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

The Rothberger game

Definition (Rothberger 1938)

A topological space X is Rothbergerif, for every sequence (U_n)n∈ω of open covers of X, there is a sequence (U_n)n∈ω satisfyingX =S

n∈ωU_n with U_n∈ U_n for all n∈ω.

Definition (Galvin 1978)

TheRothberger game in a topological spaceX is played according to the following rules: in each inningn ∈ω, One chooses an open cover U_n ofX, and then Two chooses U_n∈ U_n; the play is won by Two ifX =S

n∈ωU_n, otherwise One is the winner.

Theorem (Pawlikowski 1994)

Two ↑ Rothberger(X)⇒ One6 ↑Rothberger(X) ⇔ X is Rothberger.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 2 / 14

The Rothberger game

Definition (Rothberger 1938)

A topological space X is Rothbergerif, for every sequence (U_n)n∈ω of open covers of X, there is a sequence (U_n)n∈ω satisfyingX =S

n∈ωU_n with U_n∈ U_n for all n∈ω.

Definition (Galvin 1978)

TheRothberger game in a topological spaceX is played according to the following rules: in each inningn ∈ω, One chooses an open cover U_n ofX, and then Two chooses U_n∈ U_n; the play is won by Two ifX =S

n∈ωU_n, otherwise One is the winner.

Theorem (Pawlikowski 1994)

Two ↑ Rothberger(X)⇒ One6 ↑Rothberger(X) ⇔ X is Rothberger.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 2 / 14

The Rothberger game

Definition (Rothberger 1938)

A topological space X is Rothbergerif, for every sequence (U_n)n∈ω of open covers of X, there is a sequence (U_n)n∈ω satisfyingX =S

n∈ωU_n with U_n∈ U_n for all n∈ω.

Definition (Galvin 1978)

TheRothberger game in a topological spaceX is played according to the following rules: in each inningn ∈ω, One chooses an open cover U_n ofX, and then Two chooses U_n∈ U_n; the play is won by Two ifX =S

n∈ωU_n, otherwise One is the winner.

Theorem (Pawlikowski 1994)

Two ↑ Rothberger(X)⇒ One6 ↑Rothberger(X) ⇔ X is Rothberger.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 2 / 14

The Menger game

Definition (Hurewicz 1926)

A topological space X is Mengerif, for every sequence(U_n)n∈ω of open covers of X, there is a sequence (F_n)n∈ω satisfyingX =S S

n∈ωF_n with F_n∈[U_n]^<ℵ0 for alln∈ω.

Definition (Telgársky 1984)

TheMenger game in a topological spaceX is played as follows: in each inningn ∈ω, One chooses an open coverU_n ofX, and then Two chooses a finite subsetF_n of U_n; Two wins the play if S

n∈ωF_n is a cover ofX, otherwise One is the winner.

Theorem (Hurewicz 1926)

Two ↑ Menger(X)⇒One6 ↑ Menger(X)⇔ X is Menger.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 3 / 14

The Menger game

Definition (Hurewicz 1926)

A topological space X is Mengerif, for every sequence(U_n)n∈ω of open covers of X, there is a sequence (F_n)n∈ω satisfyingX =S S

n∈ωF_n with F_n∈[U_n]^<ℵ0 for alln∈ω.

Definition (Telgársky 1984)

TheMenger game in a topological spaceX is played as follows: in each inningn ∈ω, One chooses an open coverU_n ofX, and then Two chooses a finite subsetF_n of U_n; Two wins the play if S

n∈ωF_n is a cover ofX, otherwise One is the winner.

Theorem (Hurewicz 1926)

Two ↑ Menger(X)⇒One6 ↑ Menger(X)⇔ X is Menger.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 3 / 14

The Menger game

Definition (Hurewicz 1926)

A topological space X is Mengerif, for every sequence(U_n)n∈ω of open covers of X, there is a sequence (F_n)n∈ω satisfyingX =S S

n∈ωF_n with F_n∈[U_n]^<ℵ0 for alln∈ω.

Definition (Telgársky 1984)

TheMenger game in a topological spaceX is played as follows: in each inningn ∈ω, One chooses an open coverU_n ofX, and then Two chooses a finite subsetF_n of U_n; Two wins the play if S

n∈ωF_n is a cover ofX, otherwise One is the winner.

Theorem (Hurewicz 1926)

Two ↑ Menger(X)⇒One6 ↑ Menger(X)⇔ X is Menger.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 3 / 14

The point-open game

Definition (Galvin 1978)

Thepoint-open game in a topological spaceX is played as follows: in each inningn ∈ω, One picks a pointxn∈X, and then Two chooses an open set U_n⊆X with x_n∈U_n; the play is won by One ifX =S

n∈ωU_n, 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

The point-open game

Definition (Galvin 1978)

Thepoint-open game in a topological spaceX is played as follows: in each inningn ∈ω, One picks a pointxn∈X, and then Two chooses an open set U_n⊆X with x_n∈U_n; the play is won by One ifX =S

n∈ωU_n, 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?

