Hanák, J.

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

Toposym 3Jan HanákGame-theoretical approach to some modifications of generalized topologiesIn: Josef Novák (ed.): General Topology and its Relations to Modern Analysis and Algebra,Proceedings of the Third Prague Topological Symposium, 1971. Academia Publishing House of theCzechoslovak Academy of Sciences, Praha, 1972. pp. 173--179.Persistent URL: http://dml.cz/dmlcz/700804Terms of use:© Institute of Mathematics AS CR, 1972Institute 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.cz173 GAME-THEORETICAL APPROACH TO SOME MODIFICATIONS OF GENERALIZED TOPOLOGIES J. HANAK Brno 0. There is an interesting possibility of a use of generalized topologies in the theory of extensive games: In my works, I introduced the so-called SN-games [ = simultaneous nondeterministic games; at each nonfinal position of an SN-game all the players play mutually independently (the simultaneousness), knowing the preceding course of play, but their common influence need not determine the next position uniquely (the (local) nondeterminateness)], and I have shown that a certain reduced description (see [5], § 2.30.2 etc.) of SN-games is sufficient for the intro-duction of analogues of usual game-theoretical notions, for defining certain signi-ficant classes of SN-games, and, among other, for proving various strong game-theoretical theorems. (Cf., e.g., [3], [5], [6].) (Such a conception of SN-games has also admitted the introduction of the corresponding "descriptive theory", see [3], §9, or [5], §7, and of topological games, see [4], or [5], §8; of course, various known results concerning games with perfect information etc. are included as cor-ollaries in theorems on SN-games.) In this communication, we shall show that the game-theoretical interpretation of some partial unary operations (in particular, of those corresponding to some modifications) can be used for investigating their algebraic properties. In contradic-tion to [3] —[6], this paper contains no principal theorem: the goal is to show typical approaches, and to present a number of particular results obtained by means of them. A very short introduction of necessary auxiliary definitions, comments, and properties is given. 1.1. "*^ := 3$" means "$$ is defined to be equal to Jf". dom/ means the do-main of the mapping / . P will be a set, exp P := {A J A c p}9 F(P) := (exp P)expP; P will be usually fixed, and then we shall write, e.g., only &~ instead of ^(P). Let 1, (-1), X (where X s P) be such that 1A = A9 ( -1) A = P - A, XA = X for any A c p, <y together with the usual composition of mappings forms a monoid (the identity 1 is its unit), and F together with the usual partial ordering <; (u ^ v iff uA £ vA for any A ^ P) forms a complete lattice. ^ can be introduced in a some-what different way, too: let # be the mapping for which dom $ = «̂ ", $(u) = = {(x, A)\A^P,xe uA} (u e ^); then $ is a one-to-one ("canonical") mapping of &~ onto exp (P x exp P), but the latter set is a complete lattice with respect to £ , 174 J. HANÁK U, П» a n d this complete lattice structure can be transformed by Ф " 1 back onto ^ [thus, we shall write shortly, e.g., u я v9 U\J v instead of Ф(u) c Ф(v)9 ф-Ҷd^м) u u Ф(v))9 respectively]; it is easy to see that this (transformed) c is the same as ^ . (Besides, there is the canonical one-to-one mapping of У onto (exp exp P)p.) Cf. [12], and [5], § 2a. 1.2. If R(.)9 P i ( . ) , . . . , -Rfc(-) are some (one-variable) propositional functions defined at least on ^ , we denote ІГK : = \u\uє!