## 1 Introduction

The famous Banach–Mazur game was invented by Mazur in 1935. For the history of game theory and facts about game theory, the reader is referred to the survey [12]. Let X be a topological space and $$X=A\cup B$$ be any given decomposition of X into two disjoint sets. The game BM(XAB) is played as follows: Two players, named $$\alpha$$ and $$\beta$$, alternately choose open nonempty sets with $$U_0\supseteq V_0\supseteq U_1\supseteq V_1\supseteq \cdots$$.

$$\begin{array}{lccccr} \alpha &{}U_0&{} &{}U_1&{} &{}\\ &{} &{} &{} &{} &{}\cdots \\ \beta &{} &{}V_0&{} &{}V_1&{} \end{array}$$

Player $$\alpha$$ wins this game if $$A\cap \bigcap _{n\in \omega }U_n\ne \emptyset$$. Otherwise $$\beta$$ wins.

We study a well-known modification of this game considered by Choquet in 1958, known as Banach–Mazur game or Choquet game. Player $$\alpha$$ and $$\beta$$ alternately choose open nonempty sets with $$U_0\supseteq V_0\supseteq U_1\supseteq V_1\cdots$$. In the first round, player $$\beta$$ starts by choosing a nonempty open set $$U_0.$$

$$\begin{array}{lccccr} \beta &{}U_0&{} &{}U_1&{} &{}\\ &{} &{} &{} &{} &{}\cdots \\ \alpha &{} &{}V_0&{} &{}V_1&{} \end{array}$$

Player $$\alpha$$ wins this play if $$\bigcap _{n\in \omega }V_n\ne \emptyset$$. Otherwise $$\beta$$ wins. Denote this game by BM(X). Every finite sequence of sets $$(U_0,\ldots ,U_n)$$, obtained by the first n steps in this game is called partial play of $$\beta .$$ A strategy for player $$\alpha$$ in the game BM(X) is a map s that assigns to each partial play $$(U_0,\ldots ,U_n)$$ of $$\beta$$ a nonempty open set $$V_n\subseteq U_n$$. The strategy s is called a winning strategy for player $$\alpha$$ if player $$\alpha$$ always wins the play of the game using this strategy. The space X is called weakly$$\alpha$$-favorable (see [13]) if X admits a winning strategy for player $$\alpha$$ in the game BM(X). We say that a partial play $$(W_0,\ldots ,W_k)$$ is stronger than $$(U_0,\ldots , U_m)$$ if $$m\le k$$ and $$U_0=W_0,\ldots ,U_m=W_m$$. Notice that if $$(W_0,\ldots ,W_k)$$ is stronger than $$(U_0,\ldots , U_m)$$, then $$s(W_0,\ldots ,W_k)\subseteq s(U_0,\ldots , U_m)$$, we denote this by $$(U_0,\ldots , U_m)\preceq (W_0,\ldots ,W_k)$$. We denote a sequence $$(U_0,\ldots , U_k)$$ by $$\overrightarrow{U}(k).$$

The strong Choquet game is defined as follows:

$$\begin{array}{lccccr} \beta &{}U_0\ni x_0&{} &{}U_1\ni x_1&{} &{}\\ &{} &{} &{} &{} &{}\cdots \\ \alpha &{} &{}V_0&{} &{}V_1&{} \end{array}$$

Player $$\beta$$ and $$\alpha$$ take turns in playing nonempty open subset, similar to the Banach–Mazur game. In the first round, player $$\beta$$ starts by choosing a point $$x_0$$ and an open set $$U_0$$ containing $$x_0$$, then player $$\alpha$$ responds with an open set $$V_0$$ such that $$x_0\in V_0\subseteq U_0$$. In the n-th round, player $$\beta$$ selects a point $$x_n$$ and an open set $$U_n$$ such that $$x_n\in U_n\subseteq V_{n-1}$$ and $$\alpha$$ responds with an open set $$V_n$$ such that $$x_n\in V_n\subseteq U_n$$. Player $$\alpha$$ wins if $$\bigcap _{n\in \omega }V_n\ne \emptyset$$. Otherwise $$\beta$$ wins. We say that a partial play $$(W_0,x_0,\ldots ,W_k,x_k)$$ is stronger than $$(U_0,y_0,\ldots , U_m,y_m)$$ if $$m\le k \text{ and } U_0{=}W_0,\ldots ,U_m=W_m \text{ and } x_0=y_0,\ldots ,x_m=y_m.$$ We denote this by $$(U_0,y_0,\ldots , U_m,y_m)\preceq (W_0,x_0,\ldots , W_k,x_k)$$. We denote a sequence $$(W_0,x_0,\ldots , W_k,x_k)$$ by $$(\overrightarrow{x}\circ \overrightarrow{W})(k)$$. A topological space X is called Choquet complete if player $$\alpha$$ has a winning strategy in the strong Choquet game, and we then write Ch(X).

