Some New Star-Selection Principles in Topology

Filomat 31:13 (2017), 4041–4050 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1713041B University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat

Prasenjit Bala, Subrata Bhowmika

aDepartment of Mathematics, Tripura University Suryamaninagar, Tripura, India 799022

Abstract. Motivated by the recent works of Kocinacˇ who initiated investigation of star selection principles, we introduce and study some new types of star-selection principles. Also some open problems are posed.

1. Introduction and Preliminaries

Classical selection principles, based on the diagonalization arguments, have a long history going back to the works by Borel [2], Menger [12], Hurewicz [5], Rothberger [13], and others. Scheepers [16] be- gan a systematic investigation of selection principles, which motivated a large number of researchers for investigation on selection principles and their applications. <ω ω Throughout the paper, [X] (respectively [X]≤ ) will denote the collection of all finite (respectively countable) subsets of a set X. Let and be collections of families of subsets of an infinite set X. A B In 1925, Hurewicz [5] introduced two selection principles (in notation from [16]) f in( , ) (derived S A B from a property introduced by Menger [12]) and f in( , ). U A B f in( , ) denotes the following selection hypothesis : S A B For each sequence An : n ω of elements of , there is a sequence Bn : n ω of finite sets { ∈ } S A { ∈ } such that for each n ω, Bn An and Bn : n ω . ∈ ⊂ { ∈ } ∈ B f in( , ) denotes the following selection hypothesis : U A B For each sequence An : n ω of elements of , there is a sequence Bn : n ω of finite sets { ∈ } AS { ∈ } such that for each n ω, we have Bn An and Bn : n ω . ∈ ⊂ { ∈ } ∈ B In 1938, Rothberger [13] introduced the selection principle ( , ). S1 A B ( , ) denotes the following selection hypothesis : S1 A B For each sequence An : n ω of elements of , there is a sequence Bn : n ω such that for { ∈ } A { ∈ } each n ω, we have Bn An and Bn : n ω . ∈ ∈ { ∈ } ∈ B Scheepers [15] mentioned the selection principle ctbl( , ) as a natural companion of the above selection S A B principles, where ctbl( , ) denotes the following selection hypothesis: S A B

2010 Mathematics Subject Classiﬁcation. Primary 54D20; Secondary 37F20, 91A44 Keywords. Selection principles, star-Menger space, topological games Received: 08 March 2016; Accepted: 01 August 2016 Communicated by Ljubisaˇ D.R. Kocinacˇ Email addresses: [email protected] (Prasenjit Bal), [email protected] (Subrata Bhowmik) P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4042

For each sequence An : n ω of elements of , there is a sequence Bn : n ω such that for { ∈ } SA { ∈ } each n ω, Bn is a countable subset of An and n ω Bn . ∈ ∈ ∈ B By a space, we mean a topological space and for different notions in topology we follow [4]. For a set X, let be a collection of subsets of X and A X; then star of A with respect to is denoted U S ⊂ U and defined by St(A, ) = U : U A , . For x X, we write St(x, ) instead of St( x , ). A space X is said to beU star-compact{ ∈ Uif for every∩ open∅} cover∈ of X there existsU a finite set A {X}suchU that St(A, ) = X [3, 11]. A space X is said to be star-LindelöfUif for every open cover of X there⊂ exists a countableU set A X such that St(A, ) = X [3, 11]. From the above definitions, itU is clear that every star-compact space⊂ is star-Lindelof,¨ butU the converse is not necessarily true [1]. In 1999, Kocinacˇ [6, 7] introduced the following selection principles in connection with the star operator.

∗ ( , ) denotes the following selection hypothesis: S1 A B For each sequence n : n ω of elements of , there exists a sequence Un : n ω such that {U ∈ } A { ∈ } for each n ω, Un n and St(Un, n): n ω . ∈ ∈ U { U ∈ } ∈ B ∗ ( , ) denotes the following selection hypothesis: S f in A B

For each sequence n : n ω of elements of , there exists a sequence n : n ω such that {U ∈ } SA {V ∈ } for each n ω, n is a ﬁnite subset of n and n ω St(V, n): V n . ∈ V U ∈ { U ∈ V } ∈ B ∗ ( , ) denotes the following selection hypothesis: U f in A B

