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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