For a topological space X, let $$\tau (X)$$ denote the topology on the set X and $$\tau ^*(X)=\tau (X){\setminus }\{\emptyset \}.$$ A family $${\mathcal {P}}$$ of open nonempty sets is called a $$\pi$$-base if for every open nonempty set U, there is $$P\in {\mathcal {P}}$$ such that $$P\subseteq U.$$

A dcpo (directed complete partial order) is a poset $$(P,\sqsubseteq )$$ in which every directed set has a supremum. If $$p,q\in P$$, then we say that “p is far below q” whenever for any directed set D with $$q \sqsubseteq \sup (D)$$, there is some $$d \in D$$ with $$p\sqsubseteq d.$$ A domain is a dcpo in which every element q is the supremum of the directed set $$\{p\in P: p \text{ is } \text{ far } \text{ below } q\hbox {''}\}$$. This notion has been introduced by D. Scott as a model for the $$\lambda$$-calculus, for more information see [1, 10]. Domain representable topological spaces were introduced by Bennett and Lutzer [2]. We say that a topological space is domain representable if it is homeomorphic to the space of maximal elements of some domain topologized with the Scott topology. In 2013, Fleissner and Yengulalp [3] introduced an equivalent definition of a domain representable space for $$T_1$$ topological spaces. We do not assume the antisymmetry condition on the relation $$\ll$$. As Önal and Vural suggested in [11], if we need an additional antisymmetric property, let us consider the equivalent relation E on the set Q defined by “pEq if and only if ($$p\ll q$$ and $$q\ll p$$) or $$p=q$$”. We do not assume any separation axioms, if it is not explicitly stated.

We say that a topological space X is F-Y (Fleissner–Yengulalp) countably domain representable if there is a triple $$(Q,\ll , B)$$ such that

(D1):

$$B: Q\rightarrow \tau ^*(X)$$ and $$\{B(q):q\in Q\}$$ is a base for $$\tau (X)$$,

(D2):

$$\ll$$ is a transitive relation on Q,

(D3):

for all $$p,q\in Q$$, $$p\ll q$$ implies $$B(p)\supseteq B(q)$$,

(D4):

for all $$x\in X$$, the set $$\{q\in Q:x\in B(q)\}$$ is upward directed by $$\ll$$ (every pair of elements has an upper bound),

(D5$$_{\omega _1}$$):

if $$D\subseteq Q$$ and $$(D,\ll )$$ is countable and upward directed, then $$\bigcap \{B(q): q\in D\}\ne \emptyset$$.

If the conditions (D1)–(D4) and the condition

1. (D5)

if $$D\subseteq Q$$ and $$(D,\ll )$$ is upward directed, then $$\bigcap \{B(q): q\in D\}\ne \emptyset$$

are satisfied, we say that the space X is F-Y domain representable.

In [4], Fleissner and Yengulalp introduced the notion of a $$\pi$$-domain representable space, as this is analogous to the notion of a domain representable space.

We say that a topological space X is F-Y (Fleissner–Yengulalp) countably$$\pi$$-domain representable if there is a triple $$(Q,\ll , B)$$ such that

($$\pi$$D1):

$$B: Q\rightarrow \tau ^*(X)$$ and $$\{B(q):q\in Q\}$$ is a $$\pi$$-base for $$\tau (X)$$,

($$\pi$$D2):

$$\ll$$ is a transitive relation on Q,

($$\pi$$D3):

for all $$p,q\in Q$$, $$p\ll q$$ implies $$B(p)\supseteq B(q)$$,

($$\pi$$D4):

if $$q,p\in Q$$ satisfy $$B(q)\cap B(p)\ne \emptyset$$, there exists $$r\in Q$$ satisfying $$p,q\ll r$$,

($$\pi$$D5$$_{\omega _1}$$):

if $$D\subseteq Q$$ and $$(D,\ll )$$ is countable and upward directed, then $$\bigcap \{B(q): q\in D\}\ne \emptyset$$.