For every sequence n : n ω of members of , there exists a sequence n : n ω such that {U ∈ } A {V ∈ } for each n ω, n is a finite subset of n and St( n, n): n ω . ∈ V U { ∪V U ∈ } ∈ B Song [18–22], Kocinacˇ [6–10], Sakai [14], Tsaban [23] and many others have made investigations on these selection principles and interesting results have been obtained. Let be a family of subsets of a space X. Then: K ∗ ( , ) represents the following selection hypothesis: SSK A B For every sequence n : n ω of elements of , there exists a sequence Kn : n ω of elements {U ∈ } A { ∈ } of such that St(Kn, n): n ω (see [6]). K { U ∈ } ∈ B When is the collection of all one-point [resp., finite, compact] subsets of X, we write ∗ ( , ) [resp., K SS1 A B ∗f in( , ), comp∗ ( , )] instead of ∗ ( , ) (see [6]). SSNowA letB usSS mentionA theB definitionsSS of theK A gamesB which are naturally associated to the selection principles mentioned above. G f in( , ) denotes an infinitely long game for two players, ONE and TWO, who play a round for each A B non-negative integer. In the n-th round ONE chooses a set An , and TWO responds by choosing a finite ∈ A S set Bn An. The play A0, B0, A1, B1, ..., An, Bn, ... is won by TWO if n ω Bn ; otherwise, ONE wins (see [15]). ⊂ { } ∈ ∈ B G ( , ) denotes an infinitely long game for two players, ONE and TWO, who play a round for each 1 A B non-negative integer. In the n-th round ONE chooses a set An , and TWO responds by choosing an ∈ A element bn An. The play A0, b0, A1, b1, ..., An, bn, ... is won by TWO if bn : n ω ; otherwise, ONE wins (see [15]).∈ { } { ∈ } ∈ B G∗ ( , ) denotes an infinitely long game for two players, ONE and TWO, who play a round for each 1 A B non-negative integer. In the n-th round ONE chooses a set n , TWO responds by choosing an element U ∈ A Un n. The play 0, U0, 1, U1, ..., n, Un, ... is won by TWO if St(Un, n): n ω ; otherwise, ONE∈ winsU (see [6]). {U U U } { U ∈ } ∈ B G∗ ( , ) denotes an infinitely long game for two players, ONE and TWO, who play a round for each f in A B non-negative integer. In the n-th round ONE chooses a set n , and then TWO responds by choosing a U ∈ A S finite set n n. The play 0, 0, 1, 1, ..., n, n, ... is won by TWO if n ω St(V, n): V n ; otherwise,V ONE⊂ U wins (see [6]).{U V U V U V } ∈ { V ∈ V } ∈ B P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4043

If X is a space, then G∗ ( , ) denotes an infinitely long game for two players, ONE and TWO, who S 1 A B play a round for each non-negative integer. In the n-th round ONE chooses a set n , TWO responds by U ∈ A choosing an element xn X. The play 0, x0, 1, x1, ..., n, xn, ... is won by TWO if St(xn, n): n ω ; otherwise, ONE wins see∈ (see [6]). {U U U } { U ∈ } ∈ B If X is a space, then G∗ ( , ) denotes an infinitely long game for two players, ONE and TWO, S f in A B who play a round for each non-negative integer. In the n-th round ONE chooses a set n , TWO U ∈ A responds by choosing an finite subset Fn X. The play , F , , F , ..., n, Fn, ... is won by TWO if ⊂ {U0 0 U1 1 U } St(Fn, n): n ω ; otherwise, ONE wins (see [6]). { U ∈ } ∈ B If X is a space, then G∗ ( , ) denotes an infinitely long game for two players, ONE and TWO, S comp A B who play a round for each non-negative integer. In the n-th round ONE chooses a set n , TWO U ∈ A responds by choosing an compact subset Kn X. The play , K , , K , ..., n, Kn, ... is won by TWO if ⊂ {U0 0 U1 1 U } St(Kn, n): n ω ; otherwise, ONE wins (see [6]). { U ∈ } ∈ B

2. New Selection Principles

In this section we introduce two selection principles in connection with the star operator : ∗ ( , ) U1 A B and ∗ f in( , ). U A B

