GonzalezHrusakAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 10, No. 2, 2009

pp. 207-219

More on ultrafilters and topological games

R. A. González-Silva and M. Hrušák∗

Abstract. Two different open-point games are studied here, the
G-game (of Bouziad [4]) and the Gp-game (introduced in [11]), defined
for each p ∈ ω∗. We prove that for each p ∈ ω∗, there exists a space
in which none of the players of the Gp-game has a winning strategy.
Nevertheless a result of P. Nyikos, essentially shows that it is consis-
tent, that there exists a countable space in which all these games are
undetermined.
We construct a countably compact space in which player II of the
Gp-game is the winner, for every p ∈ ω

∗. With the same technique of
construction we built a countably compact space X, such that in X ×X
player II of the G-game is the winner. Our last result is to construct
ω1-many countably compact spaces, with player I of the G-game as a
winner in any countable product of them, but player II is the winner
in the product of all of them in the G-game.

2000 AMS Classification: Primary 54A20, 91A05: secondary 54D80, 54G20

Keywords: open-point game, ultrafilter, G-space, Gp-space, countably com-
pact

1. Introduction and preliminaries

In [15] G. Gruenhage introduced a local game on topological spaces, so called
open-point game (here denoted as the W -game). Given a topological space X
and a point x ∈ X, the rules of the open-point game are as follows: Two players
I and II play infinitely many innings, in the n-th inning player I choosing a
neighborhood Un of x and player II responding with a point xn ∈ Un. After
ω-many innings we declare a winner, using the sequence (xn)n<ω of the moves

∗The first listed author gratefully acknowledges support received from PROMEP grant no.
103.5/07/2636. The second author was supported partially by DGAPA grant no. IN108802
and partially by GAČR grant 201/00/1466.



208 R. A. González-Silva and M. Hrušák

of the second player. We say that player I wins the W (x, X)-game if the se-
quence (xn)n<ω converges to x, otherwise player II is declared a winner.

This game and its variations (see [4], [11] and [17]) have proved useful in
studying local and convergence properties of topological spaces. These variants
have the same rules and only differ from the W -game in the way a winner is
declared. Following A. Bouziad [4], we say that player I wins the G(x, X)-game
if {xn : n < ω} has an accumulation point in X, otherwise, player II is the
winner.

Here we are mainly concerned with an ultrafilter version of the open-point
game as introduced and studied in [11] and [12]. Recall the definition of the
p-limit of a sequence (R. A. Bernstein [2]). Let p be a free filter on ω. A point x
of a space X is said to be the p-limit of a sequence (xn)n<ω in X (x = p-lim xn)
if for every neighborhood U of x, {n < ω : xn ∈ V } ∈ p.

Now, we are ready to define the Gp-game, a parametrized version of the
above mentioned G-game. Let p be a free ultrafilter on ω. We say that player
I wins the Gp(x, X)-game if p-lim xn exists (in X). Otherwise, player II wins.

In what follows we are mostly concerned with the question as to whether
either player has a winning strategy in one of the above mentioned games.
A strategy for one of the players is an algorithm that specifies each move of
the player in every possible situation. More precisely, a strategy for player
I in the open-point game is any sequence of functions σ = {σn : N (x)

n
×

X n → N (x) : n < ω}. A sequence (xn)n<ω in X is called a σ-sequence if
xn+1 ∈ σn+1(〈x0, ..., xn〉; 〈V0, ..., Vn〉) = Vn+1, for each n < ω. A strategy σ for
player I is a winning strategy in the G(x, X)-game (respect. W (x, X)-game,
Gp(x, X)-game), if each σ-sequence has an accumulation point in X (respect.
xn → x, or there exist y ∈ X such that p-lim xn = y). A space X is called a
G-space (respect. W -space, Gp-space) if player I has a winning strategy in the
G(x, X)-game (resp. W (x, X)-game, Gp(x, X)-game), for every x ∈ X.

Similarly, one defines a strategy for player II. It is a sequence of functions
ρ = {ρn : X

n×N (x)
n+1

→ X : n < ω}, such that ρn(〈x0, ..., xn−1〉; 〈V0, ..., Vn〉)
∈ Vn, for each n < ω. A sequence 〈(Vn, xn) : n < ω〉 where Vn ∈ N (x) and
xn ∈ Vn is called a ρ-sequence, if xn = ρn(〈x0, ..., xn−1〉; 〈V0, ..., Vn〉) ∈ Vn, for
each n < ω. A strategy ρ for player II is a winning strategy in the G(x, X)-game
(respect. W (x, X)-game, Gp(x, X)-game), if for each ρ-sequence, 〈(Vn, xn) :
n < ω〉, the set {xn : n < ω} does not have cluster point in X (resp. xn 6→ x,
or the p-limit of the sequence {xn} does not exist).