If the conditions ($$\pi$$D1)–($$\pi$$D4) and the condition

($$\pi$$D5):

if $$D\subseteq Q$$ and $$(D,\ll )$$ is upward directed, then $$\bigcap \{B(q): q\in D\}\ne \emptyset$$

are satisfied, we say that the space X is F-Y$$\pi$$-domain representable.

## 2 $$\pi$$-domain representable spaces

In [5], Kenderov and Revalski have shown that the set $$E=\{f\in C(X):f\text { attains its minimum in }X\}$$ contains a $$G_\delta$$ dense subset of C(X) is equivalent to the existence of a winning strategy for player $$\alpha$$ in the Banach–Mazur game. Oxtoby [9] showed that if X is a metrizable space, then player $$\alpha$$ has a winning strategy in BM(X) if and only if X contains a dense completely metrizable subspace. Krawczyk and Kubiś [6] have characterized the existence of winning strategies for player $$\alpha$$ in the abstract Banach–Mazur game played with finitely generated structures instead of open sets. In [7], there has been presented a version of the Banach–Mazur game played on a partially ordered set. We give a characterization of the existence of a winning strategy for player $$\alpha$$ in the Banach–Mazur game using the notion “$$\pi$$-domain representable space” introduced by W. Fleissner and L. Yengulalp.

### Theorem 1

A topological space X is weakly $$\alpha$$-favorable if and only if X is F-Y countably $$\pi$$-domain representable.

### Proof

If X is F-Y countably $$\pi$$-domain representable, then it is easy to show that X is weakly $$\alpha$$-favorable.

Assume that X is weakly $$\alpha$$-favorable. We shall show that X is F-Y countably $$\pi$$-domain representable. Let s be a winning strategy for player $$\alpha$$ in BM(X). We consider a family Q consisting of all finite sequences $$\Big (\overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{U}_i(j_i)\Big )$$, where $$\overrightarrow{U}_m(j_m)=(U^m_0,\ldots , U^m_{j_m})$$ is a partial play and $$m\le i,$$ i.e.,

\begin{aligned} U^m_0\supseteq s(U^m_0)\supseteq U^m_1\supseteq s(U^m_0,U^m_1)\supseteq \ldots \supseteq U^m_{j_m} \supseteq s(U^m_0,\ldots ,U^m_{j_m}) \end{aligned}

and $$s(\overrightarrow{U}_0(j_0))\supseteq \ldots \supseteq s(\overrightarrow{U}_i(j_i))$$.

Let us define a relation $$\ll$$ on the family Q:

\begin{aligned}&\left( \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{U}_i(j_i)\right) \ll \left( \overrightarrow{W}_0(l_0),\ldots ,\overrightarrow{W}_k(l_k)\right) \text { iff }\\&s(\overrightarrow{U}_i(j_i))\supseteq s(\overrightarrow{W}_0(l_0))\\&\quad \& \;i\le k \; \& \; \forall \,{t\le i}\;\exists \,{r\le k}\; \overrightarrow{U}_t(j_t)\preceq \overrightarrow{W}_r(l_r). \end{aligned}

Since $$\preceq$$ is transitive, $$\ll$$ is transitive.

Let us define a map $$B:Q\rightarrow \tau ^*(X)$$ by the formula

\begin{aligned} B\left( \left( \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{U}_i(j_i)\right) \right) =s(\overrightarrow{U}_i(j_i)) \end{aligned}

for $$\left( \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{U}_i(j_i)\right) \in Q$$.

Since $$\{s(V):V\in \tau ^*(X)\}$$ is a $$\pi$$-base, $$\{B(q):q\in Q\}$$ is a $$\pi$$-base for $$\tau$$. It is easy to see that the map B satisfies the condition $$(\pi \mathrm{D}3)$$.