Deﬁnition 2.1. ∗ ( , ) denotes the following selection principle: U1 A B For each sequence n : n ω of elements of , there exists a sequence Un : n ω such that {U ∈ } S A { ∈ } for each n ω, Un n and St( i ω Ui, n): n ω . ∈ ∈ U { ∈ U ∈ } ∈ B

Deﬁnition 2.2. ∗ f in( , ) denotes the following selection principle: U A B For each sequence n : n ω of elements of , there exists a sequence n : n ω such that {U ∈ } A S S {V ∈ } for each n ω, n is a ﬁnite subset of n and St( i ω( i), n): n ω . ∈ V U { ∈ V U ∈ } ∈ B

Proposition 2.3. ∗ ( , ) ∗ f in( , ). U1 A B ⇒ U A B Proposition 2.4. If and are two collections of families of subsets of an inﬁnite set X such that , then A B A ⊂ B

∗ ( , ) ∗ ( , ) U1 B B ⇒ U1 A B

∗ ( , ) ∗ ( , ) U1 A A ⇒ U1 A B ∗ ( , ) ∗ ( , ) U1 B A ⇒ U1 A A ∗ ( , ) ∗ ( , ) U1 B A ⇒ U1 B B

Proof. Let n : n ω be a sequence of elements of . But . Therefore n : n ω is a sequence of {U ∈ } A A ⊂ B {U ∈ } elements of . Since ∗ ( , ) holds, there exists a sequence Un : n ω such that Un n for each n ω S B U1 B B { ∈ } ∈ U ∈ and St( i ω Ui, n): n ω . Therefore, ∗ 1( , ) holds. { ∈ U ∈ } ∈ B U A B Let n : n ω be a sequence of elements of . Since ∗ ( , ) holds, there exists a sequence {U ∈ } A S U1 A A Un : n ω such that Un n for each n ω and St( i ω Ui, n): n ω . But . Thus, { S ∈ } ∈ U ∈ { ∈ U ∈ } ∈ A A ⊂ B St( i ω Ui, n): n ω . Therefore, ∗ 1( , ) holds. { ∈ U ∈ } ∈ B U A B Let n : n ω be a sequence of elements of . But . Therefore, n : n ω is a sequence of {U ∈ } A A ⊂ B {U ∈ } elements of . Since ∗ ( , ) holds, there exists a sequence Un : n ω such that Un n for each n ω S B U1 B A { ∈ } ∈ U ∈ and St( i ω Ui, n): n ω . Therefore, ∗ 1( , ) holds. { ∈ U ∈ } ∈ A U A A Let n : n ω be a sequence of elements of . Since ∗ ( , ) holds, there exists a sequence {U ∈ } B S U1 B A Un : n ω such that Un n for each n ω and St( i ω Ui, n): n ω . But . Thus, { S ∈ } ∈ U ∈ { ∈ U ∈ } ∈ A A ⊂ B St( i ω Ui, n): n ω . Therefore, ∗ 1( , ) holds. { ∈ U ∈ } ∈ B U B B So, we conclude that the selection principle ∗ 1( , ) is monotonic in the second collection and is anti-monotonic in the ﬁrst collection. U A B P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4044

Figure 1: Monotonicity of ( , ) and ( , ). ∗U1 A B ∗U f in A B

Proposition 2.5. ∗ f in( , ) is monotonic in the second collection and anti-monotonic in the first collection. U A B Proof. The proof of this proposition is similar to the proof of Proposition 2.4, so omitted. In this paper, we emphasize on the cases where and are the classes of topologically significant open covers of a space X: A B - the collection of all open covers of X. ΛO - the collection of all large covers of X. An open cover of X is a large cover if each x X belongs to infinitely many members of . U ∈ Ω - the collection of all ωU-covers of X. An open cover of X is an ω-cover if every finite subset of X is contained in a member of . U Γ - the collection of all Uγ-covers of X. An open cover of X is a γ-cover if it is infinite, and each x X belongs to all but finitely many elements of . U ∈ 1p - the collection of all groupable open-coversU of X. An open cover of X is groupable if it can be O U expressed as a countable union of finite, pairwise disjoint subfamilies of n, n ω, such that each x X S U ∈ ∈ belongs to n for all but finitely many n. If the coversU are considered to be non-trivial then we have, Γ Ω Λ . Under such condition, we have the following relation diagram (Figure 2): ⊂ ⊂ ⊂ O