Let X be a regular space. TFAE: (a) Two ↑ Menger(X);

(b) One↑ compact-open(X); (c) One↑ compact-G_δ(X).

Question (Telgársky 1984)

Is Two↑ compact-open equivalent to One↑ Menger? Or, equivalently:

Does the Menger property imply Two6 ↑ compact-open?

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14

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-G_δ(X).

Question (Telgársky 1984)

Is Two↑ compact-open equivalent to One↑Menger?

Or, equivalently:

Does the Menger property imply Two6 ↑ compact-open?

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14

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-G_δ(X).

Question (Telgársky 1984)

Is Two↑ compact-open equivalent to One↑Menger?

Or, equivalently:

Does the Menger property imply Two6 ↑ compact-open?

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14

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-G_δ(X).

Question (Telgársky 1984)

Is Two↑ compact-open equivalent to One↑Menger?

Or, equivalently:

Does the Menger property imply Two6 ↑ compact-open?

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 5 / 14

Playing with k-covers

Definition

An open cover U of a topological spaceX is ak-cover (U ∈ K_X) if every compact subset ofX is included in an element ofU.

Definition

The game G₁(K,O)on a topological space X is played as follows: in each inning n∈ω, One choosesU_n∈ K_X, and then Two chooses Un∈ U_n; Two wins the play if X =S

n∈ωU_n, 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(K_X,O_X) holds — i.e., for every sequence (U_n)n∈ω of k-covers ofX, there is a sequence (U_n)n∈ω with U_n∈ U_n for all n∈ω andX =S

n∈ωU_n.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14

Playing with k-covers

Definition

An open cover U of a topological spaceX is ak-cover (U ∈ K_X) if every compact subset ofX is included in an element ofU.

Definition

The game G₁(K,O)on a topological space X is played as follows: in each inning n∈ω, One choosesU_n∈ K_X, and then Two chooses Un∈ U_n; Two wins the play if X =S

n∈ωU_n, 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(K_X,O_X) holds — i.e., for every sequence (U_n)n∈ω of k-covers ofX, there is a sequence (U_n)n∈ω with U_n∈ U_n for all n∈ω andX =S

n∈ωU_n.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14

Playing with k-covers

Definition

An open cover U of a topological spaceX is ak-cover (U ∈ K_X) if every compact subset ofX is included in an element ofU.

Definition

The game G₁(K,O)on a topological space X is played as follows: in each inning n∈ω, One choosesU_n∈ K_X, and then Two chooses Un∈ U_n; Two wins the play if X =S

n∈ωU_n, 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(K_X,O_X) holds — i.e., for every sequence (U_n)n∈ω of k-covers ofX, there is a sequence (U_n)n∈ω with U_n∈ U_n for all n∈ω andX =S

n∈ωU_n.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14

Playing with k-covers

Definition

An open cover U of a topological spaceX is ak-cover (U ∈ K_X) if every compact subset ofX is included in an element ofU.

Definition

The game G₁(K,O)on a topological space X is played as follows: in each inning n∈ω, One choosesU_n∈ K_X, and then Two chooses Un∈ U_n; Two wins the play if X =S

n∈ωU_n, 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(K_X,O_X) holds — i.e., for every sequence (U_n)n∈ω of k-covers ofX, there is a sequence (U_n)n∈ω with U_n∈ U_n for all n∈ω andX =S

n∈ωU_n.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 6 / 14

Playing with k-covers

Corollary

Two 6 ↑compact-open ⇒ S1(K,O)⇒ Menger.

Question (Telgársky 1984)

Does the Menger property imply Two6 ↑ 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

Playing with k-covers

Corollary

Two 6 ↑compact-open ⇒ S1(K,O)⇒ Menger.

Question (Telgársky 1984)

Does the Menger property imply Two6 ↑ 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

Playing with k-covers

Corollary

Two 6 ↑compact-open ⇒ S1(K,O)⇒ Menger.

