TOPOLOGICAL SPACES THAT ARE a-favorable FOR A PLAYER WITH PERFECT INFORMATION

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, July 1975 TOPOLOGICAL SPACES THAT ARE a-favorable FOR A PLAYER WITH PERFECT INFORMATION H. E. WHITE, JR. ABSTRACT. The class of spaces mentioned in the title is closely related to the class of a-favorable spaces introduced by G. Choquet \i\. For convenience, call the spaces mentioned in the title weakly a-favorable. The following statements are true: (1) every dense G g subset of a quasi-regular, weakly a-favorable space is weakly a-favorable; (2) the product of any family of weakly a-favorable spaces is weakly a-favorable; (3) any continuous, open image of a weakly a-favorable space is weakly a-favorable; (4) a quasi-regular space with a cr -disjoint pseudobase is weakly a-favorable if and only if it is pseudo-complete in the sense of J. C. Oxtoby; and (5) the product of a weakly a-favorable space and a Baire space is a Baire space. 1. Introduction. In recent years, a number of classes of topological spaces have been considered, each of which is a subclass of the class of Baire spaces, and each of which is closed under the formation of products (see [l] for a discussion of these classes). The purpose of this note is to show that the class of spaces mentioned in the title has a number of reasonable properties. The author would like to thank D. J. Lutzer for suggesting (in a letter) the desirability of finding a class of spaces satisfying statements which are essentially (2), (A), (5), (6), (7), (8), and (11) of the theorem, and for supplying the author with a copy of [l], which proved very helpful. 2. Definitions. For any function cp, let D(cf>), R(çf>) denote the domain and the range of cp, respectively. For any collection A of sets, let K = K <\. 101. A topological space (X, J ) is called weakly a-favorable (or a-favorable for a player with perfect information) if there is a sequence o - (cp ) of functions such that (2.1) D(</jj)= A*AR(cp1) and<p1((7)c U fot all U in >(<,), (2.2) for all tz in N, Received by the editors November 14, 1973 and, in revised form, April 10, 1974. AMS (MOS) subject classifications (1970). Primary 54E99, 54D99; Secondary 54E25, 54C10. Key words and phrases. Weakly a-favorable, a-favorable, pseudo-complete, ^-disjoint pseudo-base. Copyright 1975, American Mathematical Society 477

478 H. E. WHITE, JR. Dícp ) = \(U.,..., U +.)e(3"*)" + 1: ~ n + 1 I ' ' n + 1 «(^+l)cr.^d<pn+1(l/1,...,l/n D(cpn+l), and (2.3) if (U ) *, is a sequence such that n n c/v a Ui+ic^j('Uv'-'> ^for/=!,, «I, + 1)C(7n+1forall(í71,...,í7n+1)in (2.3.1) ((/,,, (7 ) e D((/j ) for all tí in /V, then fu^ : zz e NS^0. 72 There is an interesting discussion in [3, pp. 115 116] which can be used to give an interpretation, in the language of game theory, of the preceding definition. Any sequence o = (cp ) which satisfies (2.1), (2.2), and (2.3) is called a winning strategy for (X, J ). A sequence (U ) N which satisfies (2.3.1) is called an ö-sequence. A subfamily 9 of J is called a pseudo-base for J if every nonempty element of 9 contains a nonempty element of 9. A pseudo-base 9 is called (7-disjoint if 9 = Ul 9 ' n N\, where each 9 is a disjoint family. 3. Theorem. Suppose (X, 3) is a topological space. (1) If X is weakly a-favorable, then X is a Baire space. (2) If X is locally weakly a-favorable, then X is weakly a-favorable. (3) // X is either pseudo-complete [4] or a-favorable [3], then X is weakly a-favorable. (A) If, for each i in I, X. is weakly a-favorable and m > N, then the m- box product of (X.).. is weakly a-favorable. favorable. (5) // X is weakly a-favorable and U is open in X, then U is weakly a- (6) // X is quasi-regular [A] and weakly a-favorable, and X. is a dense Gg subset of X, then X is weakly a-favorable. (7) If 6 is a continuous, closed, irreducible mapping of (X, A) onto (Y, II), then X is weakly a-favorable if and only if Y is weakly a-favorable. (8) // 6 is a continuous, open mapping of (X, J ) 072Z0 (Y, ll), fltzzi X is weakly a-favorable, then Y is weakly a-favorable. is a Baire (9) // X is a Baire space and (Y, ll) is weakly a-favorable, then X x Y space. (10) // X has a o-disjoint pseudo-base 9, then X is weakly a-favorable if and only if it is a-favorable. If X is also quasi-regular, then X is weakly a-favorable if and only if X is pseudo-complete.