Figure 2: Relation Chart 1

Proposition 2.6. Every star-Lindel¨ofspace has the property ∗ ( , ). U1 O O Proof. Let X be a star-Lindelof¨ space and n : n ω be a sequence of open covers of X. Since is {U ∈ } U0 an open cover of X and X is star-Lindelof,¨ there exists a countable set, x , x , x , ..., xn, ... X such that { 0 1 2 } ⊂ St( x , x , x , ..., xn, ... , ) = X. { 0 1 2 } U0 P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4045 S For each n ω, we select Un n such that xn Un. Clearly x0, x1, x2, ..., xn, ... n ω Un. Therefore S ∈ S∈ U ∈ { } ⊂ ∈ St( n ω Un, 0) = X. Thus St( n ω Un, n): n ω is an open cover of X, i.e. ∗ 1( , ) holds for X. Hence the theorem.∈ U { ∈ U ∈ } U O O

Corollary 2.7. Compact spaces, star-compact spaces and Lindelöfspaces have the property ∗ ( , ). U1 O O Example 2.8. The converse of Proposition 2.6 is not necessarily true, i.e. there exists a space which has the property ∗ 1( , ) but is not star-Lindelof.¨ U O O + + T T Consider the space X = R R Q ]. Let A = [0, 1] ([0, 1] Q). For each x A, Ax = (n + x): n S S \{ S \ ∈ { ∈ ω A. Set Y = Ax : x A A , and define τ(X) = B : B P(Y) . τ(X) is a topology on X. } { ∈ } { } { ∈ } Now, let n : n ω be a sequence of open covers of X. Therefore, 0 is an open cover of X. By the construction{U of the space,∈ } every open set other than contains A. We selectU U such that A U and ∅ 0 ∈ U0 ⊂ 0 for i ω 0 , select Ui i . ∈ \{ } ∈ U S S Since St(U0, 0) = X, St( i ω Ui, 0) = X, so that the set St( i ω Ui, n): n ω is an open cover of X. U ∈ U { ∈ U ∈ } Hence, X has the property ∗ ( , ). U1 O O On the other hand, = Ax : x A is an uncountable open cover of X. For each x, y A, x , y, T S U { S ∈ } ∈ Ax Ay = A and x A Ax = X, but i ω Axi ( X for any countable set xi i ω A. ∈ ∈ { } ∈ ⊂ Let F = yi i ω X. For each yi X, there exists a xi A such that yi Axi . Therefore, St(F, ) = S { } ∈ ⊂ ∈ ∈ ∈ U i ω(Axi ) , X. We find X, not star-Lindelof¨ eventhough it has the property ∗ 1( , ). ∈ U O O We obtain the following diagram of implication and non-implication from the above results:

Figure 3: Relation Chart 2

Proposition 2.9. ∗ ( , ) ∗ ( , ). S1 A O ⇒ U1 A O

Proof. Let n : n ω be a sequence of elements of . Since ∗ ( , ) holds, there exists a sequence {U ∈ } A S1 A B Un : n ω such that for each n ω, Un n and St(Un, n): n ω . Hence, St(Un, n): n ω is an{ open∈ cover} for X. ∈ ∈ U { U ∈ } ∈ O { U ∈ } S S We have, for each n ω, St(Un, n) St( i ω Ui, n). Therefore, St( i ω Ui, n): n ω is also an ∈ U ⊂ ∈ U { ∈ U ∈ } open cover for X. Thus ∗ ( , ) holds. U1 A O But ∗ ( , ) ; ∗ ( , ) in general. This follows from the example given below. U1 O O S1 O O Example 2.10. Let X = (0, 3] R. We consider the topology τ(X) = (x, y]: x, y [0, 3) and x < y , X , the upper limit topology on X⊂induced from the upper limit topology{ of R. ∈ } ∪ {∅ } We construct a sequence of open covers of X as follows:

= (0, 1], (1, 2], (2, 3] , U0 { } 1 1 2 2 3 3 4 4 5 5 6 = 0, , , , , , , , , , , , U1 2 2 2 2 2 2 2 2 2 2 2 P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4046 1 1 2 2 3 11 12 = 0, , , , , , ..., , , U2 22 22 22 22 22 22 22 ...... 1 1 2 2 3 3.2n 1 3.2n n = 0, , , , , , ..., − , , U 2n 2n 2n 2n 2n 2n 2n ...... 1 For each n ω, length of each interval contained in n is n . Also, for each n ω, n is a pairwise disjoint ∈ U 2 ∈ U collection of open sets. Choose Un n, then we have St(Un, n) = Un. 1 ∈ U U So, length of St(Un, n) = n , for each n ω. If St(Un, n) covers diﬀerent portions of X for each n ω, U 2 ∈ U ∈ it will cover a length of X. The maximum length of the subset of X covered by St(Un, n): n ω is { U ∈ } 2 3 1 X 1 1 1 1 1 1 1 1 − = + + + ... = 1 + + + + ... = 1 = 2. 2n 20 1 22 2 2 2 2 n ω 2 − ∈

But length of X is 3. Hence, St(un, n): n ω can not be a cover of X. So, it is not possible to ﬁnd a { U ∈ } sequence Un : n ω such that for each n ω, Un n and St(Un, n): n ω is an open cover for X. { ∈ } ∈ ∈ U { U ∈ } This implies that X does not have the property 1∗ ( , ). We have R with the upper limit topology isS hereditarilyO O Lindelof,¨ hence X is a Lindelof¨ space. Thus, by Corollary 2.7, X has the property ∗ ( , ). U1 O O

Proposition 2.11. ∗ ( , ) ∗ f in( , ). U f in A O ⇒ U A O Proof. The proof is similar to that of Proposition 2.9, so omitted In view of the above results, we have the following relation diagram (Figure 4):

Figure 4: Relation Chart 3

Problem 2.12. Does there exists a space which has the property ∗ f in( , ) but does not have the property ∗ ( , ). U O O U f in O O

Proposition 2.13. If a space X is compact, then it has the property ∗ f in( , ). U O O <ω Proof. Let n : n ω be a sequence of open covers for X. Since X is compact, there exists n [ n] {U ∈ } A ∈ U for each n ω, such that n is a cover for X. Let x X be an arbitrary point. For each n ω, there exists ∈ A ∈S S ∈ An n n such that x An n. So, x An n An ( n) , , for each n ω. x ∈ A ⊂ U S ∈ x ∈ U ∈ x ⊂ A ⇒ S x ∩S A ∅ ∈ So, x Anx St( n, n), for each n ω, i.e. x St( i ω( i), n), for each n ω. Therefore, S S∈ ⊂ A U ∈ ∈ ∈ A U ∈ St( i ω( i), n): n ω is an open cover for X. Hence X has the property ∗ f in( , ). { ∈ A U ∈ } U O O P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4047

Proposition 2.14. If f : X Y is a continuous surjection and X has the property ∗ ( , ), then Y also has the → U1 O O property ∗ ( , ). U1 O O 1 Proof. Let n : n ω be a sequence of open covers for Y. For each n ω, let n = f − (V): V n is a {V ∈ } ∈ U { 1 ∈ V } sequence of open covers of X. Since, X has the property ∗ 1( , ), there exists a sequence f − (Vn): n ω 1 U O O S 1 { ∈ } where Vn n for all n ω such that f − (Vn) n, for each n ω and St( i ω f − (Vi), n): n ω is an open cover∈ for V X. ∈ ∈ U ∈ { ∈ U ∈ } S 1 Let y Y be an arbitrary point. Then, there exists x X such that f (x) = y. Thus, x St( i ω f − (Vi), m) ∈ 1 ∈ 1 1 ∈ S ∈ 1 U for some m ω. Therefore, there exists a f − (Vy) m such that x f − (Vy) and f − (Vy) ( i ω f − (Vi)) , . ∈ 1 T 1 ∈ U T∈ ∩ ∈ ∅ So, y Vy m and f − (Vy) f − (Vi) , for some i ω. i.e. Vy Vi , . ∈ ∈ V S S ∅ ∈ S ∅ Thus, Vy ( i ω Vi) , . ∴ y St( i ω Vi, m). Hence St( i ω Vi, n): n ω is an open cover for Y. This completes∩ the∈ proof of∅ the theorem.∈ ∈ V { ∈ V ∈ } In a similar way we prove the following result.