Towards item ($$\pi$$D4), let $$p,q\in Q$$ be such that $$B(q)\cap B(p)\ne \emptyset$$ and $$p=\left( \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{U}_i(j_i)\right)$$, $$q=\left( \overrightarrow{W}_0(l_0),\ldots ,\overrightarrow{W}_k(l_k)\right)$$. Since $$V=B(p)\cap B(q)\subseteq s(\overrightarrow{U}_0(j_0))$$ and s is a winning strategy, we find an element $$\overrightarrow{U}'_0(j'_0)$$ stronger than $$\overrightarrow{U}_0(j_0)$$ such that $$s(\overrightarrow{U}'_0(j'_0))\subseteq V$$. Step by step we find a partial play $$\overrightarrow{U}'_t(j'_t)$$ such that $$\overrightarrow{U}_t(j_t)\preceq \overrightarrow{U}'_t(j'_t)$$ and $$s(\overrightarrow{U}'_t(j'_t))\subseteq s(\overrightarrow{U}'_{t-1}(j'_{t-1}))$$ for $$t\le i$$. Since $$s(\overrightarrow{U}'_i(j'_i))\subseteq s(\overrightarrow{W}_{0}(l_0))$$, we find a partial play $$\overrightarrow{W}'_0(l'_0)$$ such that $$\overrightarrow{W}_0(l_0) \preceq \overrightarrow{W}'_0(l'_0)$$ and $$s(\overrightarrow{W}'_0(l'_0))\subseteq s(\overrightarrow{U}'_i(j'_i))$$. Similarly, as for the sequence p, for the sequence q, we define $$\overrightarrow{W}'_t(l'_t)$$ with $$\overrightarrow{W}_t(l_t)\preceq \overrightarrow{W}'_t(l'_t)$$ and $$s(\overrightarrow{W}'_t(l'_t))\subseteq s(\overrightarrow{W}'_{t-1}(l'_{t-1}))$$ for all $$t\le k$$.

Continuing in this way, we get an element $$r=\Big (\overrightarrow{U}'_0(j'_0),\ldots ,\overrightarrow{U}'_i(j'_i), \overrightarrow{W}'_0(l'_0), \ldots ,\overrightarrow{W}'_k(l'_k)\Big )$$ such that $$p,q\ll r$$ and $$r\in Q.$$

Next we show the condition ($$\pi$$D5$$_{\omega _1}$$). Let $$D\subseteq Q$$ be a countable upward directed set and let $$D=\{p_n: n\in \omega \}$$. We define a chain $$\{q_n: n\in \omega \}\subseteq D\subseteq Q$$ such that $$p_n\ll q_n$$ for $$n\in \omega$$. By the condition ($$\pi$$D3), we get $$\bigcap \{B(q_n):n\in \omega \}\subseteq \bigcap \{B(p):p\in D\}$$. Each $$q_n\in Q$$ is of the form $$q_n=\left( \overrightarrow{W}^n_0(l^n_0),\ldots , \overrightarrow{W}^n_{k_n}(l^n_{k_n})\right) .$$

Since $$q_0\ll q_1$$, there is $$j_1\le k_1$$ such that $$\overrightarrow{W}^0_0(l^0_0)\preceq \overrightarrow{W}^1_{j_1}(l^1_{j_1})$$. We have

\begin{aligned} s(\overrightarrow{W}^0_0(l^0_0))\supseteq B(q_0)=s(\overrightarrow{W}^0_{k_0}(l^0_{k_0}))\supseteq s(\overrightarrow{W}^1_{j_1}(l^1_{j_1}))\supseteq B(q_1)=s(\overrightarrow{W}^1_{k_1}(l^1_{k_1})) . \end{aligned}

Let $$\overrightarrow{U}'_0(l^0_0)=\overrightarrow{W}^0_0(l^0_0)$$ and $$\overrightarrow{U}'_1(l^1_{j_1})=\overrightarrow{W}^1_{j_1}(l^1_{j_1})$$. Inductively, we can choose a sequence $$\{s(\overrightarrow{U}'_n(l^n_{j_n})):n\in \omega \}$$ such that $$\overrightarrow{U}_n'(l^n_{j_n}) \preceq \overrightarrow{U}'_{n+1}(l^{n+1}_{j_{n+1}})$$ and

\begin{aligned} B(q_{n}) \supseteq s(\overrightarrow{U}'_{n+1}(l^{n+1}_{j_{n+1}}))\supseteq B(q_{n+1}). \end{aligned}

Since s is a winning strategy for player $$\alpha$$, we have

\begin{aligned} \emptyset \ne \bigcap \{s(\overrightarrow{U}'_n(l^n_{j_n})):n\in \omega \}=\bigcap \{B(q_n): n\in \omega \}\subseteq \bigcap \{B(p):p\in D\}. \end{aligned}

$$\square$$

We give an example of a space, which is F-Y countably domain representable, but which is not F-Y $$\pi$$-domain representable. Note that this space is F-Y countably $$\pi$$-domain representable and not F-Y domain representable.

