Some remarks on the projective properties of Menger and Hurewicz

TL;DR

This paper explores how the projective versions of Menger and Hurewicz properties in Tychonoff spaces can be characterized through their embeddings in the Čech-Stone compactification, extending known descriptions of these properties.

Contribution

It provides new characterizations of the projective Menger and Hurewicz properties in terms of their placement within the Čech-Stone compactification.

Findings

Analogous characterizations for projective Menger and Hurewicz properties

Extension of known space embedding characterizations to projective properties

Deeper understanding of the relationship between space properties and compactifications

Abstract

It is known that both the Menger and Hurewicz property of a Tychonoff space $X$ can be described by the way $X$ is placed in its \v{C}ech-Stone compactification $\beta X$. We provide analogous characterizations for the projective versions of the properties of Menger and Hurewicz.

Equations

$$\sigma (V_0,\dots ,V_{n-1})=(K_n,U_n).$$

$$f^{\#}(A)=Y\setminus f(X\setminus A).$$

$$F=\bigcup _{n\in \omega }{F}_n\subseteq \beta X\setminus X.$$

$$f:\beta X\to [0,1{]}^{\omega },$$

$$f(x)=(f_n(x){)}_{n<\omega }.$$

$$\bigcup _{n\in \omega }f(F_n)\subseteq bY\setminus Y.$$

$$\bigcup _{n\in \omega }f(F_n)\subseteq H\subseteq bY\setminus Y.$$

$$D\subseteq G\subseteq \beta X\setminus X.$$

$$H_n={\tilde{f}}^{\#}(G_n).$$

$$F\subseteq \bigcap _{n\in \omega }{H}_n\subseteq bY\setminus Y.$$

$$\sigma (\varnothing ),V_0,\sigma (V_0),V_1,\sigma (V_0,V_1),\dots$$

$$\sigma \left(V_{p|0},V_{p|1},\dots ,V_p\right)=(K,U)$$

$$K_p=\bigcap _{n\in \omega }{V}_n^{\prime },$$

$${\mathrm{cl}}_{\beta X}(V_{n+1}^{\prime })\subseteq V_n^{\prime }\subseteq {\mathrm{cl}}_{\beta X}(V_n^{\prime })\subseteq U,\text{mbox}foralln\in \omega .$$

$$Q={\omega }^{<\omega }.$$

$$f_s^{-1}(0)=K_s.$$

$$f(x)={\left(f_s(x)\right)}_{s\in Q}\in [0,1{]}^Q.$$

$$f(K_s)\subseteq bM\setminus M,\text{mbox}foreverys\in Q.$$

$$(W_{0},\ldots,W_{n})\mbox{ is an admissible $(n+1)$-tuple.}$$

$$f(K_s)=C_n\subseteq W_n\subseteq U_n.$$

$$K_s\subseteq f^{-1}(W_n).$$

$$(K_s,U_s)=\sigma (V_{s|0},V_{s|1},\dots ,V_s).$$

$$V_{s⌢k}\subseteq f^{-1}(W_n)\cap U_s$$

$$\tau (W_0,\dots ,W_n)=(f(K_{s⌢k}),W_n).$$

$$\tau (\varnothing ),W_0,\tau (W_0),W_1,\tau (W_0,W_1),\dots$$

$$\varnothing \ne \bigcap _{n\in \omega }{W}_n\subseteq bM\setminus M.$$

$$V_{s|n}\subseteq f^{-1}(W_{n-1}),\text{mbox}foreveryn\in \omega .$$

$$\sigma (\varnothing ),V_{s|0},\sigma (V_{s|0}),V_{s|1},\sigma (V_{s|0},V_{s|1}),\dots$$

$$K_0^{\prime }={\tilde{f}}^{-1}(K_0)\text{mbox}and{U}_0^{\prime }={\tilde{f}}^{-1}(U_0).$$

$$\tau (\varnothing )=(K_0^{\prime },U_0^{\prime })$$

$$\tau (V_0^{\prime },\dots ,V_{n-1}^{\prime })=\left({\tilde{f}}^{-1}(K_n),{\tilde{f}}^{-1}(U_n)\right).$$

Full text

1. Introduction

All spaces in this note are assumed to be Tychonoff topological spaces. Let $\mathcal{P}$ be a topological property. We say that a space $X$ is projectively $\mathcal{P}$ provided every separable metrizable continuous image of $X$ has property $\mathcal{P}$. This notion was studied by several authors for different topological properties $\mathcal{P}$. In this note we are concerned with the case when $\mathcal{P}$ is either the Menger property or the property of Hurewicz.