We denote the fact that player I has a winning strategy in the G(x, X)-game,
by I ↑ G(x, X). If he does not have a winning strategy we write I ↓ G(x, X).
When I ↑ G(x, X) for every x ∈ X, this is denoted by I ↑ G(X). The meaning



More on ultrafilters and topological games 209

of II ↑ G(x, X), II ↓ G(x, X) is defined analogously with the same notation
used for the W -game or Gp-game.

The following implications are easy consequences from definitions, I ↑ W (X)
=⇒ I ↑ Gp(X) =⇒ I ↑ G(X). They can not be reversed in general, as shown
for the spaces ω∗, β(ω) \ {q ∈ ω∗ : q ≤RF p}), but they are equivalent to
first countability if X is a countable space (see Proposition 2.5). Dually, II ↑
W (X) ⇐= II ↑ Gp(X) ⇐= II ↑ G(X). These implications are also strict, the
same examples work. In the next diagram, one can see relationships of these
games with other concepts of general topology (for more details, see [13]).

X is compact




y

X is first countable X is p-compact −→
X is countably

compact




y





y





y

I ↑ W (X) −→ I ↑ Gp(X) −→ I ↑ G(X)

Sharma proved in [23] that X is strongly Frechet ⇐⇒ II ↓ W (X), where a
space X is called strongly Fréchet iff for every point x ∈ X, and every sequence
(An)n<ω of subsets of X with x ∈ An for each n < ω, there exists a sequence
{xn} such that xn ∈ An for every n ∈ ω and xn → x.

The notation used here is mostly standard. The Stone-Čech compactification
βω of the countable discrete space ω is identified with the set of all ultrafilters
on ω and its remainder ω∗ = βω \ ω denotes the set of all free ultrafilters on

ω. If f : ω → X is a function into a compact space X, f̂ denotes its (unique)
extension to βω. Two ultrafilters are said to be of the same type (in βω) if

there is a permutation f of ω such that f̂ takes one to the other. The set of
ultrafilters of the same type as a fixed ultrafilter p, is denoted by T (p). For
p, q ∈ ω∗, p ≤RK q denotes that p is Rudin-Keisler bellow q and means that
there is f : ω → ω such that f̂ (q) = p. The relation p ≤RF q is the Rudin-Froĺık

order and it means that there is an embedding f : ω → βω such that f̂ (p) = q.

2. indeterminacy of the games Gp, W and G

We say that a game is determined on a space X if for every point of X one
of the players (not the same for all points) has a winning strategy, otherwise,
the game is undetermined. For nice definable spaces the games are typically
determined. However, they are not determined in general. In this section we
are going to work with the indeterminacy of the games Gp, G and W . For this,
let us introduce the following notation.

Let Y be a set. A subset T of Y <ω is a tree if whenever t ∈ T and s ∈ Y <ω

with s ⊆ t, then s ∈ T. Let t be an element of the tree T, the set of successors
of t, {y ∈ Y : t⌢y ∈ T} is denoted by succT(t). A function f : ω → Y , is said



210 R. A. González-Silva and M. Hrušák

to be a branch of T, if f ↾n∈ T for every n < ω. The set of branches of T is
denoted by [T].

Next we will show that for every p ∈ ω∗, there is a countably compact space
such that no player of the Gp-game has a winning strategy. To that end the
following lemmas will be useful.

The following fact is a standard reformulation of the existence of a winning
strategy for player II (see e.g [17]).

Lemma 2.1. Suppose that X is a topological space, x ∈ X and p ∈ ω∗. Then
the following are equivalent:

(1) II ↑ Gp(x, X).
(2) II has a wining strategy ρ′ in the Gp(x, X)-game such that x 6∈ rng(ρ

′)
(3) There exists a tree T such that

i. For every t ∈ T, x ∈ succT(t) \ {x}.
ii. For every f ∈ [T], p-lim f (n) does not exist in X.

Proof. 1=⇒2. Let ρ = {ρn : n < ω} be a winning strategy for player II in
the Gp(x, X)-game. We say that a sequence 〈V0, y0, V1, y1, ..., Vn, yn〉 is ρ-legal,
if the V0, ..., Vn are neighborhoods of x, and for each i ∈ {0, ..., n}, we have
ρi(〈y0, ..., yi−1〉, 〈V0, ..., Vi〉) = yi ∈ Vi.