### Example 1

We consider the space

\begin{aligned} X=\sigma \big (\{0,1\}^{\omega _1}\big )=\big \{x\in \{0,1\}^{\omega _1}: |\hbox {supp}\; x|\le \omega \big \}, \end{aligned}

where $$\hbox {supp}\;x=\{\alpha \in \omega _1:x(\alpha )=1\}$$ for $$x\in \{0,1\}^{\omega _1}$$, with the topology ($$\omega _1$$-box topology) generated by the base

\begin{aligned} {\mathcal {B}}=\big \{\hbox {pr}_A^{-1}(x): A\in [\omega _1]^{\le \omega }, x\in \{0,1\}^A\big \} , \end{aligned}

where $$\hbox {pr}_A: \sigma (\{0,1\}^{\omega _1})\rightarrow \{0,1\}^A$$ is a projection.

We shall define a triple $$(Q,\ll , B)$$. Let $$Q={\mathcal {B}}$$, and the map $$B:Q\rightarrow Q$$ be the identity. Define a relation $$\ll$$ in the following way:

\begin{aligned} \hbox {pr}^{-1}_A(x_A)\ll \hbox {pr}^{-1}_B(x_B) \Leftrightarrow \hbox {pr}^{-1}_A(x_A)\supseteq \hbox {pr}^{-1}_B(x_B) \end{aligned}

for any $$\hbox {pr}^{-1}_A(x_A), \hbox {pr}^{-1}_B(x_B)\in {\mathcal {B}}$$. It is easy to see that the relation $$\ll$$ is transitive and that it satisfies the condition (D3). Now, we prove the condition (D4). Let $$x\in X$$ and $$\hbox {pr}^{-1}_{A_1}(x_{A_1}), \hbox {pr}^{-1}_{A_2}(x_{A_2}) \in \{\hbox {pr}^{-1}_A(x_A)\in {\mathcal {B}}:x\in \hbox {pr}^{-1}_A(x_A)\}$$. Since $$x\in \hbox {pr}^{-1}_{A_1}(x_{A_1})\cap \hbox {pr}^{-1}_{A_2}(x_{A_2})$$, we get $$x_{A_1}\upharpoonright A_2=x_{A_2}\upharpoonright A_1$$. Set $$A_3=A_1\cup A_2$$ and let $$x_{A_3}\in \{0,1\}^{A_3}$$ be such that $$x_{A_3}\upharpoonright A_2=x_{A_2}$$ and $$x_{A_3}\upharpoonright A_1=x_{A_1}$$. We have $$x_{A_3}\in \{0,1\}^{A_3}$$ such that $$x\in \hbox {pr}^{-1}_{A_3}(x_{A_3})\subseteq \hbox {pr}^{-1}_{A_1}(x_{A_1})\cap \hbox {pr}^{-1}_{A_2}(x_{A_2})$$. Hence $$\hbox {pr}^{-1}_{A_1}(x_{A_1}),\hbox {pr}^{-1}_{A_2}(x_{A_2}) \ll \hbox {pr}^{-1}_{A_3}(x_{A_3})$$.

We prove the condition $$(D5_{\omega _1})$$. Let $$D\subseteq {\mathcal {B}}$$ be a countable upward directed family. We can construct a chain $$\{\hbox {pr}^{-1}_{A_{n}}(x_{A_{n}}): n\in \omega \}\subseteq D$$ such that for each set $$\hbox {pr}^{-1}_{A}(x_{A})\in D$$, there exists $$n\in \omega$$ such that $$\hbox {pr}^{-1}_{A}(x_{A})\ll \hbox {pr}^{-1}_{A_{n}}(x_{A_{n}})$$.

Let $$B=\bigcup \{A_n:n\in \omega \}.$$ Since $$\{\hbox {pr}^{-1}_{A_n}(x_{A_n}): n\in \omega \}$$ is a chain, there is $$x_B\in \{0,1\}^B$$ such that $$x_B\upharpoonright A_n=x_{A_n}$$ for $$n\in \omega .$$ Then

\begin{aligned} \bigcap \{\hbox {pr}^{-1}_{A_n}(x_{A_n}): n\in \omega \}=\hbox {pr}^{-1}_B(x_B)\in {\mathcal {B}}, \end{aligned}

and $$\hbox {pr}^{-1}_B(x_B)\subseteq \bigcap D.$$ This completes the proof that the space $$\sigma \big (\{0,1\}^{\omega _1}\big )$$ is F-Y countably domain representable.

