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Topological games and Alster spaces Rodrigo Roque Dias coauthor: Leandro F. Aurichi Instituto de Matemática e Estatística Universidade de São Paulo [email protected] STW 2013, São Sebastião, Brazil In honour of Ofelia T. Alas Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 1 / 14 Topological games The Rothberger game Definition (Rothberger 1938) A topological space X is Rothberger if, for every sequence S(Un )n∈ω of open covers of X , there is a sequence (Un )n∈ω satisfying X = n∈ω Un with Un ∈ Un for all n ∈ ω. Definition (Galvin 1978) The Rothberger game in a topological space X is played according to the following rules: in each inning n ∈ ω, One chooses an open cover S Un of X , and then Two chooses Un ∈ Un ; the play is won by Two if X = n∈ω Un , otherwise One is the winner. Theorem (Pawlikowski 1994) Two ↑ Rothberger(X ) ⇒ One 6 ↑ Rothberger(X ) ⇔ X is Rothberger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 2 / 14 Topological games The Rothberger game Definition (Rothberger 1938) A topological space X is Rothberger if, for every sequence S(Un )n∈ω of open covers of X , there is a sequence (Un )n∈ω satisfying X = n∈ω Un with Un ∈ Un for all n ∈ ω. Definition (Galvin 1978) The Rothberger game in a topological space X is played according to the following rules: in each inning n ∈ ω, One chooses an open cover S Un of X , and then Two chooses Un ∈ Un ; the play is won by Two if X = n∈ω Un , otherwise One is the winner. Theorem (Pawlikowski 1994) Two ↑ Rothberger(X ) ⇒ One 6 ↑ Rothberger(X ) ⇔ X is Rothberger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 2 / 14 Topological games The Rothberger game Definition (Rothberger 1938) A topological space X is Rothberger if, for every sequence S(Un )n∈ω of open covers of X , there is a sequence (Un )n∈ω satisfying X = n∈ω Un with Un ∈ Un for all n ∈ ω. Definition (Galvin 1978) The Rothberger game in a topological space X is played according to the following rules: in each inning n ∈ ω, One chooses an open cover S Un of X , and then Two chooses Un ∈ Un ; the play is won by Two if X = n∈ω Un , otherwise One is the winner. Theorem (Pawlikowski 1994) Two ↑ Rothberger(X ) ⇒ One 6 ↑ Rothberger(X ) ⇔ X is Rothberger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 2 / 14 Topological games The Menger game Definition (Hurewicz 1926) A topological space X is Menger if, for every sequence (US n )S n∈ω of open covers of X , there is a sequence (Fn )n∈ω satisfying X = n∈ω Fn with Fn ∈ [Un ]<ℵ0 for all n ∈ ω. Definition (Telgársky 1984) The Menger game in a topological space X is played as follows: in each inning n ∈ ω, One chooses an open cover UnSof X , and then Two chooses a finite subset Fn of Un ; Two wins the play if n∈ω Fn is a cover of X , otherwise One is the winner. Theorem (Hurewicz 1926) Two ↑ Menger(X ) ⇒ One 6 ↑ Menger(X ) ⇔ X is Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 3 / 14 Topological games The Menger game Definition (Hurewicz 1926) A topological space X is Menger if, for every sequence (US n )S n∈ω of open covers of X , there is a sequence (Fn )n∈ω satisfying X = n∈ω Fn with Fn ∈ [Un ]<ℵ0 for all n ∈ ω. Definition (Telgársky 1984) The Menger game in a topological space X is played as follows: in each inning n ∈ ω, One chooses an open cover UnSof X , and then Two chooses a finite subset Fn of Un ; Two wins the play if n∈ω Fn is a cover of X , otherwise One is the winner. Theorem (Hurewicz 1926) Two ↑ Menger(X ) ⇒ One 6 ↑ Menger(X ) ⇔ X is Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 3 / 14 Topological games The Menger game Definition (Hurewicz 1926) A topological space X is Menger if, for every sequence (US n )S n∈ω of open covers of X , there is a sequence (Fn )n∈ω satisfying X = n∈ω Fn with Fn ∈ [Un ]<ℵ0 for all n ∈ ω. Definition (Telgársky 1984) The Menger