Let us recall that a topological space $X$ is Menger (resp., Hurewicz) if for every sequence $(\mathscr{U}_{n})_{n\in\omega}$ of open covers of $X$, there is a sequence $(\mathscr{V}_{n})_{n\in\omega}$ such that for every $n$, $\mathscr{V}_{n}$ is a finite subfamily of $\mathscr{U}_{n}$ and the family $\bigcup_{n\in\omega}\mathscr{V}_{n}$ covers $X$ (resp., every point of $X$ is contained in $\bigcup\mathscr{V}_{n}$ for all but finitely many $n$’s).

Projective properties of Menger and Hurewicz were a subject of study before; see, e.g., Kočinac [5] or Bonanzinga et al. [2]. What makes them useful is the following fact (see [9, Proposition 2], [2, Proposition 8], [5, Theorem 3.2], [2, Proposition 31]):

Proposition 1.1**.**

A space $X$ is Menger (resp., Hurewicz) if and only if $X$ is Lindelöf and projectively Menger (resp., projectively Hurewicz).

It is known that both Menger and Hurewicz properties of a space $X$ can be conveniently characterized by the way $X$ is positioned in its Čech-Stone compactification $\beta X$. The aim of the present note is to provide analogous descriptions for their projective versions. Our interest in this sort of characterizations primarily stems from a recent work by the first author [6], [7], where they play an important role.

2. Notation

In what follows, the set of all natural numbers (including [math]) will be denoted by $\omega$. By $\omega^{<\omega}=\bigcup_{n\in\omega}\omega^{n}$ we will mean the set of all finite sequences with values in $\omega$ (including $\emptyset$), and as a natural extension, by $\omega^{\omega}$ we will denote the set of all infinite sequences with values in $\omega$. If $s=(a_{0},\ldots,a_{n-1})\in\omega^{<\omega}$ and if $k\leq n$, then $s|k=(a_{0},\ldots a_{k-1})$ is an initial segment of $s$ of length $k$. By $s|0$ we mean the empty sequence. The symbol $|s|$ stands for the length of $s$. If $s\in\omega^{<\omega}$ and $n\in\omega$ then $s\frown n$ is the sequence of length $|s|+1$ whose first $|s|$ terms is $s$ and whose last term is $n$.

For a space $X$ by $\beta X$ we denote the Čech-Stone compactification of $X$. Recall that a subset $A$ of a topological space $X$ is called a zero-set if there is a continuous map $f:X\to[0,1]$ such that $A=f^{-1}(0)$. According to Vedenissov’s lemma (see [3, 1.5.12]) if $Z$ is a compact space, then $A$ is a zero-set in $Z$ if and only if $A$ is closed $G_{\delta}$-subset of $Z$. The complement of a zero-set is called a cozero-set.

Let us describe now two topological games that will be of interest: the $k$-Porada game introduced by Telgársky in [9], and its minor modification which we shall call the $z$-Porada game.

Let $Z$ be a compact space and let $X\subseteq Z$ be a subspace of $Z$. The $k$-Porada game on $Z$ with values in $X$ is a game with $\omega$-many innings, played alternately by two players: I and II. Player I begins the game and makes the first move by choosing a pair $(K_{0},U_{0})$, where $K_{0}\subseteq X$ is a nonempty compact set and $U_{0}$ is an open set in $Z$ that contains $K_{0}$. Player II responds by choosing an open (in $Z$) set $V_{0}$ such that $K_{0}\subseteq V_{0}\subseteq U_{0}$. In the second round of the game, player I picks a pair $(K_{1},U_{1})$, where $K_{1}$ is a nonempty compact subset of $V_{0}$ and $U_{1}$ is an open subset of $Z$ with $K_{1}\subseteq U_{1}\subseteq V_{0}$. Player II responds by picking an open (in $Z$) set $V_{1}$ such that $K_{1}\subseteq V_{1}\subseteq U_{1}$. The game continues in this way and stops after $\omega$ many rounds. Player II wins the game if $\emptyset\neq\bigcap_{n\in\omega}U_{n}(=\bigcap_{n\in\omega}V_{n})\subseteq X$. Otherwise player I wins. The game described above is denoted by $kP(Z,X)$.