Now we show that $$X=\sigma \big (\{0,1\}^{\omega _1}\big )$$ is not F-Y $$\pi$$-domain representable. Suppose that there exists a triple $$(Q,\ll , B)$$ satisfying the conditions ($$\pi$$D1)–($$\pi$$D5). The family $${\mathcal {P}}=\{B(q):q\in Q\}$$ is a $$\pi$$-base. By induction, we define a sequence $$\{Q_\alpha : \alpha <\omega _1\}$$ such that the following conditions are satisfied:

1. (1)

$$Q_\alpha \in [Q]^{\le \omega }$$ and $$Q_\alpha$$ is upward directed, for $$\alpha <\omega _1$$,

2. (2)

$$\bigcap \{B(q):q\in Q_\alpha \}=\hbox {pr}^{-1}_{A_\alpha }(x_{A_\alpha }) \in {\mathcal {B}}$$ for some $$A_\alpha \in [\omega _1]^{\le \omega }$$ and some $$x_{A_\alpha }\in \{0,1\}^{A_\alpha }$$, for $$\alpha <\omega _1$$,

3. (3)

$$Q_\alpha \subseteq Q_\beta ,$$ for $$\alpha<\beta <\omega _1$$,

4. (4)

if $$\bigcap \{B(q):q\in Q_\alpha \}=\hbox {pr}^{-1}_{A_\alpha }(x_{A_\alpha })$$ and $$\bigcap \{B(q):q\in Q_\beta \}=\hbox {pr}^{-1}_{A_\beta }(x_{A_\beta })$$ for some $$A_\alpha ,A_\beta \in [\omega _1]^{\le \omega }$$ and $$x_{A_\alpha }\in \{0,1\}^{A_\alpha }$$ and $$x_{A_\beta }\in \{0,1\}^{A_\beta }$$, then $$\hbox {supp}\; x_{A_\alpha }=\{\alpha \in A_\alpha :x(\alpha )=1\}\varsubsetneq \{\alpha \in A_\beta :x(\alpha )=1\}=\hbox {supp}\; x_{A_\beta }$$, for $$\alpha<\beta <\omega _1$$.

We define a set $$Q_0$$. Take any $$r_0\in Q$$. There exist a set $$A_0\in [\omega _1]^{\le \omega }$$ and $$x_{A_0}\in \{0,1\}^{A_0}$$ such that $$\hbox {pr}_{A_0}^{-1}(x_{A_0})\subseteq B(r_0)$$. By conditions $$(\pi D1), (\pi D3),(\pi D4)$$, there exists $$r_1\in Q$$ such that $$r_0\ll r_1$$ and $$B(r_1)\subseteq \hbox {pr}_{A_0}^{-1}(x_{A_0}).$$ Assume that we have defined $$r_0\ll \ldots \ll r_n$$ and a chain $$\{A_i:i\le n\}\subseteq [\omega _1]^{\le \omega }$$ and $$x_{A_i}\in \{0,1\}^{A_i}$$ such that

\begin{aligned} \hbox {pr}^{-1}_{A_{i-1}}(x_{A_{i-1}})\supseteq B(r_i)\supseteq \hbox {pr}_{A_i}^{-1}(x_{A_i}) \text { for } i\le n. \end{aligned}

By conditions ($$\pi$$D1), ($$\pi$$D3), ($$\pi$$D4), there exists $$r_{n+1}\in Q$$ such that $$r_n\ll r_{n+1}$$ and $$B(r_{n+1})\subseteq \hbox {pr}_{A_{n}}^{-1}(x_{A_{n}}).$$ There exist a set $$A_{n+1}\in [\omega _1]^{\le \omega }$$ and $$x_{A_{n+1}}\in \{0,1\}^{A_{n+1}}$$ such that $$\hbox {pr}_{A_{n+1}}^{-1}(x_{A_{n+1}})\subseteq B(r_{n+1})$$. Let $$Q_0=\{r_n:n\in \omega \}$$. Then $$\bigcap \{B(q):q\in Q_0\}=\bigcap \{\hbox {pr}_{A_{n}}^{-1}(x_{A_{n}}):n\in \omega \}=\hbox {pr}_{A}^{-1}(x_{A})$$, where $$A=\bigcup \{A_n:n\in \omega \}$$ and $$x_A\in \{0,1\}^A$$ and $$x_A\upharpoonright A_n=x_{A_n}$$ for all $$n\in \omega .$$

Assume that we have defined $$\{Q_\alpha : \alpha < \beta \}$$ which satisfies the conditions (1)–(4).