Question (Telgársky 1984)

Does the Menger property imply Two6 ↑ 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

Playing with k-covers

Lemma

Let X be a topological space such that every compact subspace of X has an isolated point. ThenX satisfies S₁(K,O) if and only ifX 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

Playing with k-covers

Lemma

Let X be a topological space such that every compact subspace of X has an isolated point. ThenX satisfies S₁(K,O) if and only ifX 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

Playing with k-covers

Lemma

Let X be a topological space such that every compact subspace of X has an isolated point. ThenX satisfies S₁(K,O) if and only ifX 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

Playing with k-covers

Lemma

Let X be a topological space such that every compact subspace of X has an isolated point. ThenX satisfies S₁(K,O) if and only ifX 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

Definition (Alster 1988)

A topological space X is Alster if everyk_δ-coverof X — i.e., every G_δ-coverW 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 wheneverY is Lindelöf.

Theorem (Alster 1988)

Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceedingℵ₁ 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

Definition (Alster 1988)

A topological space X is Alster if everyk_δ-coverof X — i.e., every G_δ-coverW 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 wheneverY is Lindelöf.

Theorem (Alster 1988)

Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceedingℵ₁ 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

Definition (Alster 1988)

A topological space X is Alster if everyk_δ-coverof X — i.e., every G_δ-coverW 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 wheneverY is Lindelöf.

Theorem (Alster 1988)

Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceedingℵ₁ 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

Definition (Alster 1988)

A topological space X is Alster if everyk_δ-coverof X — i.e., every G_δ-coverW 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 wheneverY is Lindelöf.

Theorem (Alster 1988)

Alster spaces are productively Lindelöf. Assuming CH, productively Lindelöf regular spaces of weight not exceedingℵ₁ 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 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 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 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 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

A selective characterization

Proposition

A topological space X is Alster if and only if it satisfies S₁(K^δ,O^δ).

Proposition

The game G₁(K^δ,O^δ) and the compact-G_δ game are dual.

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 11 / 14

A selective characterization

Proposition

A topological space X is Alster if and only if it satisfies S₁(K^δ,O^δ).

Proposition

The game G₁(K^δ,O^δ) and the compact-G_δ game are dual.

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 11 / 14

A selective characterization

Proposition

A topological space X is Alster if and only if it satisfies S₁(K^δ,O^δ).

Proposition

The game G₁(K^δ,O^δ) and the compact-G_δ game are dual.

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 11 / 14

Alster 0 x 1 Two↑Menger

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Theorem (Telgársky 1983-84) Let X be a regular space. TFAE:

(a) Two ↑Menger(X);

(b) One↑ compact-open(X);

(c) One↑ compact-G_δ(X).

Corollary

If X is regular and Two ↑ Menger(X), thenX is Alster.

Example (Telgársky 1983)

There is a regular space in which the compact-G_δ 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 0 x 1 Two↑Menger

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Theorem (Telgársky 1983-84) Let X be a regular space. TFAE:

(a) Two ↑Menger(X);

(b) One↑ compact-open(X);

(c) One↑ compact-G_δ(X).

Corollary

If X is regular and Two ↑ Menger(X), thenX is Alster.

Example (Telgársky 1983)

There is a regular space in which the compact-G_δ 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 0 x 1 Two↑Menger

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Theorem (Telgársky 1983-84) Let X be a regular space. TFAE:

(a) Two ↑Menger(X);

(b) One↑ compact-open(X);

(c) One↑ compact-G_δ(X).

Corollary

If X is regular and Two ↑ Menger(X), thenX is Alster.

Example (Telgársky 1983)

There is a regular space in which the compact-G_δ 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 0 x 1 Two↑Menger

Corollary

Two 6 ↑compact-G_δ ⇒Alster.

Theorem (Telgársky 1983-84) Let X be a regular space. TFAE:

(a) Two ↑Menger(X);

(b) One↑ compact-open(X);

(c) One↑ compact-G_δ(X).

Corollary

If X is regular and Two ↑ Menger(X), thenX is Alster.

Example (Telgársky 1983)

There is a regular space in which the compact-G_δ 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?

Two ↑ Rothberger(X)⇒X_δ is Lindelöf (⇒X is Rothberger). Sketch of proof.

Let σ:^<ωτ →X be a winning strategy for One in point-open(X). Now letW be a cover ofX byG_δ subsets. For eachW ∈ W, fix a sequence (U(W,n))n∈ω of open sets withW =T

n∈ωU(W,n). Proceeding by induction on n∈ω, we shall assign to each s ∈ⁿω an elementWs ofW as follows.

First, pick W_∅ ∈ W such that σ(∅)∈W_∅. Now let n∈ω be such that W_s ∈ W has already been defined for alls ∈ⁿω. For eachs ∈ⁿω and each k ∈ω, chooseW_sak ∈ W satisfyingσ(t_s,k)∈W_sak, where t_s,k ∈ⁿ⁺¹τ is the sequence defined by t_s,k(i) =U(W_si,s(i))for alli <n and

t_s,k(n) =U(W_s,k).