We will recursively define a winning strategy ρ′ such that:

(a) x 6∈ rng(ρ′) and
(b) For every ρ′-legal sequence 〈V0, x0, V1, x1, ..., Vn, xn〉, there is a unique

ρ-legal sequence 〈V0, y0, V1, y1, ..., Vn, yn〉 such that yi = xi whenever
yi 6= x.

If n = 0, let ρ′0(V0) be equal to ρ0(V0) if ρ0(V0) 6= x otherwise ρ
′
0(V0) is any

element of V0 \ {x}.
For the inductive step, let 〈V0, x0, V1, x1, ..., xn−1, Vn〉 be sequence of moves

where the xi are played according to the strategy ρ
′. Consider 〈V0, x0, V1, x1,

..., Vn−1, xn−1〉. By the inductive hypothesis there is a unique ρ-legal sequence
〈V0, y0, V1, y1, ..., Vn−1, yn−1〉 such that yi = xi whenever yi 6= x. Define
ρ′n(〈x0, ..., xn−1〉; 〈V0, ..., Vn〉) as follows: It is equal to ρn(〈y0, ..., yn−1〉; 〈V0, ...,
Vn〉) if ρn(〈y0, ..., yn−1〉; 〈V0, ..., Vn〉) 6= x, otherwise is any point of Vn \ {x}.

It is clear that (a) holds and that ρ′ is a strategy.
Now lets see that (b) holds. Let 〈V0, x0〉 be a ρ

′-legal, then we have two
cases, x0 is equal to ρ0(V0) or not, in any case (b) holds. Now suppose
that the statement (b) is true for any ρ′-legal sequence of length n and let
〈V0, x0, V1, x1, ..., Vn, xn〉 be ρ

′-legal sequence, so the subsequence 〈V0, x0, V1, x1,
..., Vn−1, xn−1〉 holds (b), hence there is a unique ρ-legal sequence 〈V0, y0, V1, y1
, ..., Vn−1, yn−1〉 fulling (b), and ρn(〈y0, ..., yn−1〉; 〈V0, ..., Vn〉) = yn, so 〈V0, y0,
V1, y1, ..., Vn, yn〉 is the unique ρ-legal sequence.

Finally lets see that ρ′ is a winning strategy for player II, for this, let
(xn)n<ω be a sequence of moves of player II according to strategy ρ

′. Then
there is exists a unique sequence (yn)n<ω which is constructed by segments of
(xn)n<ω; the difference between (xn)n<ω and (yn)n<ω are the points yn which



More on ultrafilters and topological games 211

are x. Since the p-lim yn is not in X then the p-lim xn is not in X, so ρ
′ is a

winning strategy.
2=⇒3. Let T′′ = {l ∈ (N (x) × X)<ω : l is a ρ′−legal sequence} and define

T
′ = {g↾n : g ∈ [T

′′] and g is inf inite}. Note that each f ∈ [T′] is a Gp-play
a cording to strategy ρ′, hence T′ 6= ∅ and if sf = (xfn)n<ω is the subsequence
generated by the points of f , then this sequence does not have a p-limit in X.
Set T = {sf↾n: f ∈ [T

′]}, with sf↾n⊆ s
g↾m if and only if f↾n⊆ g↾m.

To see that i holds, pick t ∈ T and a neighborhood U of x. From the construc-
tion of T, choose a branch f ∈ [T′] such that t = sf↾n. Let (V

f
n )n<ω be the sub-

sequence generated by the neighborhoods of f . Then ρ′|t|(〈t(0), t(1), ..., t(|t| −

1)〉; 〈V
f
0 , V

f
1 , ..., V

f

|t|−1
, U〉) ∈ U , hence U ∩ (succT(t) \ {x}) 6= ∅. Finally, if

g ∈ [T], then g = sf for some f ∈ [T′], so p-lim g(n) does not exist in X, this
fulfilling condition ii.

3=⇒1. Take a tree T fulfilling clauses i and ii. For each n ∈ ω, define
ρn : X

n × N (x)
n+1

→ X, such that

ρn(〈x0, ..., xn−1〉; 〈V0, ..., Vn〉) ∈ Vn ∩ succT(〈x0, ..., xn−1〉).