Let $${\mathcal {R}}_\beta = \bigcup \{Q_\alpha :\alpha < \beta \}.$$ The set $${\mathcal {R}}_\beta$$ is upward directed by conditions (3), (1). Let $${\mathcal {R}}_\beta =\{p_n:n\in \omega \}$$. By (2) and (3), we get $$\bigcap \big \{B(p_n): n\in \omega \big \}=\hbox {pr}^{-1}_{A_\beta }(x_{A_\beta })\in {\mathcal {B}}$$ for some set $$A_\beta \in [\omega _1]^{\le \omega }$$ and $$x_{A_\beta }\in \{0,1\}^{A_\beta }.$$ There exist a set $$A\in [\omega _1]^{\le \omega }$$ and $$x_{A}\in \{0,1\}^{A}$$ such that $$\hbox {pr}_{A}^{-1}(x_{A})\varsubsetneq \hbox {pr}_{A_\beta }^{-1}(x_{A_\beta })$$ and $$\hbox {supp}\; x_{A_\beta }\varsubsetneq \hbox {supp}\; x_{A}.$$ Since $${\mathcal {P}}$$ is a $$\pi$$-base, we can find $$r_\beta \in Q$$ such that $$B(r_\beta )\subseteq \hbox {pr}_{A}^{-1}(x_{A})$$. Inductively, we can define a sequence $$\{q_n:n\in \omega \}\subseteq Q$$, a chain $$\{A_n:n\in \omega \}\subseteq [\omega _1]^{\le \omega }$$, and a sequence $$\{x_{A_n}\in \{0,1\}^{A_n}:n\in \omega \}$$ such that $$r_\beta , p_0\ll q_0$$, $$q_{n-1},p_n\ll q_n$$, and

\begin{aligned} B(q_n)\supseteq \hbox {pr}_{A_n}^{-1}(x_{A_n})\supseteq B(q_{n+1}) \text { for } n\in \omega . \end{aligned}

Let $$Q_\beta ={\mathcal {R}}_\beta \cup \{q_n:n\in \omega \}.$$ The set $$Q_\beta$$ satisfies conditions (1)–(4), so we finish the induction. The set $$\bigcup \{Q_\alpha :\alpha < \omega _1\}$$ is upward directed.

By conditions (2), (3), we have

\begin{aligned}&\bigcap \{B(q):q\in \bigcup \{Q_\alpha :\alpha< \omega _1\}\big \}=\bigcap \{\hbox {pr}_{A_\alpha }^{-1}(x_{A_\alpha }): \alpha<\omega _1\}=\\&\quad =\pi _A^{-1}(x_A), \text { for } A=\bigcup \{A_\alpha :\alpha<\omega _1\} \text { and } x_A\in \{0,1\}^A \\&\quad \text { such that } x_A\upharpoonright A_\alpha =x_{A_\alpha } \text { for } \alpha <\omega _1, \end{aligned}

where $$\pi _A:\{0,1\}^{\omega _1}\rightarrow \{0,1\}^A$$ is the projection. By condition (4), we get $$|\hbox {supp}\; x_A|=\omega _1$$. Hence $$\pi _A^{-1}(x_A)\cap \sigma \big (\{0,1\}^{\omega _1}\big )=\emptyset$$, a contradiction. $$\square$$

Note that by the proof of [4, Proposition 8.3] it follows that if there exists a triple $$(Q, \ll , B)$$, which satisfies the conditions of the definition of F-Y countably $$\pi$$-domain representable and $$|\bigcap \{B(q): q\in D\}|=1$$ for every countable and upward directed set $$D\subseteq Q$$, then the space X is F-Y $$\pi$$-domain representable by this triple.

### Theorem 2

The Cartesian product of any family of F-Y countably $$\pi$$-domain representable spaces is F-Y countably $$\pi$$-domain representable.