a-favorable TOPOLOGICAL SPACES 479 (11) // X is a T space with a base of countable order [5], then X is weakly a-favorable if and only if there is a dense G g subset which is metrically topologically complete. Remarks, (i) It follows from (10) that the concepts of weakly a-favorable, a-favorable, and pseudo-complete coincide for the class of quasiregular spaces which have dense metrizable subspaces. In particular, they coincide for quasi-regular, semi-metrizable spaces (since every semi-metrizable Baire space has a dense metrizable subspace). (ii) It follows from (A) and (8) that X x Y is weakly a-favorable if and only if both X and Y are weakly a-favorable. (iii) Statement (9) generalizes 4.2 of [2]. The proof given here is shorter and simpler than the proof of 4.2 that is given in [2]. (iv) Statement (11) is very similar to the corollary to Theorem 2.4 of [2]. (v) A generalization of Theorem 2.4 of [2] can be obtained by combining (3), (8), and (10). Proof. The proofs of (1), (2), (3), and (5) are easy and are omitted. The proof of (A) is quite similar to the proof of theorem 7.12(iv) of [3]» and is omitted. (6) Suppose o = (cp ) en isa winning strategy for (X, J ). Since J is quasi-regular, the family Ä of all regular elements of J is a pseudo-base for J ; hence we may assume that \j\r(<f> )' n N\ C J{. Suppose X. = (]\G : n N\, where each G is open in X. Denote the r r u n n * relative topology on X by 3", and define y: AQ * J so that XQ n y(u) = (7 for all U in 3". Define, by induction, a sequence S = (if/ ) which 0 U n n ZV satisfies (2.1) and (2.2) relative to (X, 3 n), and such that it n N and (Í7.,, U ) e D(if/ ), then (G, D y((7 ),, G D y(u )) D(<p ) and xl ' n n I ' 1 ' n ' n ' n (3.1) if) ([/,,, U ) = Xn cp ÍG.n y([/. ), -, G n y(v )). nl ' n Ü rnl ' 1 ' ' n ' n In detail: Suppose 72 > 1, and that, for / = 1,, 72-1,<A- has been defined. Suppose (Uy,---, Un) e (3"*)" and C/.+ 1 C />.(t71, - - -, (7.) for / = 1,,?2-1. Then XnyiU )Cifj Au.,---, u A 0 ' 72 ' 72 1 1 ' 72 1 CcS,(G, n y(//,),---, G. n yiu A). 'n l 1 1 zz 1 rz 1 Since R(c/j _.)C Ji and X is dense in X, y(u ) is contained in ^-l(gin WO* * G -In yiun-l})- Theref re ^ iuv---, UJ can be defined by (3.1).

