Chvalina, Jan

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJan ChvalinaStackbases in power sets of neighbourhood spaces preserving the continuity ofmappingsArchivum Mathematicum, Vol. 17 (1981), No. 2, 81--86Persistent URL: http://dml.cz/dmlcz/107095Terms of use:© Masaryk University, 1981Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCH. MATH. 2, SCRIPTA FAC SCI. NAT. UJEP BRUNENSIS XVII: 81—86, 1981 STACKBASES IN POWER SETS OF NEIGHBOURHOOD SPACES PRESERVING THE CONTINUITY OF MAPPINGS JAN CHVALINA, Brno (Received September 29, 1980) A generalized filter base in a set M is defined in [7] as a nonempty family of nonempty subsets of M. The generalized filter base is called a proper family in [4] and a stackbase in [3]. We shall use the last term. If (M, cx)9 (M, c2) are topological spaces determined by Kuratowski closure operations ct, c2 then there exist stackbases at , <T2 in M such that for each continuous mapping/: (M, cx) -• -> (M, c2) whenever Xea2 then/"l(X)e tr, or/"X(X) = 0. Further, for each xeM and each X2ea2r\ [/(*)) (where \x) = {X c M : x e X}) there exists a set Xt e eat n [*) with/(Xt) c X2. Moreover, the assignment c -> a is one-to-one. Indeed, assigning to a Kuratowski closure operation c on M the system a of all nonempty open subsets of the topological space (M, c) we obtain that a n [*) is a neighbourhood base at the point x and the above statements follow e.g. from [6] Theorem 1.4.6. The just formulated continuity condition at JC e M can be written in the form a2 n n [f(x)) <f(ax n [*)), where A, -< X2 for Xu X2 cz exp M means, by [4], that for each Xe X1 there exists YeX2 with Y cz X This note aims to show that the above described assertion does not hold in the case of neighbourhood spaces ([5], [7]) which are not topological, i.e. corresponding closure operations are the so called Fr6chet—Cech closure operations ([1], [5] — satisfying the following three axioms only: 1° c0 = 0, 2° X c cX, 3° X cr Yimplies cX c cY). Further we shall prove the existence of an assignment of a stackbase $f(t) in exp M to an arbitrary Frechet-Cech closure operation t on M with the following properties: If tt ^ t2 then Sf(tx) # £f(t2) and for every continuous mapping/of the neighbourhood space (M, tx) into the neighbourhood space (M, t2) the corresponding self-map/of exp M satisfies the condition/_1(X) e Sf(tx) u {0} for each Xe Sf(f%\ consequently $f(t2) n [/(X)) <f(&(tx) n [X)) for each Xeexp M, where [X) « = { l c exp M : X e A}. In what follows we denote by exp' M the system of all non-void subsets of M (including M). The system of all Frechet-Cech closure ope-rations on a set M will be denoted by (£(M). 81 Definition. Suppose Xi9 X2 c exp M. A self-map / of the set M is said to be (^i* A2)-continuous if for every XeX2 we havef~~l(X) € Xx u {0}. It is to be noted that it is inessential to use in our considerations only one fixed setM instead of two sets MX9M2 (or spaces (M19tx)9 (M29t2)) and mappings between them. The corresponding changes are only of a formal character. Then the below constructed mapping if becoms in fact an object function of a functor from the category of all neighbourhood spaces and continuous mappings into the category of sets endowed with stackbases with morphisms—mappings compatible in the sense of the above definition. Hence from the main theorem there follows the proof of Proposition 5 from [2]. Proposition 1. Let M be a set of the cardinality at least 3, F : <£(M) -> exp' exp' M a mapping such that for each pair tt9 t2ed(M)9 every cotinuous mapping f: (M, tx) -» (M, t2) is (F(tx)9 F(t^ycontinuous. Then F is not injective. Proof. Let