### Proof

Let X be a product of a family $$\{X_a:a\in A\}$$ of F-Y countably $$\pi$$-domain representable spaces. Let $$(Q_a,\ll _a, B_a)$$ be a triple which satisfies conditions ($$\pi$$D1)–($$\pi$$D4) and ($$\pi$$D5$$_{\omega _1}$$) for the space $$X_a.$$ Any basic nonempty open subset U in X is of the form $$U=\prod \{U_a:a\in A\}$$, where $$U_a$$ is nonempty open subset of $$X_a$$ and $$U_a=X_a$$ for all but a finite number of $$a\in A$$. We may assume that $$0_a\in Q_a$$ is the least element in $$Q_a$$ and $$B_a(0_a)=X_a$$ for each $$a\in A$$. Put

\begin{aligned} Q=\left\{ p\in \prod \{Q_a:a\in A\}:|\{a\in A:p(a)\ne 0_a\}|<\omega \right\} . \end{aligned}

Define a relation $$\ll$$ on Q by the formula

\begin{aligned} p\ll q\Longleftrightarrow p(a)\ll _a q(a) \text { for all } a\in A, \end{aligned}

where $$p,q\in Q$$. Let us define a map $$B:Q\rightarrow \tau ^*(X)$$ by $$B(p)=\prod \{B_a(p(a)):a\in A\}$$, where $$p\in Q$$. It is easy to check that $$(Q,\ll ,B)$$ is a F-Y countably $$\pi$$-domain representing X. $$\square$$

In a similar way, one can prove the above theorem also for F-Y countably domain representable, F-Y $$\pi$$-domain representable, and F-Y domain representable.

## 3 Domain representable spaces

In 2003, Martin [8] showed that if a space is domain representable, then player $$\alpha$$ has a winning strategy in the strong Choquet game. In 2015, Fleissner and Yengulalp [4] showed that it is sufficient that a space is F-Y countably domain representable. Now, we shall show that the property of being F-Y countably domain representable is necessary. For this purpose, we can use a triple $$(Q,\ll , B)$$ defined in [4, Proposition 8.3] or we can use a similar triple to the triple defined in the Theorem 1. Namely, if s is a winning strategy for player $$\alpha$$, we consider a family Q consisting of all finite sequences $$(\overrightarrow{x_0}\circ \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i))$$, where $$\overrightarrow{x_m}\circ \overrightarrow{U}_m(j_m)=(U^m_0,x^m_0,\ldots , U^m_{j_m},x^m_{j_m})$$ is a partial play in the strong Choquet game for all $$m\le i$$, i.e.,

\begin{aligned}&U^m_0\supseteq s(U^m_0,x^m_0)\supseteq U^m_1\supseteq s(U^m_0,x^m_0,U^m_1,x^m_1)\supseteq \ldots \supseteq U^m_{j_m}\\&\quad \supseteq s(U^m_0,x^m_0,\ldots ,U^m_{j_m},x^m_{j_m}) \end{aligned}

and $$s(\overrightarrow{x_0}\circ \overrightarrow{U}_0(j_0))\supseteq \ldots \supseteq s(\overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i))$$.

Let us define a relation $$\ll$$ on the family Q:

\begin{aligned}&\left( \overrightarrow{x_0}\circ \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i)\right) \ll \left( \overrightarrow{y_0}\circ \overrightarrow{W}_0(l_0),\ldots ,\overrightarrow{y_k}\circ \overrightarrow{W}_k(l_k)\right) \\&\text { iff } s\left( \overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i)\right) \supseteq s\left( \overrightarrow{y_0}\circ \overrightarrow{W}_0(l_0)\right) \& \; i\le k\; \& \\&\forall \,{t\le i}\;\exists \,{r\le k}\; \overrightarrow{x_t}\circ \overrightarrow{U}_t(j_t)\preceq \overrightarrow{y_r}\circ \overrightarrow{W}_r(l_r). \end{aligned}

We define a map $$B:Q\rightarrow \tau ^*$$ by the formula

\begin{aligned} B \left( \left( \overrightarrow{x_0}\circ \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i)\right) \right) =s\left( \overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i)\right) \end{aligned}

for each $$\left( \overrightarrow{x_0}\circ \overrightarrow{U}_0(j_0),\ldots ,\overrightarrow{x_i}\circ \overrightarrow{U}_i(j_i)\right) \in Q$$.

As a consequence, we obtain:

### Theorem 3

A topological space X is Choquet complete if and only if it is F-Y countably domain representable.