Let ρ = {ρn : n < ω}. It is straightforward to see that ρ is a winning strategy
for player II in the Gp(x, X)-game, as in any play the resulting sequence is a
branch of the tree T, and by ii, it does not have a p-limit in X. �

The next result, due to Z. Froĺık, is used in the proof of Lemma 2.3 and also
later on in the text.

Lemma 2.2 (Froĺık). If f, g : ω → ω∗ are embeddings and p ∈ ω∗. Then,

f̂ (p) = ĝ(p) if and only if {n < ω : f (n) = g(n)} ∈ p.

Lemma 2.3. Let p ∈ ω∗ and T ⊆ (ω∗)<ω be a countable tree, such that

(1) For each t ∈ T, |succT(t)| ≥ 2.
(2) For each f ∈ [T], f is an embedding in ω∗.
(3) If f, g ∈ [T], f 6= g, then |f ∩ g| < ℵ0.

Then, f̂ (p) 6= ĝ(p) for any two elements f, g ∈ [T], and in particular the set
p[T] = {p-lim f (n) : f ∈ [T]} has cardinality c.

Proof. Follows from clauses 2 and 3, and Lemma 2.2. �

The idea to construct a space X in which the Gp-game is undetermined (for
p ∈ ω∗ fixed), is to construct recursively a space X ⊂ ω∗, diagonalizing across
all the possible strategies for players I and II. There are two obvious obstacles
to doing this. If we don’t know X, then we can’t say too much about the
strategies. Another obstacle, is that there are going to be at least 2|X| possible
strategies. Fortunately Lemma 2.3 can be used to overcome both obstacles.
The space X is going to be constructed in c-many steps, so the cardinality of
{T ⊆ X <ω : T satisfies the conditions of Lemma 2.3 } is at most c.



212 R. A. González-Silva and M. Hrušák

Theorem 2.4. For each p ∈ ω∗, there exists a countably compact space X
such that for every x ∈ X, I ↓ Gp(x, X) and II ↓ Gp(x, X).

Proof. Fix a bijection Φ : c → c × c such that, for Φ(α) = (Φ0(α), Φ1(α)), we
have Φ0(α), Φ1(α) ≤ α, for each α < c. By recursion we are going to construct
for each ν < c, spaces Xν , Yν and a sequence of trees {T

ν
α : α < c}, such that

(1) X0 ⊂ ω
∗ is countable and dense in itself, and Y0 = ∅.

(2) Xη ⊂ Xµ y Yη ⊂ Yµ, for all η < µ < ν.
(3) |Xµ| ≤ |µ + ω| y |Yµ| ≤ |µ|, for all µ < ν.
(4) Xµ ∩ Yη = ∅, for all η < µ < ν.
(5) {Tνα : α < c} is an enumeration of all trees in X

<ω
ν satisfying the

conditions of Lemma 2.3.
(6) If µ + 1 < ν, then Xµ+1 ∩ p[T

Φ0(µ)
Φ1(µ)

] 6= ∅ and Yµ+1 ∩ p[T
Φ0(µ)
Φ1(µ)

] 6= ∅.

The construction of the space X0 can be done using Theorem 1.4.7 of [18].
For a limit ordinal ν, define Xν =

⋃

µ<ν
Xµ and Yν =

⋃

µ<ν
Yµ. When ν =

µ + 1, define Xν = Xµ ∪ {pµ} and Yν = Yµ ∪ {qµ}, where pµ, qµ ∈ ω
∗ have the

property that pµ 6= qµ and

pµ, qµ ∈ p[T
Φ0(µ)
Φ1(µ)

] \ (Xµ ∪ Yµ).

Let X =
⋃

ν<c
Xν . Note that if T ⊆ X

<ω is a tree which satisfies the condition
of Lemma 2.3, then there exists a ν < c such that T ⊆ X <ων , hence there exists
α < c such that Tνα = T. And by the fact that Φ is onto, then there is γ < c

with T
Φ0(γ)
Φ1(γ)

= T.

Let’s see that X is countably compact. Take a countable subset Y of X,
without loss of generality we can assume that it is discrete. It is easy to
construct a tree T contained in Y <ω with the properties of Lemma 2.3. Hence

by the observation before, there exists γ < c with T
Φ0(γ)
Φ1(γ)

= T. So pγ is cluster

point of Y , which is in X.