Proposition 2.15. If f : X Y is a continuous surjection and X has the property ∗ f in( , ), then Y also has the → U O O property ∗ f in( , ). U O O

Corollary 2.16. If the product of two spaces belongs to the class ∗ ( , ), then each of them belongs to the class U1 O O ∗ ( , ). Similarly, if the product of two spaces belong to the class ∗ f in( , ), then each of them belongs to the U1 O O U O O class ∗ f in( , ). U O O

Problem 2.17. Does there exist spaces which have the property ∗ ( , ) but their product do not have the property. U1 O O

Proposition 2.18. Let , and are any collection of subsets of X and if is a cover for X. If ∗ ( , ) and A B C C U1 A B ∗ ( , ) holds, then X . U1 B C { } ∈ C

Proof. n : n ω be a sequence of elements of . Since ∗ ( , ) holds, there exists a sequence {U ∈ } AS U1 A B Un : n ω such that for each n ω, Un n and St( i ω Ui, n): n ω . { ∈ } S ∈ ∈ U { ∈ U ∈ } ∈ B Suppose Vn = St( i ω Ui, n) for each n ω and = Vn : n ω . Now, choose a sequence n : n ω ∈ U ∈ V { ∈ } {V ∈ } such that n = , for each n ω. Then n : n ω is sequence of elements of . Since ∗ 1( , ) holds, V V 0 ∈ {V ∈ } 0 S B 0 U B C there exists a sequence Vn : n ω such that for each n, Vn n = and St( i ω Vi , n): n ω . We have { ∈ } ∈ V V { ∈ V ∈ } ∈ C           [    [  [  St  V0 ,  : n ω St  V0 ,  St  V0 ,  = X,   i V ∈  ∈ C ⇒   i V ∈ C ⇒  i V  i ω   i ω  i ω ∈ ∈ ∈ i.e. X . { } ∈ C k Theorem 2.19. If X have the property ∗ ( , ) for any ﬁnite k, then X has the property ∗ f in( , Ω). U1 O O U O S S S Proof. Let n : n ω be a sequence of open covers of X and let ω = N N N .... be a countable {U ∈ } 1 2 3 partition of ω into countable subsets. For each k ω and each m Nk, let m = U1 U2 ... Uk : ∈ ∈ kW { × × × U1, U2, U3, ..., Uk m . Then m : m Nk is a sequence of open covers of X . ∈ U } {Wk ∈ } Since ∗ 1( , ) holds for X , we can choose a sequence Hm : m Nk such that for each m, Hm m SU O O k { ∈ } ∈ W and St( i N Hi, m): m Nk is an open cover of X . For every m Nk and Hm m. Let, Hm = { ∈ k W ∈ } ∈ ∈ W U1(Hm) U2(Hm) U3(Hm) ... Uk(Hm), where Ui(Hm) m for i k. × × × × ∈ U ≤ s Let F = x , x , x , ...xs be a ﬁnite subset of X. Then (x , x , x , ..., xs) X , so there exists n Ns such { 1 2 3 }S 1 2 3 ∈ ∈ that (x1, x2, x3, ..., xs) St( i Ns Hi, n), where Hi i and i Ns. So, there exists a W n such that ∈ T∈ S W ∈ W ∈ ∈ W (x1, x2, x3, ..., xs) W and W ( i Ns Hi) , . Let W = U1(W) U2(W) ... Us(W). Ui(W) n, i s. ∈ ∈ ∅ × × × T S∈ U ≤ Thus x1 U1(W), x2 U2(W), ..., xs Us(W) and (U1(W) U2(W) ... Us(W)) ( i Ns Hi) , . i.e. ∈ ∈ T S ∈ × × × ∈ ∅ (U1(W) U2(W) ... Us(W)) ( i Ns (U1(Hi) U2(Hi) U3(Hi) ... Us(Hi))) , which implies (U1(W) × × T× S ∈ S × ×S × × S ∅ × U2(W) ... Us(W)) (( i Ns U1(Hi)) ( i Ns U2(Hi)) ( i Ns U3(Hi))... ( i Ns Us(Hi))) , . × × ∈ T S × ∈ × ∈ × ∈ T S Ss∅ Thus, for each j s, Uj(W) ( i Ns Uj(Hi)) , . Hence, for each j s, Uj(W) ( i Ns ( j=1 Uj(Hi))) , . ≤ ∈ ∅ ≤ ∈ ∅ P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4048