Since the strategy σ is winning, it follows that (...) {W_s :s ∈^<ωω} ⊆ 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)⇒X_δ is Lindelöf (⇒X is Rothberger).

Sketch of proof.

Let σ:^<ωτ →X be a winning strategy for One in point-open(X).

Now letW be a cover ofX byG_δ subsets. For eachW ∈ W, fix a sequence (U(W,n))n∈ω of open sets withW =T

n∈ωU(W,n).

Proceeding by induction on n∈ω, we shall assign to each s ∈ⁿω an elementWs ofW as follows.

First, pick W_∅ ∈ W such that σ(∅)∈W_∅. Now let n∈ω be such that W_s ∈ W has already been defined for alls ∈ⁿω. For eachs ∈ⁿω and each k ∈ω, chooseW_sak ∈ W satisfyingσ(t_s,k)∈W_sak, where t_s,k ∈ⁿ⁺¹τ is the sequence defined by t_s,k(i) =U(W_si,s(i))for alli <n and

t_s,k(n) =U(W_s,k).

Since the strategy σ is winning, it follows that (...) {W_s :s ∈^<ωω} ⊆ W is a cover ofX.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14

How about Two Rothberger?

Theorem

Two ↑ Rothberger(X)⇒X_δ is Lindelöf (⇒X is Rothberger).

Sketch of proof.

Let σ:^<ωτ →X be a winning strategy for One in point-open(X).

Now letW be a cover ofX byG_δ subsets. For eachW ∈ W, fix a sequence (U(W,n))n∈ω of open sets withW =T

n∈ωU(W,n).

Proceeding by induction on n∈ω, we shall assign to each s ∈ⁿω an elementWs ofW as follows.

First, pick W_∅ ∈ W such that σ(∅)∈W_∅. Now let n∈ω be such that W_s ∈ W has already been defined for alls ∈ⁿω. For eachs ∈ⁿω and each k ∈ω, chooseW_sak ∈ W satisfyingσ(t_s,k)∈W_sak, where t_s,k ∈ⁿ⁺¹τ is the sequence defined by t_s,k(i) =U(W_si,s(i))for alli <n and

t_s,k(n) =U(W_s,k).

Since the strategy σ is winning, it follows that (...) {W_s :s ∈^<ωω} ⊆ W is a cover ofX.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14

How about Two Rothberger?

Theorem

Two ↑ Rothberger(X)⇒X_δ is Lindelöf (⇒X is Rothberger).

Sketch of proof.

Let σ:^<ωτ →X be a winning strategy for One in point-open(X).

Now letW be a cover ofX byG_δ subsets. For eachW ∈ W, fix a sequence (U(W,n))n∈ω of open sets withW =T

n∈ωU(W,n).

Proceeding by induction on n∈ω, we shall assign to each s ∈ⁿω an elementWs ofW as follows.

First, pick W_∅ ∈ W such that σ(∅)∈W_∅. Now let n∈ω be such that W_s ∈ W has already been defined for alls ∈ⁿω. For eachs ∈ⁿω and each k ∈ω, chooseW_sak ∈ W satisfyingσ(t_s,k)∈W_sak, where t_s,k ∈ⁿ⁺¹τ is the sequence defined by t_s,k(i) =U(W_si,s(i))for alli <n and

t_s,k(n) =U(W_s,k).

Since the strategy σ is winning, it follows that (...) {W_s :s ∈^<ωω} ⊆ W is a cover ofX.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14

How about Two Rothberger?

Theorem

Two ↑ Rothberger(X)⇒X_δ is Lindelöf (⇒X is Rothberger).

Sketch of proof.

Let σ:^<ωτ →X be a winning strategy for One in point-open(X).

Now letW be a cover ofX byG_δ subsets. For eachW ∈ W, fix a sequence (U(W,n))n∈ω of open sets withW =T

n∈ωU(W,n).

Proceeding by induction on n∈ω, we shall assign to each s ∈ⁿω an elementWs ofW as follows.

First, pick W_∅ ∈ W such that σ(∅)∈W_∅. Now let n∈ω be such that W_s ∈ W has already been defined for alls ∈ⁿω. For eachs ∈ⁿω and each k ∈ω, chooseW_sak ∈ W satisfyingσ(t_s,k)∈W_sak, where t_s,k ∈ⁿ⁺¹τ is the sequence defined by t_s,k(i) =U(W_si,s(i))for alli <n and

t_s,k(n) =U(W_s,k).

Since the strategy σ is winning, it follows that (...) {W_s :s ∈^<ωω} ⊆ W is a cover ofX.

Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 13 / 14

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Rodrigo Roque Dias (IME–USP) Topological games and Alster spaces STW 2013 14 / 14