Let us introduce the following modification of the $k$-Porada game. As above, $Z$ is a compact space and $X$ is a subspace of $Z$. The $z$-Porada game on $Z$ with values in $X$, denoted by $zP(Z,X)$, is played as $kP(Z,X)$ with the only difference that compact sets $K_{n}$ played by player I are required to be zero-sets in $Z$ (i.e. compact $G_{\delta}$). We keep the requirement that these sets are contained in $X$.

For a space $Y$ we denote by $\mathcal{T}_{Y}$ (resp., $\mathcal{K}(Y)$) the collection of all nonempty open (resp., compact) subsets of $Y$. For a space $Y$ and its subspace $X$ we denote by $\mathcal{Z}(Y,X)$ the collection of all nonempty zero-sets in $Y$ contained in $X$. A strategy of player I in the game $kP(Z,X)$ (resp., in the game $zP(Z,X)$) is a map $\sigma$ defined inductively as follows: Set $V_{-1}=Z$. If the strategy $\sigma$ is defined for the first $n$ moves, $n\in\omega$, then an $n$-tuple $(V_{0},V_{1},\ldots,V_{n-1})\in\mathcal{T}_{Z}^{n}$ is called admissible if either the tuple is empty (i.e. $n=0$) or else $K_{0}\subseteq V_{0}\subseteq U_{0}$ and both $K_{i}\subseteq V_{i}\subseteq U_{i}$, and $(K_{i},U_{i})=\sigma(V_{0},\ldots V_{i-1})$ for $i\in\{1,\ldots,n-1\}$. For any admissible $n$-tuple $(V_{0},\ldots,V_{n-1})$ we choose a pair $(K_{n},U_{n})\in\mathcal{K}(V_{n-1}\cap X)\times\mathcal{T}_{V_{n-1}}$ (resp., $(K_{n},U_{n})\in\mathcal{Z}(Z,V_{n-1}\cap X)\times\mathcal{T}_{V_{n-1}}$) with $K_{n}\subseteq U_{n}$ and we set

[TABLE]

A strategy $\sigma$ of player I in either of the games $kP(Z,X)$ or $zP(Z,X)$, is called winning if player I wins every run of the game in which she plays according to the strategy $\sigma$.

If $f:X\rightarrow Y$ is a function, then for $A\subseteq X$ we set

[TABLE]

The following lemma is immediate.

Lemma 2.1**.**

Suppose that $f:X\rightarrow Y$ is a continuous surjection between spaces $X$ and $Y$. We have:

1. (i)

If $X$ is compact and $U\subseteq X$ is open, then $f^{\#}(U)$ is open in $Y$ 2. (ii)

For any $A\subseteq X$ and $y\in Y$ we have $y\in f^{\#}(A)$ if and only if $f^{-1}(y)\subseteq A$. 3. (iii)

For any $A\subseteq X$ and $B\subseteq Y$, if $A\subseteq f^{-1}(B)$, then $f^{\#}(A)\subseteq B$.

We will use a standard notation for the closure operator, i.e. if $X$ is a space and $A\subseteq X$, then the closure of $A$ in $X$ will be denoted by $\operatorname{cl}_{X}(A)$.

3. The projective Hurewicz property

The following characterization of the Hurewicz property was established by Just et al. [4] (for the subsets of the real line), Banakh and Zdomskyy [1] (for separable metrizable spaces) and Tall [8] (the general case).

Theorem 3.1**.**

Let $bX$ be a compactification of $X$. The following two conditions are equivalent:

1. (A)

$X$* has the Hurewicz property* 2. (B)

For every $\sigma$-compact subset $F$ of the remainder $bX\setminus X$, there exists a $G_{\delta}$-subset $G$ of $bX$ such that $F\subseteq G\subseteq bX\setminus X$.

Recall that a space $X$ is projectively Hurewicz provided every separable metrizable continuous image of $X$ is Hurewicz. The following result was obtained by Bonanzinga et al. (see [2, Theorem 30]).

Theorem 3.2**.**

(Bonanzinga, Cammaroto, Matveev) The following conditions are equivalent:

1. (A)

X is projectively Hurewicz, 2. (B)

For every sequence $(\mathscr{U}_{n})_{n\in\omega}$ of countable covers of $X$ by cozero-sets, there is a sequence $(\mathscr{V}_{n})_{n\in\omega}$ such that for every $n$, $\mathscr{V}_{n}$ is a finite subfamily of $\mathscr{U}_{n}$ and for all $x\in X$, point $x$ belongs to $\bigcup\mathscr{V}_{n}$, for all but finitely many $n$’s (i.e. the family $\{\bigcup\mathscr{V}_{n}\;:\;n\in\omega\}$ is a $\gamma$-cover of $X$).