Claim 1: For each x ∈ X, I ↓ Gp(x, X).
Fix x ∈ X and suppose that player I has a winning strategy σ = {σn : n <

ω} at x in the Gp-game. For each s ∈ 2
<ω inductively pick xs ∈ X and a clopen

neighborhood Wn of x, such that

xs ∈ σ|s|(〈xs|0 , xs|1 , ..., xs||s|−1〉; 〈V0, ..., V|s|−1〉) = V|s|,
xs 6∈ {xr : r ∈ 2

≤|s| and r 6= s},
xs ∈ Wn for all s ∈ 2

n and
xs 6∈ Wn for all s ∈ 2

<n.

Define ts = 〈xs|0 , xs|1 , xs|2 , ..., xs〉 and T = {ts : s ∈ 2
<ω}. From our

construction it follows that T is a tree which satisfies the premises of Lemma

2.3. Hence there exists γ < c with T
Φ0(γ)
Φ1(γ)

= T. And f ∈ [T] such that p-

lim f (n) = qγ . Note that rng(f ) is a σ-sequence which does not have a p-limit
in X. So the strategy σ is not winning.



More on ultrafilters and topological games 213

Claim 2: For each x ∈ X, II ↓ Gp(x, X).
Suppose that there exist x ∈ X and a tree T ⊆ X <ω with the properties

of Lemma 2.1. We are going to construct a countable subtree T of T which is
going to satisfy the conditions of Lemma 2.3. Fix t ∈ T. For each s ∈ 2<ω and
n > 0, pick inductively points xs⌢0 6= xs⌢1 in X and a clopen neighborhood
Vn of x with the followings properties:

xs⌢ 0, xs⌢1 ∈ succT (t
⌢xs|1

⌢xs|2
⌢...⌢xs),

xs⌢ 0, xs⌢1 6∈ {xr : r ∈ 2
≤|s|+1 \ {s⌢0, s⌢1}},

xs⌢ 0, xs⌢1 ∈ Vn, for each s ∈ 2
n−1 and n − 1 ≥ 0,

xs⌢ 0, xs⌢1 6∈ Vn, for each s ∈ 2
<n−1 and n − 1 > 0.

Let ts = t
⌢〈xs|1 , xs|2 , ..., xs〉, and define T = {ts : s ∈ 2

<ω}. So T is a

subtree of T like Lemma 2.3. Hence there exists γ < c with T
Φ0(γ)
Φ1(γ)

= T.

However from this fact, there is a branch f ∈ [T] with p-lim f (n) ∈ X. �

The proof of the following fact is analogous to the proof given in [15, Theorem
3.3] for the W-game. We have already mentioned that, for a countable space
X, the existence of a winning strategy for player I in the G-game on X is
equivalent to X being first countable.

Proposition 2.5. In a Tychonoff countable space X, the following statements

are equivalent for a fixed element x in X:

(1) χ(x, X) = ℵ0.
(2) I ↑ G(x, X).

Proof. 1 =⇒ 2. It is straightforward to define a winning strategy for player I
using a countable local base.

2 =⇒ 1. Suppose that χ(x, X) > ℵ0. Let σ be any strategy for player I.
Enumerate the range of σ as {Vn : n < ω}. As X is zero-dimensional, we can
get for each n < ω, a clopen subset Un such that

(1) Un+1 ⊂ Un, for every n < ω.
(2)

⋂

n<ω
Un = {x}.

(3) Un ⊂ Vn, for every n < ω.

Since χ(x, X) > ℵ0, there exists a neighborhood V of x such that |Un \ V | =

ℵ0 for each n < ω. Take xn ∈ Un \ V for each n < ω. Then x 6∈ {xn : n < ω}.
Now, if y ∈ X \{x}, then there exist n < ω with y 6∈ Un, hence X \ Un ∈ N (y),

so |(X \ Un) ∩ {xn : n < ω}| < ℵ0, i.e. y 6∈ {xn : n < ω}. Hence the sequence
{xn : n < ω} does not have cluster points. It is easy to see that it contains a
subsequence which is σ-sequence without cluster points. Therefore the strategy
σ is not winning. �

Theorem 1.12 of [21] essentially says that it is consistent that there exist
countable dense-in-themselves spaces on which our three games are undeter-
mined. We will need the following version of this result.



214 R. A. González-Silva and M. Hrušák

Theorem 2.6 (P. Nyikos). Assume p > ω1. If D is a countable dense subset
of 2ω1 , then I ↓ G(D) and II ↓ W (D).

From this Theorem and the implications between the games W , Gp and G,
we have the next corollary.

Corollary 2.7 (p > ω1). There exists a topological countable group G such
that the games W , G and Gp are undetermined in G.