M be a set with card M = 3. Denote by Mx an arbitrary three element subset of M containing elements xx, x2, xz and put M2 = M\MX. Let ux, u2 be Frichet-Cech closure operations on Mx defined by ux{x^ = {xi9xi+x}9 u2{xt} = = {xi9 xi+2} where addition of indices is modulo 3 and / = 1 , 2, 3. Denote by t* the discrete topology on M2 and put (M, tt) = (Ml9 ut) + (M2, t*)9 i = 1, 2 [i.e. (M, tx) is the disjoint sum of neighbourhood spaces (Mx, ux)9 (M2, t*) and similarly for (M, /2)]. Evidently tx # t2. Define permutations/l5/2,f3 : M-» M as follows: For x e M2 we put f(x) = x and f(Xi) = xi9 if i ^ j then /,(xy) = xk9 where k e € {1, 2, 3}, i ? - i t #1 , i, ye {1, 2, 3}. It is easy to verify that every/, i = 1, 2, 3 is a homeomorphism of the space (M, tx) onto (M, t2) and also (M, t2) onto (M, fj) f o r / = / f \ / = U X 3. Then these mappings are (F(^), F(f2))-continuous as well as (F(t2)9 F(t-))-continuous. We are going to show F(tx) = F(t2). Suppose l e F(t2). If Mx c X or X c M2 we have /(X) = X = /_ 1 W for i = 1, 2, 3 thus X = = /r1(X)€F(f1). Suppose card (XnMx) = 1 and Xn Mx = {^J, y0 e {1, 2, 3}. Consider a mapping fj9 i.e. the permutation of M with the fixed point XJQ. Then //o(X) = X = ///(X) hence Xe F(rx). If card (X n M t) = 2, say I n M , - - {*,., xk}, j9 k € {1, 2, 3}, we use the homeomorphism / where / e {1, 2, 3}, f # * ^ fc. Then X « / f *(X) e Ffo), consequently we have F(t2) <= F(tx). Using the equality /f1 = / for each 16 {1, 2, 3} we get in the same way as above F(tx) e F(t2)9 i.e. F(tx) = = F(t2). This completes the proof. Consider ax, <r2 e exp' exp M. In regard with [4] 1.1 we put ax(u)a2 = «- {•?! u S2: Ste a1? «S2 e <r2}. Let (M, 0 be a neighbourhood space with M g£ 0, ' Jfe exp' M. We put ^ f P0 = {exp M} e exp' exp M if the set X is dense in the space (M9t)9 i.e. tX = M and » t(J0 - {Wu) exp' X) u exp' X : a e exp' exp (M \ *.*) otherwise. Further we put «^M(0= U **(*)• Xeexp'M 82 From the definition of &>M(t) it follows immediately 0 6 $PM(t) (for M # 0), exp Mn € S?M(t) hence SfM(t)& 0, i.e. ^ M ( 0 is a stackbase in exp M. Lemma 1. Let tt9t2 be arbitrary different Frichet-Cechclosure operations on a non-void set M. Then S?M(tt) # S?M(t2). Proof. There exists a non-void set X0 c M, such that txX0 # t2^o- Exactly one of the following cases is possible: ( i ) * ^ ^ ^ o » 0 0 ^ o # ̂ i^o»(»0't^o II h%o (i.e. ttX0i t2X0 are incomparable with respect to the set inclusion). In the case (i) we put o == (exp' (M \ ^Xo) (KJ) exp' X0) u exp' X0. There is o € S^M(tt) and o # %t2(X0). Admit o 6 ^M(t2). Since 0 & o # exp Af, there exists a non-void set Xt c: M such that <x = (A(u) exp' Xx) u exp' Xt, where 0 # # A c exp (M \ t2Xt). Since exp' Xt c or, we have Xt n X0 # 0. In the opposite case we would have Xt £ exp' X0 and simultaneously X! ^ exp' (M \ t} X0) (u) exp' X0, Assume Xt <£ X0. Then Xx n (M \ X0) ^ 0 thus there exists a nonempty set S cz Xtn n(M\ X0) with S# exp' X0, S# exp' (i*T\ r-Xo) (u) exp' X0 for S n X0 * 0. Then 5 # a which contradicts Setxp' Xt a o. Hence Xt c X0. Simultaneously exp' X0 c (A(u) exp' Xt) u exp' Xx. If exp' X0 c A(u) exp' Xt then for every point xeX0 we have { J C } = I U F where X e A, Y c Xt and thus X, F are disjoint, Y =£ 0. Then x e F c J . , i.e. X0 c Xt. (The same follows also from the inclusion exp'Xo c exp'Xt). Hence Xt =- X0, which means cre^2(X0)~-a contradiction. Consequently <r $ S?M(t2) in the considered case. Since the case (ii) is analogous to (i) we consider (iii). Put again o = (exp' (M \ ttX0) (u) exp' X0) u e*P' -J-o- Since by the above consideration the* assumption oeS?M(t2) implies o € Vtl(X0) which is impossible for M \ ttX0 # exp' (M \ t2X0), we have o $ S?M(t2). Consequently yM(h) # &M(h\ Lemma 2. Let f be a continuous mapping of the neighbourhood space (M9tx) into the neighbourhood space (M, t2). The induced mapping f: exp M -> exp M is (&M(h)> S?