game in a topological space X is played as follows: in each inning n ∈ ω, One chooses an open cover UnSof X , and then Two chooses a finite subset Fn of Un ; Two wins the play if n∈ω Fn is a cover of X , otherwise One is the winner. Theorem (Hurewicz 1926) Two ↑ Menger(X ) ⇒ One 6 ↑ Menger(X ) ⇔ X is Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 3 / 14 Topological games The point-open game Definition (Galvin 1978) The point-open game in a topological space X is played as follows: in each inning n ∈ ω, One picks a point xn ∈ X , and then TwoSchooses an open set Un ⊆ X with xn ∈ Un ; the play is won by One if X = n∈ω Un , otherwise Two is the winner. Theorem (Galvin 1978) · One ↑ Rothberger(X ) ⇔ Two ↑ point-open(X ); · Two ↑ Rothberger(X ) ⇔ One ↑ point-open(X ). Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 4 / 14 Topological games The point-open game Definition (Galvin 1978) The point-open game in a topological space X is played as follows: in each inning n ∈ ω, One picks a point xn ∈ X , and then TwoSchooses an open set Un ⊆ X with xn ∈ Un ; the play is won by One if X = n∈ω Un , otherwise Two is the winner. Theorem (Galvin 1978) · One ↑ Rothberger(X ) ⇔ Two ↑ point-open(X ); · Two ↑ Rothberger(X ) ⇔ One ↑ point-open(X ). Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 4 / 14 Topological games What about the Menger game? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14 Topological games What about the Menger game? Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Question (Telgársky 1984) Is Two ↑ compact-open equivalent to One ↑ Menger? Or, equivalently: Does the Menger property imply Two 6 ↑ compact-open? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14 Topological games What about the Menger game? Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Question (Telgársky 1984) Is Two ↑ compact-open equivalent to One ↑ Menger? Or, equivalently: Does the Menger property imply Two 6 ↑ compact-open? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14 Topological games What about the Menger game? Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Question (Telgársky 1984) Is Two ↑ compact-open equivalent to One ↑ Menger? Or, equivalently: Does the Menger property imply Two 6 ↑ compact-open? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14 Topological games Playing with k-covers Definition An open cover U of a topological space X is a k-cover (U ∈ KX ) if every compact subset of X is included in an element of U. Definition The game G1 (K, O) on a topological space X is played as follows: in each inning n ∈ ω, One chooses Un ∈ KX , and then Two chooses Un ∈ Un ; Two S wins the play if X = n∈ω Un , otherwise One is the winner. Proposition (Telgársky 1983, Galvin 1978) The game G1 (K, O) and the compact-open game are dual. Corollary If Two 6 ↑ compact-open(X ), then S1 (KX , OX ) holds — i.e., for every sequence (Un )n∈ω of k-covers ofSX , there is a sequence (Un )n∈ω with Un ∈ Un for all n ∈ ω and X = n∈ω Un . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14 Topological games Playing with k-covers Definition An open cover U of a topological space X is a k-cover (U ∈ KX ) if every compact subset of X is included in an element of U. Definition The game G1 (K, O) on a topological space X is played as follows: in each inning n ∈ ω, One chooses Un ∈ KX , and then Two chooses Un ∈ Un ; Two S wins the play if X = n∈ω Un , otherwise One is the winner. Proposition (Telgársky 1983, Galvin 1978) The game G1 (K, O) and the compact-open game are dual. Corollary If Two 6 ↑ compact-open(X ), then S1 (KX , OX ) holds — i.e., for every sequence (Un )n∈ω of k-covers ofSX , there is a sequence (Un )n∈ω with Un ∈ Un for all n ∈ ω and X = n∈ω Un . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14 Topological games Playing with k-covers Definition An open cover U of a topological space X is a k-cover (U ∈ KX ) if every compact subset of X is included in an element of U. Definition The game G1 (K, O) on a topological space X is played as follows: in each inning n ∈ ω, One chooses Un ∈ KX , and then Two chooses Un ∈ Un ; Two S wins the play if X = n∈ω Un , otherwise One is the winner. Proposition (Telgársky 1983, Galvin 1978) The game G1 (K, O) and the compact-open game are dual. Corollary If Two 6 ↑ compact-open(X ), then S1 (KX , OX ) holds — i.e., for every sequence (Un )n∈ω of k-covers ofSX , there is a sequence (Un )n∈ω with Un ∈ Un for all n ∈ ω and X = n∈ω Un . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14 Topological games Playing with k-covers Definition An open cover U of a topological space X is a k-cover (U ∈ KX ) if every compact subset of X is included in an element of U. Definition The game G1 (K, O) on a topological space X is played as follows: in each inning n ∈ ω, One chooses Un ∈ KX , and then Two chooses Un ∈ Un ; Two S wins the play if X = n∈ω Un , otherwise One is the winner. Proposition (Telgársky 1983, Galvin 1978) The game G1 (K, O) and the compact-open game are dual. Corollary If Two 6 ↑ compact-open(X ), then S1 (KX , OX ) holds — i.e., for every sequence (Un )n∈ω of k-covers ofSX , there is a sequence (Un )n∈ω with Un ∈ Un for all n ∈ ω and X = n∈ω Un . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14 Topological games Playing with k-covers Corollary Two 6 ↑ compact-open ⇒ S1 (K, O) ⇒ Menger. Question (Telgársky 1984) Does the Menger property imply Two 6 ↑ compact-open? We will now partially answer this question in the negative by giving consistent examples of Menger regular spaces that do not satisfy S1 (K, O). Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 7 / 14 Topological games Playing with k-covers Corollary Two 6 ↑ compact-open ⇒ S1 (K, O) ⇒ Menger. Question (Telgársky 1984) Does the Menger property imply Two 6 ↑ compact-open? We will now partially answer this question in the negative by giving consistent examples of Menger regular spaces that do not satisfy S1 (K, O). Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 7 / 14 Topological games Playing with k-covers Corollary Two 6 ↑ compact-open ⇒ S1 (K, O) ⇒ Menger. Question (Telgársky 1984) Does the Menger property imply Two 6 ↑ compact-open? We will now partially answer this question in the negative by giving consistent examples of Menger regular spaces that do not satisfy S1 (K, O). Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 7 / 14 Topological games Playing with k-covers Lemma Let X be a topological space such that every compact subspace of X has an isolated point. Then X satisfies S1 (K, O) if and only if X is Rothberger. Example If cov(M) < d, then any non-Rothberger subspace of R of size cov(M) is Menger but does not satisfy S1 (K, O). Example Any Sierpiński subset of the real line (which exists e.g. under CH) endowed with the Sorgenfrey topology is a Menger regular space that does not satisfy S1 (K, O). Problem Is it consistent with ZFC that every Menger regular space satisfies S1 (K, O)? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 8 / 14 Topological games Playing with k-covers Lemma Let X be a topological space such that every compact subspace of X has an isolated point. Then X satisfies S1 (K, O) if and only if X is Rothberger. Example If cov(M) < d, then any non-Rothberger subspace of R of size cov(M) is Menger but does not satisfy S1 (K, O). Example Any Sierpiński subset of the real line (which exists e.g. under CH) endowed with the Sorgenfrey topology is a Menger regular space that does not satisfy S1 (K, O). Problem Is it consistent with ZFC that every Menger regular space satisfies S1 (K, O)? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 8 / 14 Topological games Playing with k-covers Lemma Let X be a topological space such that every compact subspace of X has an isolated point. Then X satisfies S1 (K, O) if and only if X is Rothberger. Example If cov(M) < d, then any non-Rothberger subspace of R of size cov(M) is Menger but does not satisfy S1 (K, O). Example Any Sierpiński subset of the real line (which exists e.g. under CH) endowed with the Sorgenfrey topology is a Menger regular space that does not satisfy S1 (K, O). Problem Is it consistent with ZFC that every Menger regular space satisfies S1 (K, O)? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 8 / 14 Topological games Playing with k-covers Lemma Let X be a topological space such that every compact subspace of X has an isolated point. Then X satisfies S1 (K, O) if and only if X is Rothberger. Example If cov(M) < d, then any non-Rothberger subspace of R of size cov(M) is Menger but does not satisfy S1 (K, O). Example Any Sierpiński subset of the real line (which exists e.g. under CH) endowed with the Sorgenfrey topology is a Menger regular space that does not satisfy S1 (K, O). Problem Is it consistent with ZFC that every Menger regular space satisfies S1 (K, O)? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 8 / 14 Alster spaces Definition (Alster 1988) A topological space X is Alster if every kδ -cover of X — i.e., every Gδ -cover W of X such that for every compact K ⊆ X there is W ∈ W with K ⊆ W — has a countable subcover. Problem (Tamano) Characterize (internally) the productively Lindelöf spaces, i.e. the spaces X such that X × Y is Lindelöf whenever Y is Lindelöf. Theorem (Alster 1988) Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceeding ℵ1 are Alster. Problem (Alster 1988) Is every productively Lindelöf (regular) space Alster? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 9 / 14 Alster spaces Definition (Alster 1988) A topological space X is Alster if every kδ -cover of X — i.e., every Gδ -cover W of X such that for every compact K ⊆ X there is W ∈ W with K ⊆ W — has a countable subcover. Problem (Tamano) Characterize (internally) the productively Lindelöf spaces, i.e. the spaces X such that X × Y is Lindelöf whenever Y is Lindelöf. Theorem (Alster 1988) Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceeding ℵ1 are Alster. Problem (Alster 1988) Is every productively Lindelöf (regular) space Alster? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 9 / 14 Alster spaces Definition (Alster 1988) A topological space X is Alster if every kδ -cover of X — i.e., every Gδ -cover W of X such that for every compact K ⊆ X there is W ∈ W with K ⊆ W — has a countable subcover. Problem (Tamano) Characterize (internally) the productively Lindelöf spaces, i.e. the spaces X such that X × Y is Lindelöf whenever Y is Lindelöf. Theorem (Alster 1988) Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceeding ℵ1 are Alster. Problem (Alster 1988) Is every productively Lindelöf (regular) space Alster? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 9 / 14 Alster spaces Definition (Alster 1988) A topological space X is Alster if every kδ -cover of X — i.e., every Gδ -cover W of X such that for every compact K ⊆ X there is W ∈ W with K ⊆ W — has a countable subcover. Problem (Tamano) Characterize (internally) the productively Lindelöf spaces, i.e. the spaces X such that X × Y is Lindelöf whenever Y is Lindelöf. Theorem (Alster 1988) Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceeding ℵ1 are Alster. Problem (Alster 1988) Is every productively Lindelöf (regular) space Alster? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 9 / 14 Alster spaces Alster vs. Two ↑ Menger Of course, σ-compact spaces are Alster. Theorem (Aurichi, Tall 2012) Alster spaces are Menger. In view of the fact that σ-compact ⇒ Two ↑ Menger ⇒ Menger, Question (Tall 2013) Does Two ↑ Menger imply Alster? Does the converse hold? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 10 / 14 Alster spaces Alster vs. Two ↑ Menger Of course, σ-compact spaces are Alster. Theorem (Aurichi, Tall 2012) Alster spaces are Menger. In view of the fact that σ-compact ⇒ Two ↑ Menger ⇒ Menger, Question (Tall 2013) Does Two ↑ Menger imply Alster? Does the converse hold? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 10 / 14 Alster spaces Alster vs. Two ↑ Menger Of course, σ-compact spaces are Alster. Theorem (Aurichi, Tall 2012) Alster spaces are Menger. In view of the fact that σ-compact ⇒ Two ↑ Menger ⇒ Menger, Question (Tall 2013) Does Two ↑ Menger imply Alster? Does the converse hold? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 10 / 14 Alster spaces Alster vs. Two ↑ Menger Of course, σ-compact spaces are Alster. Theorem (Aurichi, Tall 2012) Alster spaces are Menger. In view of the fact that σ-compact ⇒ Two ↑ Menger ⇒ Menger, Question (Tall 2013) Does Two ↑ Menger imply Alster? Does the converse hold? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 10 / 14 Alster spaces A selective characterization Proposition A topological space X is Alster if and only if it satisfies S1 (Kδ , Oδ ). Proposition The game G1 (Kδ , Oδ ) and the compact-Gδ game are dual. Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 11 / 14 Alster spaces A selective characterization Proposition A topological space X is Alster if and only if it satisfies S1 (Kδ , Oδ ). Proposition The game G1 (Kδ , Oδ ) and the compact-Gδ game are dual. Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 11 / 14 Alster spaces A selective characterization Proposition A topological space X is Alster if and only if it satisfies S1 (Kδ , Oδ ). Proposition The game G1 (Kδ , Oδ ) and the compact-Gδ game are dual. Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 11 / 14 Alster spaces Alster 0 x 1 Two ↑ Menger Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Corollary If X is regular and Two ↑ Menger(X ), then X is Alster. Example (Telgársky 1983) There is a regular space in which the compact-Gδ game is undetermined — hence a regular Alster space in which Two 6 ↑ Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 12 / 14 Alster spaces Alster 0 x 1 Two ↑ Menger Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Corollary If X is regular and Two ↑ Menger(X ), then X is Alster. Example (Telgársky 1983) There is a regular space in which the compact-Gδ game is undetermined — hence a regular Alster space in which Two 6 ↑ Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 12 / 14 Alster spaces Alster 0 x 1 Two ↑ Menger Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Corollary If X is regular and Two ↑ Menger(X ), then X is Alster. Example (Telgársky 1983) There is a regular space in which the compact-Gδ game is undetermined — hence a regular Alster space in which Two 6 ↑ Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 12 / 14 Alster spaces Alster 0 x 1 Two ↑ Menger Corollary Two 6 ↑ compact-Gδ ⇒ Alster. Theorem (Telgársky 1983-84) Let X be a regular space. TFAE: (a) Two ↑ Menger(X ); (b) One ↑ compact-open(X ); (c) One ↑ compact-Gδ (X ). Corollary If X is regular and Two ↑ Menger(X ), then X is Alster. Example (Telgársky 1983) There is a regular space in which the compact-Gδ game is undetermined — hence a regular Alster space in which Two 6 ↑ Menger. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 12 / 14 How about Two ↑ Rothberger? Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14 How about Two ↑ Rothberger? Theorem Two ↑ Rothberger(X ) ⇒ Xδ is Lindelöf (⇒ X is Rothberger). Sketch of proof. Let σ : <ω τ → X be a winning strategy for One in point-open(X ). Now let W be a cover of X by Gδ subsets. For each T W ∈ W, fix a sequence (U(W , n))n∈ω of open sets with W = n∈ω U(W , n). Proceeding by induction on n ∈ ω, we shall assign to each s ∈ n ω an element Ws of W as follows. First, pick W∅ ∈ W such that σ(∅) ∈ W∅ . Now let n ∈ ω be such that Ws ∈ W has already been defined for all s ∈ n ω. For each s ∈ n ω and each k ∈ ω, choose Ws a k ∈ W satisfying σ(ts,k ) ∈ Ws a k , where ts,k ∈ n+1 τ is the sequence defined by ts,k (i) = U(Wsi , s(i)) for all i < n and ts,k (n) = U(Ws , k). Since the strategy σ is winning, it follows that (...) {Ws : s ∈ <ω ω} ⊆ W is a cover of X . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14 How about Two ↑ Rothberger? Theorem Two ↑ Rothberger(X ) ⇒ Xδ is Lindelöf (⇒ X is Rothberger). Sketch of proof. Let σ : <ω τ → X be a winning strategy for One in point-open(X ). Now let W be a cover of X by Gδ subsets. For each T W ∈ W, fix a sequence (U(W , n))n∈ω of open sets with W = n∈ω U(W , n). Proceeding by induction on n ∈ ω, we shall assign to each s ∈ n ω an element Ws of W as follows. First, pick W∅ ∈ W such that σ(∅) ∈ W∅ . Now let n ∈ ω be such that Ws ∈ W has already been defined for all s ∈ n ω. For each s ∈ n ω and each k ∈ ω, choose Ws a k ∈ W satisfying σ(ts,k ) ∈ Ws a k , where ts,k ∈ n+1 τ is the sequence defined by ts,k (i) = U(Wsi , s(i)) for all i < n and ts,k (n) = U(Ws , k). Since the strategy σ is winning, it follows that (...) {Ws : s ∈ <ω ω} ⊆ W is a cover of X . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14 How about Two ↑ Rothberger? Theorem Two ↑ Rothberger(X ) ⇒ Xδ is Lindelöf (⇒ X is Rothberger). Sketch of proof. Let σ : <ω τ → X be a winning strategy for One in point-open(X ). Now let W be a cover of X by Gδ subsets. For each T W ∈ W, fix a sequence (U(W , n))n∈ω of open sets with W = n∈ω U(W , n). Proceeding by induction on n ∈ ω, we shall assign to each s ∈ n ω an element Ws of W as follows. First, pick W∅ ∈ W such that σ(∅) ∈ W∅ . Now let n ∈ ω be such that Ws ∈ W has already been defined for all s ∈ n ω. For each s ∈ n ω and each k ∈ ω, choose Ws a k ∈ W satisfying σ(ts,k ) ∈ Ws a k , where ts,k ∈ n+1 τ is the sequence defined by ts,k (i) = U(Wsi , s(i)) for all i < n and ts,k (n) = U(Ws , k). Since the strategy σ is winning, it follows that (...) {Ws : s ∈ <ω ω} ⊆ W is a cover of X . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14 How about Two ↑ Rothberger? Theorem Two ↑ Rothberger(X ) ⇒ Xδ is Lindelöf (⇒ X is Rothberger). Sketch of proof. Let σ : <ω τ → X be a winning strategy for One in point-open(X ). Now let W be a cover of X by Gδ subsets. For each T W ∈ W, fix a sequence (U(W , n))n∈ω of open sets with W = n∈ω U(W , n). Proceeding by induction on n ∈ ω, we shall assign to each s ∈ n ω an element Ws of W as follows. First, pick W∅ ∈ W such that σ(∅) ∈ W∅ . Now let n ∈ ω be such that Ws ∈ W has already been defined for all s ∈ n ω. For each s ∈ n ω and each k ∈ ω, choose Ws a k ∈ W satisfying σ(ts,k ) ∈ Ws a k , where ts,k ∈ n+1 τ is the sequence defined by ts,k (i) = U(Wsi , s(i)) for all i < n and ts,k (n) = U(Ws , k). Since the strategy σ is winning, it follows that (...) {Ws : s ∈ <ω ω} ⊆ W is a cover of X . Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14 References • K. Alster, On the class of all spaces of weight not greater than ω1 whose cartesian product with every Lindelöf space is Lindelöf, Fund. Math. 129 (1988), 133–140. • L. F. Aurichi and R. R. Dias, Topological games and Alster spaces, arXiv:1306.5463 [math.GN] • L. F. Aurichi and F. D. Tall, Lindelöf spaces which are indestructible, productive, or D, Topology Appl. 159 (2012), 331–340. • F. Galvin, Indeterminacy of point-open games, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 26 (1978), 445–449. • F. D. Tall, Productively Lindelöf spaces may all be D, Canad. Math. Bull. 56 (2013), 203–212. • R. Telgársky, Spaces defined by topological games, II, Fund. Math. 116 (1983), 189–207. • R. Telgársky, On games of Topsøe, Math. Scand. 54 (1984), 170–176. Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 14 / 14