3. player II and countable compactness

If X is countably compact, player I has a (trivial) winning strategy in the
G-game. This is no longer true for the Gp-game. In fact, it is easy to construct
(for a fixed p ∈ ω∗) a countably compact space X such that II ↑ Gp(X). Now,
we will construct a countably compact space X such that II ↑ Gp(X) for every
p ∈ X and then show that there is a countably compact space X such that
II ↑ G(X × X), which is a strengthening of results of Novak and Terasaka’s
examples (see [24, Lemma 3.1]).

Recall the definition of the relative type, introduced by Z. Froĺık. Let Y ∈

[ω∗]ω be discrete and p ∈ Y ∗ = Y
βω

\ Y . The relative type of p with respect

to Y is T (ĥ(p)), where h : Y → ω is an embedding. It is going to be denote by
T (p, Y ). Now, for a subset S of βω and p ∈ ω∗, define T [p, S] = {T (p, Y ) : Y ∈
[S]ω and Y is homeomorphic to ω}. Froĺık proved that T [p, ω∗] has cardinality
c.

Theorem 3.1. There exists a countably compact space X such that II ↑
Gp(x, X) for every p ∈ ω

∗ and x ∈ X.

Proof. The space X is going to be the union of {Xν : ν < ω1}, where each Xν
is constructed recursively. Suppose that for each µ < ν < ω1 we have Xµ such
that

(1) X0 ⊆ ω
∗ is countable and dense in it self.

(2) X0 is a dense subset of Xµ, for each µ < ν.
(3) |Xµ| ≤ c, for each µ < ν.
(4) Xη ⊂ Xµ, if η < µ < ν.
(5) If µ + 1 < ν, then every countable discrete subset of Xµ has a cluster

point in Xµ+1.
(6) For each x ∈ Xµ \ X0, {y ∈ Xµ : T [x, X0] ∩ T [y, X0] 6= ∅} = {x}.

We can assume the existence of the space X0, using Theorem 1.4.7 of [18].
Now, we show how to construct Xν . When ν is a limit ordinal, define Xν =
⋃

µ<ν Xµ. If ν is a successor ordinal, say ν = µ + 1 then we have from clause 3,

that the set of all embeddings from ω to Xµ has size c, let {fα : α < c} be an

enumeration of this set. For each α < c, pick a point pα ∈ fα[ω]
β(ω)

such that



More on ultrafilters and topological games 215

for all y ∈ Xµ, T [pα, X0] ∩ T [y, X0] = ∅, and also T [pα, X0] ∩ T [pβ, X0] = ∅,
for all β < α. Define Xµ+1 = Xµ ∪ {pα : α < c}.

Notice that our space X =
⋃

ν<ω1
Xν is countably compact and also for each

p ∈ ω∗, |{y ∈ X \ X0 : T (p) ∈ T [y, X0]}| ≤ 1. Therefore, |{y ∈ X : T (p) ∈
T [y, X0]}| ≤ ω.

Fix p ∈ ω∗ and x ∈ X. Let’s see that II ↑ Gp(x, X). It follows from the
previous observation that the set A = {q ∈ X : T (p) ∈ T [q, X0]} is countable.
Enumerate it as {qi : i < ω}. For each i < ω fix an embedding fi : ω → X0
such that f̂i(p) = qi. The strategy of player II is to choose in the n-th step
g(n) ∈ X0 \ {f0(n), f1(n), ..., fn(n)} such that the function g : ω → X0 defined
in this way is an embedding. From Lemma 2.2, we have ĝ(p) 6∈ A. And hence
T (p) ∈ T [ĝ(p), X0], then ĝ(p) 6∈ X. So this is a winning strategy for player II
in the Gp(x, X)-game. �

In the construction of the next example, we use a space which is countable,
dense in itself and extremally disconnected. This space is defined for a fixed
ultrafilter p ∈ ω∗ and it is denoted by Seq(p), its underlying set is ω<ω, the
set of all finite sequences in ω. A set U ⊂ ω<ω is open if and only if for every
t ∈ U , {n < ω : t⌢n ∈ U} ∈ p (see [7], [20], [5], [25]).

Lemma 3.2. There exists a countable dense-in-itself space X ⊂ ω∗ such that,
for any x ∈ X there exists a sequence {Vn : n < ω} ⊂ N (x) with the following
property: if {xn : n < ω} ⊂ X and xn ∈

⋂

m≤n Vm for each n < ω then

x 6∈ {xn : n < ω}.