Ғ9 R(u)}9 3~Rl...Rk := &'Rí n n . . . n ЗГRìí. The following propositional functions are often used (we write " /? :" instead of "Я(м) iff"): R: мø = 0; M: A £ B c p => UA я uB; E: 1 s м; I: м c 1; U: м2 = м; A: Лj, Л 2 -^ P =* м ( ^ i u ^2) = и*4i u мЛ2. Now, ^ " R E U A ІS the s e t of (all) topologies (on P). Various kinds of generalized topologies obtained by replacing the Kuratowski axioms by some weaker ones have been investigated (let us quote several of many papers concerning these problems: [1], [7] —[14]; especially, closure spaces (ЗГRЋA, see [2], Sec. 14) and Čech topological spaces (^REM — - n the sense of [1]; of course, SГ A ^ ^ " м ) Һave been often investigated (in particular, the problems of modifying Čech topologies and the properties of the constellations of Čech topologies have been studied circumstantially; see, e.g., [1], [8], [9], [11]), and some topological considerations have been performed even for general elements of -У itself (Koutský topologies, or "topologies without axioms", see [7], [13], and a part of [14]). In this communication, we shall consider another kind of generalized topologies, namely «^~RM; they will be called game topologies (on P), cf. [5], §§2.8, 2.26.3. Under a type we shall mean (P, P0) where P 0 c p (in [5], § 1.1, P Ф 0 was supposed besides); «íГRM(P, P 0 ) : = {м | м є 3~KM(P)9 uP = P - P 0 } will be the set of game topologies of the type (P, P 0 ) . ([5], §2.13 etc.) 1.3. Under a game system (on P) we shall mean a non-empty system Щ = = (uj)jєJ є (^ Г RM) J s u c h that all the Uj have the same type, and П -4j = 0 implies П UjAj = 0 for every (Aj)J&J є (exp P)J. [For an SN-game 0, let P (P 0 , J) be the set of its positions (final positions, players, respectively), let (for j є J) Uj є ^(P) be such that UjA = {x \ x є P, in 0, j can guarantee at x that the next position will (exist and) belong to A} for any A £ p , let Z : = P — P 0 ; then ^ : = ( м ^ j is a game system, UjP = Z for each j є J, and Z is the set of nonfinal positions of *§. On the other hand, if some Чl is a game system, then ^ can be obtained in the above men-tioned way, to a suitable 0. See [5], § 2.29-30, § 2 (47), (26) e t c ] 1.4. Under an operator we shall mean a partial unary operation in 3~9 i.e., a mapping of a subset of ^" into -Ӯ*. We shall need, in particular, these operators: м -* м* (fc = 0,1,...), defined as the fcth power in the monoid T; u -> м : = P—мPu u м; м -• м : = ( - 1 ) . м . ( - 1 ) ; м -» м' : = мP n м (i.e., м'Л =- мP - м(P - A)); м -» м where йA = uA v (A — uP). J. HANAK 175 It holds: if u e ^KM(P, P0) (where P 0 s p), then u ' e ^"RM(P, P0), u" = u; if Uj e«^"RM(P, P0), then u' is the greatest (under c ) element of {u2 \ u2 G ^ R M , (u^u2) is a game system}. (See [5], §4 (26)-(28), §4.8.4 etc.) An SN-game # is said to be complete iff card J = 2 and (uJ2)' = ujl for {ji,j2} = J (where J, Uj etc. have the meaning given by the remark in Sec. 1.3). Complete games have very signi-ficant properties, similarly as their particular case — two-player Bergean games with perfect information. (Cf. [5], §§ 4d, 6c, and [6].) 1.5. For /?!(.), . . . , /?k(.) (Sec. 1.2), under the upper [lower] Rt . . . Rk-modi-fication of u (e 2T) we mean v e &~Ri...Rk such that (i) u £ t? [u 2 v], and (ii) v £ w [v 3 w] whenever w 6 ^Rl...Rk and u £ w [u ^ w]; of course, there exists at most one upper [lower] Rt ... ^-modification of u, thus the forming of upper [lower] JRi ... /^-modifications may be considered an operator. (Cf. [2], Sec. 31 B.) Let 1 Jjf u G « "̂M; the well-known construction of the transfinite powers of u can be expressed in such a way: u° := 1, u* := lim u . un (£>0 is an ordinal