an 0o € S(p) F s-accumulates to an 0o i Sip) 8 p X ct € F 8 F a p, c 0o. p p) if for each shrinkable ±<fs for each a F such that s of open sets containing 0 o fl F such that F (denoted by : 3 € A} p of open sets containing and some p S(p) = {0 family : 6 € A> p p, there exists a ( A , Convergence and accumulation of filterbases in the usual sense, of course, imply s-convergence and s-accumulation, respectively. How ever, the converses do not hold as the next example shows. Let Example 2.2. I = [0,1] have as a subbase the usual open sets together with the set F - {F n Z = { r : 0 < r < l : n £ N} (where irrational}) and let F = {a; : -rr < x < n 2 n +1 2 p = h • Then late in the usual sense to to the point r and p but F F is rational}. n+1 x and Let is does not converge or accumu s-converges and s-accumulates p. There are a number of theorems concerning s-convergence and s-accumulation of filterbases whose statements parallel those of convergence and accumulation of filterbases in the usual sense. We state a few of these theorems but omit their straightforward proofs. In a topological space Theorem 2.3. (a) If is a filterbase in F then (b) Let Fj than Fj. and F2 Then the following properties hold : X such that F s-accumulates to p. converges to p, then X F If X s~converges to p € X, is regular and if F F s-accumulates at no point other than p. be two filterbases on F 1 s-accumulates to X where pi X if F2 is stronger F2 s-accumulates to p. (c) A maximal filterbase (d) only if M s-converges to p. If X is a T^ normal space, then a filterbase M (accumulates) to p € X in X s-accumulates to p t X if and only if 169 if and F converges F s-converges (s-accumulates) to p. Definition 2.4. Let X. be a net in J Then be a topological space and let s-converges 0 if for each shrinkable family Q(T^) c 0 to p $X a (denoted by £ S(p) c If : a € A) p) of open sets 0^ £ S(p) and some d £D 0 such that s-accumulates p) if for each shrinkable family s of open sets containing p, there exists an 0 0 fl Q(T J t 0 a a such that 0: D — *■X (denoted by 0 = {o £ D : d < c}) . The net (where S(p) = {0^ : a € A} 0 S(p) = {0 p, there exists an containing to p € X 0: D — * X X, then the family is a net in F(0) = {0 (T^) : d € D} d 6 D. for each X is a filterbase on and it is routine to verify that : (a) F(0) — *■p € X s (b) F(0) « p € X if and only if if and only if S Conversely, every filterbase 0: D — * X F 0 — *■p. : > 0 * p. 5 in X determines a net such that : p € X (a) 0 (b) 0 « p £ X if and only if if and only if p. F F « p. The construction of such a net is the same as that of [3, p.213]. In a topological space Theorem 2.5. (a) If 0 is a net in X s-accumulates to p. X the following properties hold: that s-converges to p £ X3 then If X converges to p € X, then 0 is a regular space and if 0 0 s-accumulates at no point other than p. (b) A universal net 0 s-accumulates to p £ X if and only if 0 s-converges to p. 3. Filterbase and net characterizations of regular-closed spaces An open filterbase F in X 170 is a regular filterbase if for U € F, there exists a each Let space A A = {Z7CX : a £ J. and if Then 8 S a) such that cl(V) c £/ [l]. F ( F and 3 = {7Q : 8 £ I} be covers of a p is a regular refinement of if is a shrinkable refinement of itself. shrinkable refinement of itself if for each € 8 A V 8 (8 refines is a € B, there exists a P An open cover is regular if it such that cl(V^) c y ). has an open regular refinement. Lemma 3.1. Let A[F) = y4g (F) of F be an open regular filterbase on (where ^ S (F) X. Then denotes the set of s-accumulation points F). Proof. ^ S (F) c A{F). We only need to show that p j: j4(F]. suppose that containing p and some Let p € X Then there exists an open set U i F such that and G(p) U (] G(p) = 0 . Then S(p) = {X-cl(V) : V € F and cl(V) c [/} forms a shrinkable family of open sets containing p and has the property that for each (X-d(V)) € S(p )j (X-cl(V)) fl V = 0. conclude that Consequently, p j: A o (F). We -4(F) = A [F]. We next characterize regular-closed spaces in terms of arbitrary filterbases. Theorem 3.2. In a regular space (b) X is regular-closed. Each filterbase in X (c) Each maximal filterbase in (a) Proof, X X the following are equivalent: s-accumulates to some point p € X. X s-converges to some point p £ X. (a) implies (b). Suppose there exists a filterbase, that has no s-accumulation point. • exists a shrinkable family S(x) Then for each x € X3 there of open sets containing 171 ~ 0 x such ’ It follows that collection of open sets A = {U(x) : U(x) € S(x)s x t X} that for each U(x) € S(x)s U(x) fl F, in f°r son,‘ e Fy(x) * X. forms a regular open cover of Consequently, by Theorem 4 of [l, p.104], there exists a finite subcollection, 6 = {U.