Proof. Let p ∈ ω∗ be not a P-point and consider the space Seq(p). Using
Theorem 1.4.7 of [18], we can take an homeomorphic copy of Seq(p) inside of
ω∗. So, now it is sufficient to prove that Seq(p) is the desired space.

Since p is not a P-point, there exists a sequence {Un : n < ω} ⊂ p without
pseudointersection in p. Take x ∈ Seq(p) and define Vn = {t ∈ Seq(p) : x ⊆
t and t(|x|) ∈ Un}. If (xn)n<ω is a sequence such that xn ∈

⋂

m≤n Vm, then

xn(|x|) ∈ Um, for every n > m. Hence W = {xn(|x|) : n < ω} 6∈ p. So
U = ω \ W ∈ p, this implies that V = {t ∈ X : x ⊆ t and t(|x|) ∈ U} is a
neighborhood of x, disjoint from {xn : n < ω}. �

It is easy to see that the product of at most ω-many W -spaces (Gp-spaces), is
also a W -space (Gp-space). However, this is not true for G-spaces, as we will see
in the next example. An application of the following example, is the existence
of a countably compact space whose product is not countably compact.

Example 3.3. There exists a countably compact space X such that II ↑
G(X × X).

Proof. Let X be the space constructed in Theorem 3.1, with the condition that,
the space X0 is homeomorphic to Seq(p) where the free ultrafilter p is not a



216 R. A. González-Silva and M. Hrušák

P-point. Let’s see that II ↑ G((x, y), X × X), for a fixed point (x, y) ∈ X × X.
By ∆ we denote the set {(x, x) : x ∈ X0}.

Case (i): (x, y) ∈ X × X \ (X0 × X0). Let {(xn, yn) : n < ω} be an
enumeration of all the points in X0×X0. For each n < ω, let Wn ∈ N ((xn, yn))\
N ((x, y)) clopen such that X0 × X0 \

⋃

m≤n Wm is infinite for every n < ω and

also (xm, ym) 6∈ Wn for every m < n (this is possible because the space X0 is a
subspace of ω∗ dense in it self). Let V0 be the first move of player I, player II
responds with a point (g(0), h(0)) ∈ V0 ∩ (X0 × X0), and at the same time he
chooses clopen sets A0 ∈ N (g(0)) \ N (x) and B0 ∈ N (h(0)) \ N (y), such that

(X0 × X0) \ [(A0 × X0) ∪ (X0 × B0) ∪ ∆)] is infinite.

Inductively players I and II produce a sequence of points in X0 × X0,
{(g(n), h(n)) : n < ω}, and sequences of clopen sets {An : n < ω} and {Bn :
n < ω}, such that, if the moves of player I are denoted by Vn

′
s then:

(1) (g(0), h(0)) ∈ V0,
(2) (g(n), h(n)) ∈ Vn ∩ (X0 × X0 \ [

⋃

m≤n Wm ∪
⋃

m<n(Am × X0) ∪ (X0 ×
Bm) ∪ ∆]), for all n < ω,

(3) An ∈ N (g(n)) \ N (x), for all n < ω,
(4) Bn ∈ N (h(n)) \ N (y), for all n < ω, and
(5) X0 × X0 \ [

⋃

m≤n Wm ∪
⋃

m≤n(Am × X0) ∪ (X0 × Bm) ∪ ∆] is infinite,
for all n < ω.

From the construction of the space X, it is possible that player II play
inductively in this way, choosing the An

′
s and Bn

′
s, fulfilling the previous

conditions.
So at the end the resulting sequence S(g, h) = {(g(n), h(n)) : n < ω} is dis-

crete and also the functions in each coordinate, g, h : ω → X0 are embeddings.
Now, since (xn, yn) ∈ Wn for each n < ω, no element of X0 × X0 is a cluster
point of the sequence S(g, h). Note that if (a, b) ∈ X × X \ X0 × X0 is a cluster
point of S(g, h), then T [a, X0] ∩ T [b, X0] 6= ∅ but from the construction of X,
this implies that a = b while S(g, h) ∩ ∆ = ∅, so S(g, h) is closed and discrete.