480 H. E. WHITE, JR. Now, if (U ) m is an d -sequence, then (G C\ y(u ))., is an o- ' v n ners U n ' n n N sequence. Hence Ç\\U : n N\ =C\\G n n y(u ): n N\ 4 0, n n and Sisa u winning strategy for (X, J Q). r. (7) Define 6*: 3"* U* by letting 0*((7) = Y -v 6[X ^ U] for each U in If (X, J ) is weakly a-favorable and ö = (c/j ),, is a winning strategy for (X, J ), then we define, by induction, a sequence b = (if/ ) which satisfies (2.1) and (2.2) relative to (Y, ll) and such that, if n N and (V,,---,V )ed(ifj ), then (0~ l\v, 1,, ^-I!V ]) e D(cp ) and I '72 72 1 ' 72 72 ÍV.,---, V )=6*Í4> Í6-l[V.],-.-, 6-1[V ])). 'ni1 ' n r n i ' ' n If (V )., is an «-sequence, then (6~ [V ])., is an d-sequence, so C\\Vn: zz e N\ = ftcttd-^vj: n N\] 4 0. If (Y, ll) is weakly a-favorable and 0 = (ifj) en is a winning strategy for (Y, ll), then we define, by induction, a sequence d = (cp ) en which satisfies (2.1) and (2.2) and such that, it n N and (U.,- -, U ) D(<f, ), then «fd^), -, 6*(Un)) D(if/n) and <p ((/,,, U )=6-l[ifj Í6*ÍU,),---, 6*ÍU ))]. ~ n 1' 'rz rn 1 n If ('/ ),., is an «-sequence, then (6 (U )),., is an t>.-sequence, so n n N ^ ' x n n N 0 ^ Die/ :zz N DfliÖ-1^^ )]:» enur'fni^u ):«e/v},é0. 72 72 72 (8) Suppose b = ((fy ) is a winning strategy for (X, J ). Define, by induction, sequences vn = (ib )... (y ) ^., of functions such that (a) o satisfies (2.1) and (2.2) relative to (Y, ll), (b) D(yx) = D(ifj J and, if V D(yx), then y1(v)= Ö_1[V] and-7>1(v)= 6[<p y(y y(v))], and (c) for all 72 in N, (n.l) D(yn+l)= D(if/n+1) and R(yn + l)c D(cpn+1), and (n.2) if (V1,.--,Vn + 1) D(yn + 1)andyn(Vv...,Vn)=(Ul,...,Un),then and Yn±i{vv->vn+l) = iuv---,un,6-'[vn+l]ncpn(uv.-.,un)) Now, if (V ) N is an ö -sequence, then there is an d-sequence (Í7 ) CK1 such that 6[U ]=V for all 77 in N. Hence C\\V : n N\ A nn^n n n n 6{fl\Un:n e N\]/0. (9) Suppose o = (zi ) is a winning strategy for (Y, ll) and that

a-favorable TOPOLOGICAL SPACES 481 (G )., is a sequence v n n N L of dense, open subsets of X x Y. It suffices to show that C\\G : zz e N\ 4 0. n We define, by induction, a sequence (y ).. of functions such that, for each 72 in N, (n.l) D(y ) is a disjoint subfamily of J, R(y ) C ll, and H x y (f/) C G for ail // in D(y ), (n.2) D(y, ) refines />(y ), (n.3) if ' n 72 72 72+1 72 tfy e 0(y;) for j = 1,, zz, and fíy + 1 C H. for j = 1, - -, n - 1, then (y.ctf,), ""' yn-hrp ë D(-^J' and ^n-4)ud(y ) is dense in X. In detail: At the 72th step, let A denote the set of all functions y such that (n.l) and (n.3) (and, if 7Z > 1, ((n - 1).2)), with y replaced by y, hold. Order J by inclusion. Since Zorn's lemma is applicable, there is a maximal element y in J. If y does not satisfy (n.4) and 72 > 1, then, for 7 = 1,, 72-1, there is H. in D(y.) such that H.+ lc H. fot j = 1,---, n- 2, and Hn n [X ^ cl[u/)(yn)]] 40. So there are H in 3"*, K in U such that H x K is contained in G«{[^-in[X'X'Cl[UD(y")]]]X0"-l(yiWl),'"'y"-l(f/"-l))}' Then y u (//, K) e A. Therefore y satisfies (n.4). ' n ' n ' n Since X is a Baire space, (\\{jd(cp ): n N\ 4 0. Suppose x (\\KJD(<p ): 72 N\. Then there is a sequence (// ) N such that for each 77 in N, h\ D(yn) and H^ C /^ By ((n.3))^, "(,(»>,.«* is an S- sequence; hence Diy (# ): n N\ 4 0. And, if y e D!y (W ): n N\, then («i y)efllf/ xy?!(//n): 72 e N C f lg : n N\. (10) Suppose d = (cp ) en is a winning strategy for (X, J) and 9 = Ui^ : n N\ where, for each n in N, 9 is a disjoint family and U > is ^^ n 7 ' n ' J n dense in X. We define, by induction, a sequence (y ) en of functions.such that, for each?2 in zv, (n.l) D(y ) is a disjoint subfamily of J that refines 9 and R(yn) C 3"*, (n.2) D(y" + j) refines DQ^), (n.3) if H. D(Yj) for 7 = 1,, 72, and H.+ l C H. for 7= 1,, 77-1, then (y/r/j), -, y^f/^)) e D(r/jn) and <Pn(yl(Hl ),, Y Wn)) = #. and (n.4)ud(y ) is dense in X. In detail: At the 77th step, let y be a maximal element of J, where we r' 'n n' define A verbatim as in (9). If y does not satisfy (n.4) and 72 > 1, then there is P in 9 and, for / = 1,, n - 1, there is H. in D(y ) such that n n ' ' ' ' ' 7 ' j H., C H. foi j = 1,,72-2 and 7+1 7 ' ' K=P«"Hn_ln[x^cl[\jDiyn)JJ40.