(x.) Z A : i = 1,2,3,..., m 3 T' property that m n(i) U U ^ j Since 8 . = 0. Choose covers contradiction. FQ € F such that X, there exists a (b) implies (a). Let F X n(i) U (x7) fl F., , , j- 0 p k Vp'xiJ A (F) s which is a 0.. be an open regular filterbase on * By Lemma 3.1 and hypothesis (b) we have that Therefore m c fl fl F . i=l J = 1 j i * 8 with the property Therefore, We conclude that U-(x .) € 8 t i F n D U (x,) i 0. U p k that Now for each FU .(x.) £ F such that 3 ^ there exists a corresponding . j = l,2,3,... ,n(i) }, with the U.(x.) = X. 3 t i=1 j=l U.(xJ fl F and X. A(F) = A (F) j^0. s is regular-closed according to Theorem 4.14 of [1, p.104]. (b) M implies (a). Let s-accumulates to some point M be a maximal filterbase in p tX and hence s-converges to X. Then p by Theorem 2.3 (c). (o) implies (b). Let F exists a maximal filterbase, M, in s-converges to some point be a filterbase in X stronger than F. p £ X, F s-accumulates to p X. Then there Since according to Theorem 2.3. Since filterbases and nets are "equivalent" in the sense of s-convergence and s-accumulation, we can now characterize regularclosed spaces in terms of nets. Theorem 3.3. In a regular space X (a) X (b) Each net in (c) Each universal net s-converges. the following are equivalent: is regular-alosed. X has an s-aocumulation point. 172 M If a regular space Lemma 3.4. filterbase in Suppose that point. Fix p that has the property that every X with a unique s-accumulation point is convergent then every filterbase in Proof. X p £X F X has an s-accumulation point. X is a filterbase in and define with no s-accumulation = {F U {p} : F ( F}. is the unique s-accumulation point of converge to F^ and It follows F^ does not p. If S(A) and S(B) are shrinkable families containi-ag Lemma 3.5. A and B respectivelys then S(A U B) = {U U V : U £ S(A) and V € S(B)} is a shrinkable family containing Proof. A U B. The straightforward proof is omitted. Remark 3.6. Let F be a filterbase on X A c X. shrinkable family on open sets containing S(A) and let be a We say that F misses S(A) if for each U € S(A) , there exists a corresponding Fy £ F with the property that cl (U) fl F^ = 0. A Suppose that p j: A. is a closed subset in a regular space Furthermore, assume that F is a filterbase on misses a shrinkable family S(A) that contains exists a shrinkable family S q (A) containing p f cl(U) has the property that let V(p) define p that Then there that misses U € S q (A). such that and F and To see this, A fl V(p) = 0 and S 0(A) = {U fl (X-zl(H(p)) : U £ S(A), Hip) is open and p £ H(p) c cl (H(p)) c V(p)}. It follows that contains for each be an open set containing A A. X X S q (A) A, misses is a shrinkable family of open sets that F, and has the property that p £ cl[i/ fl (X-cl(H(p)))] for each [jj fl (X-cKH(p)))] £ S Q(A). In view of the preceding discussion, we now give the following definition. Definition 3.7. Let F be a filterbase on a regular space 173 X, p € X, and let C = (SiA) : S(A) is a shrinkable family of open sets containing A c X-{p} }. a closed set We define S = {S(A) t C : S(A) misses F and p ^ cl (U(A)) for each U(A) € S(A)}. Theorem 2 of [2, p.454] characterizes minimal regular spaces in terms of open regular filterbases. In our next theorem we character ize minimal regular spaces in terms of arbitrary filterbases. Theorem 3.8. A regular space (X3i) is minimal regular if and only if eaoh filterbase in Proof. X with a unique s-accumulation point is convergent. Suppose the condition is given and let filterbase on * 3.1, p X with a unique accumulation point p. converges to p ( I. F. is the unique s-accumulation point of hypothesis, F F be an open regular This shows that By Lemma Consequently, by X is minimal regular according to Theorem 2 of [2, p.454]. Conversely, assume that suppose that point p X is a minimal regular space and F is a filterbase on X with a unique s-accumulation to which it does not converge. Let S be the set given in Definition 3.7 and define S = {H c X-{p} : H (Xjj)} is open in U {(X-cl(U(A))) U G(p) : U(A) € SiA), S(A) * S, G(p) € t, and p € G(p)}. {H c X-{p} : H € t) It is clear that the collection for a topology on X-{p}. Now let (X-cKUfA^)) U G(p) be elements in B forms a base and (X-cl (U(AZ))) U Hip) containing the point p. We note that p € [X-cl (U(Al) U U(A2))] U (G(p) fl Hip)) C [(X-cliUiA^)) U Gip)] n [ a - c l (U(AZ))) U Hip)]. But according to Lemma 3.5 and Remark 3.6, 174 [X-cliUiA^ U U(A2)j ] U (G(p) fl H(p)) t B . Consequently, it follows that the collection a base for a topology 8 xQ X on with t 0 c xQ x. containing