The set U1(W), U2(W), ..., Us(W) = n is a finite subset of n and for each j s, xj Uj(W) S Ss{ } V S US ≤ S∈ S ⊂ St(( i Ns ( j=1 Uj(Hi))), n), i.e. for each j s, xj Uj(W) St(( i Ns ( i)), n). Thus F St(( i Ns ( i)), n), ∈ S S U ≤ ∈ ⊂ ∈ V U ⊂ ∈ V U i.e. F St(( i ω( i)), n). ⊂ ∈ V U For each n, n is a finite subset of n satisfying: for each finite set F X there is an n such that S S V S S U ⊂ F St(( i ω( i)), n) and St( i ω( i), n): n ω Ω. This implies that X satisfies ∗ f in( , Ω). ⊂ ∈ V U { ∈ V U ∈ } ∈ U O T Note 2.20. For a finite collection of open covers i : i = 1, 2, 3, ..., n we define i : i = 1, 2, 3, ..., n = {U } {U } U U U ... Un : U , U , U , ..., Un n . { 1 ∩ 2 ∩ 3 ∩ ∩ 1 ∈ U1 2 ∈ U2 3 ∈ U3 ∈ U } 1p Theorem 2.21. If a space has the property ∗ ( , Γ), then it has the property ∗ ( , ). U1 O U1 O O

Proof. Let n : n ω be a sequence of open covers of X. We construct new open covers follows. {U ∈ } ( ) \ n(n + 1) (n + 1)(n + 2) n = i : i < , for each n ω. V U 2 ≤ 2 ∈

So, n : n ω is also a sequence of open covers of X. Since X has the property ∗ ( , Γ), we can find a {V ∈ } U1 O sequence Wn : n ω such that Wn n for each n ω and every x X belongs to all but finitely many { S∈ } ∈ V ∈ ∈ members of St( i ω Wi, n): n ω . { ∈ V ∈ } i(i+1) (i+1)(i+2) For each i ω, Wi Uj, for some Uj j with 2 j < 2 . We consider the set of non-negative ∈ ⊂ ∈ U p p ≤ integers n < n < ... < n < ... defined by n = ( +1) . 0 1 pS p 2 S If x X belongs to St( i ω Wi, k) for some k ω, then x belongs to St( i ω Wi, l), for each l such that ∈ S ∈ SV ∈ ∈ U nk l < nk+1. i.e. x n l

3. Topological Games Related to ∗ ( , ) and ∗ ( , ) U1 A B Ufin A B In this section, we introduce two topological games which are naturally associated with the selection principles introduced in Section 2. The game related to the selection principle ∗ ( , ) is denoted by U1 A B ∗G ( , ) and the game related to the selection principle ∗ f in( , ) is denoted by ∗G f in( , ). 1 A B U A B A B Two games, say P and P0 , are equivalent if: ONE has a winning strategy in P if, and only if, ONE has a winning strategy in P0 , and TWO has a winning strategy in P if, and only if, TWO has a winning strategy in P0 [17]. Two games, P and P0 , are dual if: ONE has a winning strategy in P if, and only if, TWO has a winning strategy in P0 , and TWO has a winning strategy in P if, and only if, ONE has a winning strategy in P0 [17]. Let and be collections of a families of subsets of a set X. A B

Deﬁnition 3.1. ∗G ( , ) denotes an inﬁnitely long game for two players, ONE and TWO, who play a 1 A B round for each non-negative integer. In the n-th round ONE chooses n , TWO responds by choosing U ∈ A S an element Un n. The play 0, U0, 1, U1, ..., n, Un, ... is won by TWO if St( i ω Ui, n): n ω ; otherwise, ONE∈ wins.U {U U U } { ∈ U ∈ } ∈ B