(t2))-continous. Proof. Assume oeS?M(t2). We are going to show f"l(^)^S^M(tt) whenever / - V ) # 0. If ^ = exp M then f~x(a) = exp M e SfM(tt), Let X c M be a non-empty set which is not dense in the space (M, t2). Suppose o -» (A(u)exp' X) u u exp' X, 0 # A c exp (M \ t2X). It holds/" V ) * { S c M :/(S) == /(S) e <y}. We show that iff~*(o) # 0 then there exists a non-void set Y c M such that (i) / - l(a) c (exp (M \ tt Y) (u) exp' F) u exp' F, (ii) if S'€/""1(o ,)\exp'F and T c F is an arbitrary non-void subset then (5 n ( M \ txF) u r e / ' V ) \exp ' F. ( i i i ) e x p ' F c / ~ » . Assume / " V ) # 0 and put F **f~l(X)9 admit F * 0. There is a non-void set Sef~l(o\ thus either /(S) c For;there exist non-void disjoint subsets Q%>Qz c Mf - « . . • ' • " 8» Qx 6 exp (M \ t2X)f feci such that f(S) = QxvQ2. Thus f(S) n X # 0. On the other hand/(S) c / (M) and/""J(X) = 0, i.e./(M) n X = 0 hencef(S) n X = 0 which is a contradiction. Consequently Y =*fl(X) ?-= 0. We shall prove (i). Suppose Sef"l(a). Thus /(S) € <r, i.e. either /OS) c X or /(S) = gx u g 2 , where Qx, <?2 are suitable subsets of M with the above mentioned properties. If f(S) c X we have S czf%f(S) czfl(X) = F, thus Seexp' F. In the second case 5 aflf(S) = = /~3(Gi) u / " " 1 ^ ) and there exist non-void sets Pt cfx(Q^ / = 1, 2 such that Px u P2 = S. (Indeed, since Qx n Q2 = 0, Qljz%^Q1 we have fl(Qx) r\ nfx(Q2) = 0. fx(Qi) * 0 ^fKQd and putting P, = Sn/"1^), / = 1, 2 we get the just used statement). Since the mapping/: (M, tx) -* (M, t2) is continuous we have/ft F) c r2/(Y) = t2ff\X) = *2(Xn/(M)) c r2X Then r j c / - *f(tt Y) c c>fl(t2X)9 thus M\fx(t2X) c M\ttY. Further, with respect to the inclusion QxcM\t2X9 we have P, c / -1 ( Q i ) c : / ^ ( M \ r 2 X ) = / -1 ( M ) \ r 1 ( r 2 X ) = = M\fl(t2X) c M\txX and P2 c/~*(Q2) czfl(X) = F which implies S = = P, u P2 e exp (M \ f j Y) (u) exp' F. Hence S e (exp (M \ tx Y) (u) exp' F) u u exp' 7, therefore (i) holds. We shall prove (ii). If 5 efl(a) \ exp' Y then by (i) S belongs to exp (M \ tx Y) (u) exp' Y and by the above considerations there exist non-void disjoint sets Px, P2 with Px cf^M \ t2X% P2 c Y. Since Pxc M\txY we have S n (M \ txY) = (Px u P2) n (M \ t{Y) = \PX n (M \ txYj] u u [P2 n(M\txF)] = P!. Since f(S) e exp' X implies 5 e exp' F, the set /(S) = = /(Px) u/(P2) belongs to a and <r \ exp' X = A (u) exp' X for A c exp(M\f2X), we have f(Px)eL Let fi c F, be a non-void subset. There is f(Q) <=• f(Y) = = X n /(M) c X, i.e. f(Q) e exp' X, f((X n(M\ txY)) u Q) = /(PO u f(Q) e e A (u) exp' X = <r \ exp' X. This means (Sn(M\ tx Y)) u Q e /~ V \ exp' X) = » / " l W X/^Hexp' X) =/~1(a) \exp' F since /""HX) ^ 0 implies /_1(exp X) = = exp ' / ^X ) . The family/""H*) satisfies condition (ii). The inclusion (iii) can be easily verified. Indeed, if Seexp' Ythmf(S) cf(Y) = Xnf(M) thus/(-?)€exp'A c c er, i.e. 5 belongsAo/'V). Now, if we put £ = {Sn(M\tx Y) : 5,e/"1((r)\exp' F} we get with respect to (i), (ii) and (iii) the equality f~\°) = (£ M exp' F) u exp' F. Consequently/""1^) belongs to f?M(tx). Theorem. Let M be a non-void set. There exists an infective mapping $f : £(M) -» -* exp' exp' M of the system of all Frechet—Cech closure operations on M into the system of all stackbases in exp M such that for every continuous mapping f: (M, tx) -* -* (M, f2), wi*h tr, t2 € G(M), the induced self-map f of exp M is (&(tx), S?