S(x) € 5 shrinkable family J(x) € tq We next show that Kix) c X-{p} be Let F « x3 there exists a Since x. that contains Then there exists an open set of open sets forms . t X. forms a base for a regular topology on an open set in 8 U(x) € S(x). Let x containing with the property that a: € J(x) c cl ^iJix)) c \_K(X) fl U(x)~\ c K(x) c !q (J(x )) (where (We note that Thus, Tq J(x) denotes the closure of p ^ cl Q(J(x)) iX3x^)). J(x) D (X-cl(U(x))) =0.) since is regular at each point in x £ X-{p}. Now let (X-cKU(A))) U H(p) p. be a basic open set containing G(p) exists an open set € (X3\) Since containing t p is regular, there such that p $ G(p) c cl (G(p)) c H(p). (We note that clQ (G(p)) = cl (G(p)).) Now since A, there exists a shrinkable family containing S(A) 6 S V(A) € S(A) is a with the property that cl (U(A)) c V(A) c c l(V(A)). Consequently, p € [X-cl(V(A))] U G(p) c cl0[ a - c lfVfAjJJ U C-fpJ] c [X-cl(U(A)j] U H(p) showing that tq p. is regular at open set U(p) F i F . But for each basic open set € t containing Pj there exists a corresponding [j-cl(7J(A))~\ 3 Fu . that T0 i x Thus showing that p By hypothesis, there exists an such that F U(p) for each [Z-cl (I)(A)) ] U Hip) containing F^ £ F with the property that [X-cl(U(A))] U Hip) <f. Uip). We conclude (X3t) is not minimal regular. 175 4. First countable regular spaces. (X3x) A space if t is called first countable and minimal regular is first countable and regular, and if no first countable X topology on which is strictly weaker than is first countable and regular-closed if (Xs t ) regular, and t x is regular. (Xs\) is first countable and is a closed subspace of every first countable regular space in which it can be embedded [4 ]. A first countable regular space Theorem 4.1. X is first countable and regular-closed if each countable filterbase on X s-accumulates to some point p £ X. Let Proof. F X. be a countable open regular filterbase on By Lemma 3.1, A[¥) = ^4^{F ) / 0 X which implies that is first countable and regular-closed according to Theorem 2.6 of [4, p.116]. A first countable regular space Theorem 4.2. and regular-closed if each sequence in X X is first countable s-accumulates to some point p £ X. Suppose that Proof. X is not regular-closed. Y a first countable regular space h : X — * h(X) c y exists a sequence, p £ Y-h(X). Since is not closed in f : N — + h(X) s in h(X) h(X) is homeomorphic to z € h(X). Therefore some point and a homeomorphism h(X) such that Then there exists z = p Y. Thus there converging to some point X, f s-accumulates to according to Theorem 2.5 (a) which is a contradiction. Theorem 2.6 of [4, p.116] shows that a first countable regular space X is first countable and minimal regular if every countable open regular filterbase on convergent. X with a unique accumulation point is In our final theorem we show that a space countable and minimal regular if each sequence in 176 X X is first with a unique s-accumulation point is convergent. Lemma 4.3. If a regular space sequence in Suppose that tion point. n X has an s-accumulation point. (x^) p £X Fix is odd and z^ is a sequence in X with no s-accumula and define a sequence (z if (z^) s-accumulation point of Theorem 4.4. has the property that every X with a unique s-accumulation point is convergent then every sequence in Proof. X n is even. Then p by z^ = p if is the unique to which it does not converge. A first countable regular space (X>x) countable and minimal regular if every sequence in is first X with a unique s-accumulation point is convergent. Proof. Suppose that mapping of (X,x) h 1 is continuous.) converging to a point point of the sequence that h(p) is a continuous bijective onto a first countable regular space need to show that Y h: (X,x) — *■(Yyo) y € Y. (y ) Let If p 6 X Y be a sequence in (h~^(y ))s then the continuity of is regular, h(p) = y. (We is an s-accumulation is an s-accumulation point of the sequence Therefore, since (Yso). h shows (y ). Consequently, h l(y) the unique s-accumulation point of the sequence hypothesis, the sequence (h~l(y )) converges to -1 n that h is continuous. (h ^(y^)). h l(y) By showing REFERENCES 1. M.P. Berri, J.R. Porter and R.M. Stephenson, A Survey of Minimal Topological Spacess General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Kanpur Topological Conference (1968), 93^114. 2. M.P. Berri and R.H. Sorgenfrey, Minimal Regular Spaces, Proc. 177 is Amer. Math. Soc, 14(1963), 454-458. 3. J. Dugundji, Topology, Allyn and Bacon, Boston, Mass. 1966. 4. R.H. Stephenson, Jr., Minimal First Countable Topologies> Trans. Amer. Math. Soc. 138(1969), 115-127. University of Arkansas at Pine Bluff