number); we put formally u00 := u* for (any) £ such that u* = u*+1 (oo is not an ordinal number). It is known that u00 is the upper [lower] U-modification of u if u e ^"EM [u e -fTnJ. For each u e « "̂M, l u t i [1 n u] belongs to « "̂EM [ ^ i M ] , and uA : = := (1 u u)00 [uv : = ( l n u)00] is the upper UEM-modification [lower UlM-modific-ation] of u. The modifications A , v, and also the operators u -* uA := (u)v, u -+ -> uv : = (u)A (where u G ,yM; of course, then u G ^ * M ) are important in game con-siderations. For ue^M it holds: u£ = (u)v, u^ = (u)A, uA = (u7)^, «v = ( M % ; if, moreover, UG^RM, then uA = (u')A, uv = (u')v. ([5], §5.18.) 2.1. To a given type (P, P0) we introduce: Z := P - P0, Z := U Z{0 '}, 0£Z<o> 0 S : = ( ( e x p P ) - { 0 } ) z , and p := (J {x = (xfc| 0 ^ k < 1 + /) I xk e Z if 0 g fc < /, xk G P 0 if fc = /} , where co0 is the first infinite ordinal number. For z = (z0, . . . , z , ) e Z we define K(Z) : = z„ /(z) : = /. For x = (xk \ 0 <; fc < 1 4- /) e P we write shortly x = (xk) and define Z(x) : = /. For aeS, xeP we put S(<T) : = {x = (xk) | X G P, xk+1 eo(x0,..., xfc) if 0 ^ /c < /(x)} , s(x, <r) : = {x = (xk) | x G S(<F), X0 = x} ( # 0) . 2.2. Let ueyRM(P,P0) in this section. We put S(u) : - {* | * e (exp P)z , K(Z)G u(az) for every z e Z J l e S ) , Further, let ue(expp)ex->p be given by uA : = : = {x | x G P, s(x, (T) £ A for some a e S(u)}. There holds (see [5], § 3a): «0 = 0; A c B c P = > 0 A c i r B ; i f (wy)I6j i s a game system such that UjP = Z (cf. Sec. 1.3), 176 J. HANÁK then П Aj = 0 implies П UJAJ = 9 for any (Aj)JєJ є (exp P)J ([5], § 4.18). But if JєJ JєJ v := iľ (cf. Sec. 1.4), it may happen there is A - P such that vA = uP — u(P — A) does not hold ([5], §§4.21.5, 4.23—24); this very important fact concerns immedi-ately some connections with axiomatic set theories. [In terms of the game inter-pretation mentioned in the remarks in Sec. 1.3, P (exp P) is the set of variants (aims) at 0, S(uj) is the set of player/s strategies; x є UjA means thatj can enforce A from x, etc. Cf. [5], §§2c, Зa.] 3.1. In Sec. 3, let (P, P0) be a type, u є Укù(P, P0), v : = u'. For p є (exp P)e x p P (= the set of aim-mappings at (P, P0)) we define pє(expP)c x p P by pA = P — - p(P - Л) (cf. Sec. 1.4!). Similarly as in [5] (§ 5.8, or [3], too), we shall denote a general aim-mapping by symbol pв and we shall write pe: = p* (this symbolism admits various concrete forms, e.g.: pв = pA, pD, p, pe = pA, pп9 p, respectively). To such pE we define uв: = u. pв; E itself is then considered an operator (dome = = &~RM(P> Po))> a n d pв is said to be the aim-meaning ofв. pв and e are said to be normaliff(cf. Sec. 2.2!) (vвA =) v . pвA = иP - »(P - f>£Л) (= P - u . p e(^ - ^) = = P - we(P - A) = iv4), i.e., iff (u')в = г̂  (identically). In general, always (u')в c c ue, and c is normal iff e is normal. 3.2. Some operators (e.g., u -> u° (=1)) have simple normal aim-meanings. The union (intersection, product) of two operators having normal aim-meanings need not have an aim-meaning. But if u -> uв is an operator having a [normal] aim-meaning, then u -+ u . uв, u -» й. uв, and u -* й . uE have [normal] aim-mean-ings. 3.3. (Cf. [3], §§ 2.8.1, 6.9.4, or [5], part IV.) Let pв, pe (s = Д, V, D) be such that x є pàA [x e p&Ä\ oxkeA for some [each] k, p*A = P ғ u p*A , pvA = (P - Pғ) n păA , x є paA o for each k there exists r = fc such that xre A9 x e pQA o there exists k such that xre A for each r = k, where x = (x f c)єP, Л c P , 0<£fc, r < l + l(x), P ғ : = { x | x є P , l(x) < æ0} ; it is easy to see that, indeed, pe = jŕ for в = Д, V, • (cf. Sec. 3.1). Now, uв and ue are defined twice for є = Д, V (and u e « "̂RM), namely by Sec. 3.1 and 3.3, and by 1.5; nevertheless, it can be proved that these two definitions yield the same concepts. Moreover, it can be shown, among others, that pв9 pe J. HANÁK 177 (e = Д , V, D) are normal. From the definitions it follows trivially that мe, м8 e e ^M(P), мл0 = 0, мдP = P, and мv ç мд c мD c мa c мл c мv , мд s 1 = c мл , мv = мл . (1 u PJ-,), мv = мд . (1 n Z), etc. 3.4. It holds м = Í Є ^ R ^ P , 0); we shall write (cf. Sec. 3.2!) м~£ := м.