(t2))-conti-nuous. Proof. Consider SM : C(M) -* exp' exp' M defined above and apply Lemma 1 and Lemma 2. 84 Remark. From the above theorem it follows that for every continuous mapping of a neighbourhood space (M, tt) into a neighbourhood space (M, $2) &&d every non-empty subset X c M we have (£f(t2) n [/(X))) <f(Sf(tl) n [JT)). As the proof of this statement there can be used the proof of implications (3) *>-> (4) «-*> (2) from Theorem 1.4.6 [6]. There is a quite natural construction of a stackbase in exp M determined by a Frechet-- tech closure operation t on M and a subset A of Mas follows: For <r G exp' exp M we put T(a) = <T u {tX : XB<J}, T(0) = 0 and &"A(0 = {^eexp' exp M : ̂  eexp M \ T(exp M \ a ) } . It is easy to verify that (exp M, T) is a neighbourhood space moreover with the completely additive closure operation (i.e. each point Xeexp M possesses the least ^neighbourhood). Hence by [4] 1.6, 7.1 and 7,6, the stackbase fA(t) is in fact a filter (the T-neighbourhood filter of A) on exp M. But as the following example shows 3T A does not preserve the continuity of mappings. Consider the topological space (to, t *), where <o is the set of all positive integers and t*{n} = {n, n -f- 1, n -F 2,...} for n e co, t*X = (J **{*} for a nonempty subset xeX X a <o and t*0 = 0. Define a mapping f: co -> co by f(«) = w for n even andf(«) -* = n — 1 for « odd. Evidently f is a continuous self-map of (co, **). But denoting by T* the Frechet—tech closure operation assigned as above to t* we have coe e {{1, 2}, co} = F*{{1, 2}}, /(co) = {2n - I :neco}<f: {{{}, co} = r*{{l}} « = r*{/({l,2})}. Further see [4] 1.6 and 1.12. Jt is easy to see that the stackbase £f(t) constructed above is not a stack in general (i.e. the condition aeSf(t), <r c t => x eif(t) is not satisfied). Thus to assign a Fr&het-Cech closure operation 7V(f) to the stackbass^(^) is possible in the follow-ing way: For A cz M we put i^(Al) — {T c exp M :1 <re ¥(t) n [.4) with tea} and T^(t)(cr) = {A c M : <r nt ^ 0 for every t € i^(A)}, where a c exp M. Then by the definition of £f(t) and [4] 1.6,1.12, 7.1, 7.6, we have (exp M, Ty(t)) is a neigh-bourhood space and for an arbitrary continuous mapping f: (M, tt) -+ (M, t2) the mapping/of the space (exp M, 2>(tl)) into (exp M, 7V(t2)) is also continuous. REFERENCES [1] Čech, E.: Topological Spaces, Topological Papers of Eduard Čecћ, Academia Prague (1968), 436-472. [2] Chvalina, J. and Sekanina, M.: Realizations of Closure Spaces Ьy Set Systems.Ћcю. Thirđ Sym. Gen. Topology and its Rel. to Modern Analysîs and Algebra, Pгague 1971. Academia Prague (1972), 85—87. [3] Grimeisen, G.: Convergence Structures, Mimeographed notes of lectures given atthe Шîversity of Colorado in the Spring Semester 1970, 42 pp. • : . ' : t ó : [4] Hamburg, P.: Konvergenz in verallgemeinerten topologischen Raurnen. Portugal. Math. 35, 3 (1976), 137—168. [5] Mamuzič, Z. P.: Introduction to General Topology. Nordhoff, Groningen (1963). [6J Preuss, G.: Allgemeine Topologie, Springer-Verlag, Berlin—Heidelberg—New York (1975). [7] Thampuran, D. V.: Neighbourhood Spaces a**d Convergence, Portugal. Math. 33, 1 (1974), 43-49. /. Chvattna 662 95 Brno, Janáčkovo nám, 2a Czechoslovakia 

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Stackbases in power sets of neighbourhood spaces preserving the continuity of mappings

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Stackbases in power sets of neighbourhood spaces preserving the continuity of mappings

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Institute of Mathematics AS CR, v. v. i.

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