482 H. E. WHITE, JR. Then y»uí^1o.i)»-'vi(vi)'jo',f)iey<,- Therefore y satisfies (n.4). Let X0"=n<UD(yn):72e N\ and 9>Q =! H O X0: // e U!D(y ):» 6 /Vi. Then J$0 is a base for a topology J. on X Define the pseudo-metric d on X. by letting dix, y) = inf Ît2~ ': there is H in Diy ) such that x, y H i for all x, y in X Then J is the pseudo-metric topology induced by d. In fact, if H e D(y ) and x H O X n, then \y X.: d(x, y)< n~l\ = ' zz zz zz (r ^ U ' ^ f/ O X And (X d) is complete. For suppose (x ) is a Cauchy sequence in (X, d). We may assume that, for all 72 in N, d(x, x.,)< «~. Then there is a sequence (H ),., such that, for all 72 in N, x H D(y ) ' zz n N n n ' n and Hn + l C tf^. By ((n.3))nen, (yn(hn))n N is an S-sequence; hence fliy (W ): zz /V z 0. But, since y. (//, ) C // for each 72, Hi W : ' zz zz zz+izz+1 zz zz n N\ 4 0 And, if * eol H : n N\, then (x ),.. converges to x. Hence zz n n en " (X,.) ) is both a-favorable and pseudo-complete. Since J _ is a pseudobase for the relative topology J(^0) on^0' ^0' ^0^ *s ^>otn a"favorable and pseudo-complete. Since X. is dense in X, (X, J ) is a-favorable and, if J is quasi-regular, (X, J ) is pseudo-complete. (11) The proof of (11) is very similar to the proof of (10). In fact, if (X, 3) s a T space without any isolated points, and Jo is a base of countable order for J, then we can choose the sequence (9 )... so that \j\ J : ' ^ * n ncn ^ n?2 N\ C 9> and 9 C\ 9 = 0 for all 72 in N. Then it is easily verified n n + i ' that J 0 = 3 (X ); hence X is the required metrically topologically complete dense Gg. REFERENCES 1. J. M. Aarts and D. J. Lutzer, Completeness properties designed for recognizing Baire spaces (to appear). 2. -, Pseudo-completeness and the product of Baire spaces, Pacific J. Math. 48 (1973), 1-10. 3. G. Choquet, Lectures on analysis. I: Integration and topological vector spaces, Benjamin, New York, 1969. MR 40 #3252. 4. J. C. Oxtoby, Cartesian products of Baire spaces, Fund. Math. 49 (1960/61), 157-166. MR 25 #4055; erratum, 26, 1453. 5. H. H. Wicke and J. M. Worrell, Jr., Open continuous mappings of spaces having bases of countable order, Duke Math. J. 34 (1967), 255-272. MR 35 #979. 25I NORTH BLACKBURN ROAD, RT. 5, ATHENS, OHIO 45701