мe, м_e := й . мe (then м_8 = ( < p if м£ = (ӮJ5). Clearly, (м)v = (м)л, (м)v = (ő)д. The following important equalities hold: (м)£ = мe, (м)c = мe for s = Д, D , " D = ("~Л)~л > uD = ( 0 ~A . (To derive the propositions presented in this paper, it is natural to use the latter two equalities to prove the normality of D and D ; nevertheless, there are other ways, cf. [3],§6a, or§7b,or§9b.) 4.1. Theorem. Let (P, P0) be a type, u e ^RM(P9 P0). Then: A. (1) иv . u* = иv for e = д,v (-) иA . u" = u' for e = Д , V . D (з) u" • "a = "a • "a - "a for e = Д , D (4) "e . MA . t lд = И A . Mд for є = Д , D (5) « 8 . MA . u v = ИA . м v for e = V, Д, D (6) иv . мп . uv = ua . иv (?) tty . иp . мv = ИD . uv (8) " д . иv . uD = иv . uD (9) "« . ИV . U д = ИV . Ид for e = Д , D (10) "« V V . u . uv = u . uv for e = Д , D B. For j = 1,..., 10, let (f) be obtained from (j) by the replacement of each ub [ u j by ub [ud], 5 = A, V, D, e. Then (f) holds, too. C. If (e) is obtained from the equality in (j) or (f) (j = 1, 3, 4,9,10) by choosing s := s0 e {A> V, D} where e0 does not belong to those written in (j), then for suitable (P, P0) and u the equality (e) does not hold. 4.2. Part B is based on the normality of the operators under consideration. Counter-examples for proving Part C can be easily presented. The theorem contains 178 J. HANAK assertions which follow immediately from the other ones (by means of the simple properties mentioned in Sec. 3.3: (10) is a corollary of (9); in (3), (4), (5), (9), (10), always only one case is "essential" while the others follow from it and (2); (1) and (2) can be "reduced", too). 4.3. (We suppose the same as in Sec. 3.1; recall that symbol pe admits also pQ etc. as concrete "values".) For x = (xk) є P and 0 = m < 1 -F í(x), the sequence x[m] := (xm+k \0 = k < i + (l(x) - m)) (here co0 - m = co0) belongs to P and is said to be a remainder of x, while x is called an extension of x[w]. We say that A c P has the property I0 [l°] iff A is closed urкłer forming remainders [extensions]. Clearly, A has 1° [ï0] iíf P — A has I0 [I0] . We say that an aim-mapping pe has property K є {1°, I0} iff peA has this property for any A c P. Thus, if pE has 1° [ I 0 ] , then pP has I 0 [ I0 ] . In particular, pE [p,] Һas the property 1° [ l 0 ] for є = Д , V, D-Lєmma 1. Let pғ either (l) be pA, or (2) have the property I 0 . Let G Є S(U), x є P, A e P, x = (xfc) є s(x, <ғ) c pEA, 0 < m < 1 + l(x). Then xm є uєA (1) if {x0, ..., xm-i} n A = 0, (2) a/wajs (respectively). Lemma 2. Lé?í pc Лш;e the property 1° [ I 0 ] . TЛen мд . u£ = w£ [uA . мE = wß]. (Cf. [3], §§ 2.1.2, 2.4.2, 2.6.2, 2 (14) —(15), 6.2, 6.5.1, 6.5.3.) 