Deﬁnition 3.2. ∗G f in( , ) denotes an inﬁnitely long game for two players, ONE and TWO, who play a A B round for each non-negative integer. In the n-th round ONE chooses n , TWO responds by choosing <ω U ∈ AS S n [ n] . The play 0, 0, 1, 1, ..., n, n, ... is won by TWO if St( i ω( i), n): n ω ; otherwise,V ∈ U ONE wins. {U V U V U V } { ∈ V U ∈ } ∈ B

Proposition 3.3. If for a space X, TWO has a winning strategy in the game G∗ ( , ), then TWO has a winning 1 O O strategy in ∗G ( , ). 1 O O P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4049

Proof. Suppose TWO has a winning strategy σ in the game ∗G ( , ). Use σ to define then a strategy ϕ for 1 O O TWO in the game G∗ ( , ) on X. Suppose that first move of ONE in the game ∗G ( , ) is an open cover 1 O O 1 O O of X. If TWO responds in G∗ ( , ) by σ( ) = U , then TWO plays ϕ( ) = σ( ) = U . Assume U1 1 O O U1 1 ∈ U1 U1 U1 1 that then ONE plays 2 in the game G1∗ ( , ), and TWO responds by σ( 1, 2) = U2, then TWO plays ϕ( , ) = U . AndU so∈ on. O O O U U U1 U2 2 As σ is a winning strategy for TWO in the game G∗ ( , ), consider a σ-play 1 O O , σ( ); , σ( , ), ... U1 U1 U2 U1 U2 won by TWO, i.e. [ St (Ui, i) = X. U i ω ∈ By the definition of ϕ, the ϕ-play

, ϕ( ); , ϕ( , ), ... U1 U1 U2 U1 U2 is won by TWO, since   [ [  [ St  Ui, i St (Ui, i) .  U  ⊃ U i ω i ω i ω ∈ ∈ ∈

Now we show that there exists a space X in which TWO has a winning strategy in the game ∗G ( , ), 1 O O but no winning strategy in G∗ ( , ). 1 O O Example 3.4. Consider the space X constructed in Example 2.10. This space is Lindelof¨ because the real line R with upper limit topology is a hereditarily Lindeof¨ space. Suppose ONE and TWO are playing the game G ( , ) and ONE chooses for the 0-th innings. Clearly there exists [ ] ω such that S ∗ 1 0 0 ≤ X = . SupposeO O = W , W , WU , .... . TWO, according to his strategy σ,W responds ∈ U by choosing W W { 0 1 2 } σ( ) = W . After that for n-th innings (n ω 0 ), whenever ONE chooses n , TWO U0 0 ∈ W ⊂ U0 ∈ \{ } U ∈ O responds by choosing a Un n such that Un Wn , . We observe that Wn St(Un, ) for each n ω. S ∈ U ∩ ∅S S ⊂ U0 ∈ So, Wn St( i ω Ui, 0) for each n ω. Therefore, X = St( i ω Ui, 0),. which means that σ is a ⊂ ∈ U ∈ {W} ⊂ ∈ U winning strategy for TWO in the game ∗G ( , ). 1 O O Now suppose ONE and TWO are playing the game G1∗ ( , ) on the same space X. If ONE chooses n 1 i 1 2 i 2 3 i 3.22 1 3.2n io O O n = 0, n , n , n , n , n , ..., n− , n for each inning (i.e. for each n ω), then for any choice U 2 2 2 2 2 2 2 ∈ O ∈ Un n by TWO, St(Un, n): n ω < . So, TWO does not have a winning strategy in G∗ ( , ). ∈ U { U ∈ } O 1 O O

From the above results we conclude that the games ∗G1( , ) and G1∗ ( , ) are neither equivalent nor dual to each other, which intern reﬂects the signiﬁcance of ourO study.O O O

Proposition 3.5. If for a space X, TWO has a winning strategy in ∗G ( , ), then TWO has a winning strategy in 1 A B ∗G f in( , ). A B

Proposition 3.6. If for a space X, TWO has no winning strategy in ∗G f in( , ), then TWO has no winning strategy A B in ∗G ( , ). 1 A B

Acknowledgements

The authors are thankful to Lj.D.R. Kocinacˇ who helped us in many ways to understand the selection principles. The authors are also thankful to the reviewer for his valuable suggestions regarding the improvement of the paper. P. Bal, S. Bhowmik / Filomat 31:13 (2017), 4041–4050 4050

References

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