Case (ii): (x, y) ∈ X0 × X0. Let {(xn, yn) : n < ω} be an enumeration
of all the points in X0 × X0 \ {(x, y)} and let {Wn : n < ω} a sequence of
neighborhoods as in Case (i). Let {U xn : n < ω} and {U

y
n : n < ω} sequence

of neighborhoods of x and y respectively, like in Lemma 3.2. If Vn is the n-th
move of player I, then player II is going to play as before, but with the clause
2 strengthened as:

(g(n), h(n)) ∈ Vn ∩ (
⋂

m≤n U
x
m × U

y
m) ∩ (X0 × X0 \ [

⋃

m≤n Wm∪
⋃

m<n(Am × X0) ∪ (X0 × Bm) ∪ ∆]).
Then as before, player II gets a discrete set S(g, h), with g, h embeddings.

And in this case, the only possible cluster point is (x, y), but from the choice
of the sequences g and h, and the properties of the points of X0, it follows that
(x, y) is not a cluster point of S(g, h). �



More on ultrafilters and topological games 217

Question 3.4. Is there for each n ≥ 2, a space X such that X n is countably
compact and II ↑ G(X n+1)?.

The next example will show a family of countably compact spaces, such that
the product of countably many of them is a G-space but in the product of all
them, player II of the G-game, has a winning strategy. The example shows that
the converse of Theorem 2.2 of the paper [12] is not true which establishes that,
if player I has a winning strategy in the G-game in the product of ω1-many
spaces, then all but countably many of them are countably compact.

To make this example, we generalize the space from [11, Theorem 2.3]. For

p ∈ ω∗, let R(p) = {f̂ (p) : f : ω → ω is strictly increasing}.
Let ∅ 6= M ⊆ ω∗. Put M0 = ω and M1 =

⋃

p∈M R(p). Let ν < ω1. If ν is

limit ordinal, then Mν =
⋃

µ<ν
Mµ. If ν = µ + 1, then we define

Mν = {f̂ (p) : f : ω → Mµ is an embedding, f|Af is strictly increasing and p ∈ M}

where Af = {n < ω : f (n) ∈ ω} for a function f : ω → βω. Then define
Ω(M ) =

⋃

ν<ω1
Mν . By using arguments similar to those used in the paper

[11], we can prove that, for ∅ 6= M ⊆ ω∗, the space Ω(M ) is a countably
compact Gp-space, for all p ∈ M .

Example 3.5. There is a family {Xν : ν < ω1} of countably compact spaces
such that I ↑ G(

∏

ν<µ
Xν ), for every µ < ω1, but II ↑ G(

∏

ν<ω1
Xµ).

Proof. We start fixing a family {pν : ν < ω1} of free ultrafilters on ω which are
pairwise RK-incomparable (see [22]). For ν < ω1, define Xν = Ω({pµ : µ ≥ ν}).
We know that Xν is a countably compact space and I ↑ Gpν (Xν ), for all ν < ω1.
In fact I ↑ Gpµ (Xν ), for all µ > ν. Then by Theorem 2.6 of [12], we obtain
that I ↑ Gpµ (

∏

ν<µ
Xν ), this shows the first part of the theorem. Notice that

from the linearity of the RF-order and the properties of the ultrafilters pν ’s,
it follows that

⋂

ν<ω1
Xν = ω. Now, fix x ∈

∏

ν<ω1
Xν , we will show that

II ↑ G(x,
∏

ν<ω1
Xν ). Indeed, assume that player I has chosen at the n-th step

Vn =
⋂

α∈Fn
[α, Vα], where Fn ∈ [ω1]

<ω, Vα ∈ N (x(α)) for each α ∈ Fn and
[α, Vα] = {f ∈

∏

ν<ω1
Xν : f (α) ∈ Vα}. The strategy of player II is to choose

at the n-th step xn ∈
∏

ν<ω1
Xν such that

xn(α) =

{

x(α) if α ∈ Fn,
n if α ∈ ω1 \ (

⋃

m≤n Fm).

From the fact that
⋂

ν<ω1
Xν = ω, we have that for β = sup(

⋃

n<ω
Fn) the

set {xn|[β,ω1) = n : n < ω} does not have a cluster point. So {xn : n < ω} is
close and discrete and hence player II wins. �



218 R. A. González-Silva and M. Hrušák

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Received December 2008

Accepted September 2009



More on ultrafilters and topological games 219

R. A. González-Silva (rgonzalez@culagos.udg.mx)
Departamento de Ciencias Exactas y Tecnológicas (UdG) Enrique Dı́az de León
1144, Col. Paseos de la Montaña, 47460, Lagos de Moreno Jalisco, México

M. Hrušák (mhrusak@matmor.unam.mx)
Instituto de Matemáticas (UNAM) A.P. 61-3 Xangari, 58089 Morelia, Mi-
choacán, México