4.4. Now, assertions (2) and (3) of the theorem follow immediately by means of Sec. 4.3. In the other "essential" equalities, one inclusion follows trivially from ue c 1 cz uв (e = Д , V); the other inclusion can be proved by means of Sec. 4.3 and 3.3. As an example, let us present the idea of the proof of (8). Let A я P; there exists cғj є S(u) [a2 є S(u)~\ such that s(x, cr̂ ) c pпA [s(x, cr2) £ pvuпÄ] for each xєгPA [x є uvuuÄ\. We choose a є S(u) in the following way: If z = = (z0, ..., zt) є Z, then if zt ф uпA, then cгz = <г2z, if zř є uпAj then GZ = -= <Fi(zr, ..., z,) where r = min {k | 0 ^ k ^ /, {zfc, ..., z j <= u°A}. Let x є wvwDAl, x = (xfc) є s(x, <т). Using the evident relations / l п P 0 Ç uDA and P0 u uuA c := uvwDA[, and the equality uvuDA = uд(P0 u WDУ4) (Sec. 3.4), and applying both the cases — successively (2) (for x є uuA firstly) and (1) — of Lemma 1, we obtain that x є pA(uvu°A). Hence, xє uAuvuuA for any / l ç P and xєuvu°A. Consequently, uvu°A e uAuvuDA c uvuDA. 5. In this paper, only a minor part of the results on SN-games ([3], [5] etc.) was used; especially, the min-max results (which were proved in versions being in several ways more geneгal, by vaгious approaches and in connection with the investigation of the game-theoretical meaning of extгeme properties of certain operators or their generalizations) have been reduced to the normality of some J. HANĂK 179 operators having aim-meanings, graphs of SN-games have not been used, etc* [Further, game-theoretical considerations can be performed even for ^"м; I have studied the latter possibility (cf. [6]), using some new operators (e.g., u -* u*9 u*A = = мØ -r u(P — A) -r uP (-f* is the symmetric diíference)).] References [1] E. Čech: Topologické prostory. Časopis Pěst. Mat. Fys. 66 (1937), 225—264. [2] E. Čech: Topological spaces. (Revised by Z. Frolik and M. Kat tov.) Academia, Prague, 1966. [3] /. Hanåk: Simultaneous nondeterministic games. Mathem. Theory of Polit. Decisions, Research Memorandum No. 3, J. E. Purkyn University, Brno. (December 1967. Cyclo-styled, 76 pp.) [4] J. Hanák: Topological SN-games. Mathem. Theory of Polit. Decisions, Research Memo-randum No. 7, J. E. Purkyné University, Brno. (April 1968. Cyclostyled, 20 pp.) [5] / . Hanåk: Simultaneous nondeterministic games (I)—(IV). Arch. Math.; I: 5 (1969), 29—60; II: 6 (1970), 115-144; III: 7 (1971), 123-144; IV: to appear. [6] / . Hanåk: Equilibrium points in some SN-games. (To appear.) [7] K. Koutskў: Určenost topologických prostorů pomocí úplných systémů okolí bodů. Publ. Fac. Sci. Univ. Masaryk 374 (1956), 153-163. [8] K. Koutský, V. Polák and M. Sekanina: On the commutativity of modification. Spisy Příгod. Fak. Univ. Brno 454 (1964), 275-292. [9] /. Lihová: About the fîrst and the second constellations belonging to the topology u. Acta Fac. Rerum Natur. Univ. Comenian. mimoriadne číslo 1971, 57—62. [10] Z. P. Mamuzić: Note sur les espaces de voisinages (V) et les ordres semi-topogènes. General Topology and its Relations to Modern Analysis and Algebra, II. (Proc. Second Prague Topological Sympos., 1966). Academia, Pгague, 1967, 246—247. [11] V. Polák, N. Poláková and M. Sekanina: Remarks to inner constellation in topological spaces. Spisy Přírod. Fak. Univ. Brno 507 (1969), 317-332. [12] /. Schmidt: Symmetric approach to the fundamental notions of general topology. General Topology and its Relations to Modeгn Analysis and Algebra, II. (Proc. Second Prague Topological Sympos., 1966). Academia, Prague, 1967, 308—319. [13] M. Sekanina: Űplné systémy okolî množin v obecných topologických prostorech. Publ. Fac. Sci. Univ. Masaryk 374 (1956), 185-192. [14] M. Ceкaнuнa: Cиcтeмы тoпoлoгий нa дaннoм мнoжecтвe. Чexocлoв. мaтeм. жypнaл 15 (90) (1965), 9—29. J. E. PURKYNË UNIVERSITY, BRNO 

Game-theoretical approach to some modifications of generalized topologies

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Game-theoretical approach to some modifications of generalized topologies

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