The above theorem suggests the following counterpart of Theorem 3.1.

Theorem 3.3**.**

For a space $X$, the following conditions are equivalent:

1. (A)

$X$* is projectively Hurewicz,* 2. (B)

For any set $F\subseteq\beta X\setminus X$ being a countable union of zero-sets in $\beta X$, there exists a $G_{\delta}$-subset $G$ of $\beta X$ such that $F\subseteq G\subseteq\beta X\setminus X$.

Proof.

Let us assume that $X$ is projectively Hurewicz and let $(F_{n})_{n\in\omega}$ be a sequence of zero-sets in $\beta X$ such that

[TABLE]

For $n\in\omega$ let $f_{n}:\beta X\to[0,1]$ be a continuous map witnessing $F_{n}$ being a zero-set, i.e. $F_{n}=f_{n}^{-1}(0)$. Let

[TABLE]

be the diagonal map given by the family $\{f_{n}:n\in\omega\}$, i.e.

[TABLE]

Denote $f(X)=Y\subseteq[0,1]^{\omega}$ and $f(\beta X)=bY$. Observe that

[TABLE]

Indeed, if $y\in f(F_{n})$, then $y(n)=0$ so $f^{-1}(y)\subseteq f_{n}^{-1}(0)=F_{n}\subseteq\beta X\setminus X$. On the other hand $y\in Y$ would give $f^{-1}(y)\cap X\neq\emptyset$, because $f$ maps $X$ onto $Y$.

By our assumption, $X$ is projectively Hurewicz, so $Y$ has the Hurewicz property being a separable metrizable continuous image of $X$. It follows from (1) and Theorem 3.1 that there is a $G_{\delta}$-subset $H$ of $bY$ with

[TABLE]

We set $G=f^{-1}(H)$. Since $f$ maps $X$ onto $Y$, we get $F\subseteq G\subseteq\beta X\setminus X.$

Conversely, assume condition $(B)$. We need to show that $X$ is projectively Hurewicz. Let $Y$ be a separable metric space and let $f:X\to Y$ be a continuous surjection. Let $bY$ be a metrizable compactification of $Y$.

The map $f$ can be uniquely extended to the continuous map $\widetilde{f}:\beta X\rightarrow bY$. In order to prove that $Y$ has the Hurewicz property, we will apply Theorem 3.1. To this end, let us fix a $\sigma$-compact subset $F$ of $bY\setminus Y$. We need to find a $G_{\delta}$-set $H$ in $bY$, such that $F\subseteq H\subseteq bY\setminus Y$.

Let $D=\widetilde{f}^{-1}(F)$. Since $F$ is a $\sigma$-compact subset of a metric space $bY$, it is a countable union of zero-sets so the set $D$ is a countable union of zero-sets too. Moreover, since $\widetilde{f}$ maps $X$ onto $Y$ and $F\cap Y=\emptyset$, we have $D\subseteq\beta X\setminus X$. Now, condition $(B)$ provides a $G_{\delta}$-set $G$ in $\beta X$ such that

[TABLE]

Write $G=\bigcap_{n\in\omega}G_{n}$, where $G_{n}$ is open for every $n\in\omega$. For each $n\in\omega$ we set

[TABLE]

By Lemma 2.1 the set $H_{n}$ is open, for every $n\in\omega$. We claim that $H=\bigcap_{n\in\omega}H_{n}$ is a desired $G_{\delta}$-subset of $bY$, i.e.

[TABLE]

Indeed, if $y\in F$, then $\widetilde{f}^{-1}(y)\subseteq D\subseteq G$, so $\widetilde{f}^{-1}(y)\subseteq G_{n}$, for every $n\in\omega$. This gives $y\in H_{n}$, for every $n\in\omega$ (cf. Lemma 2.1). For the second inclusion, fix $y\in\bigcap_{n\in\omega}H_{n}$. It follows from Lemma 2.1 that $\widetilde{f}^{-1}(y)\subseteq\bigcap_{n\in\omega}G_{n}=G$. Since $G$ is disjoint from $X$ and $\widetilde{f}$ maps $X$ onto $Y$ we have $y\in bY\setminus Y$. ∎

4. The projective Menger property

The theorem below, characterizing the Menger property, is due to Telgársky [9, Theorem 2].

Theorem 4.1**.**

(Telgársky) Let $bX$ be a compactification of $X$. The following two conditions are equivalent.

1. (A)

$X$* has the Menger property* 2. (B)

Player I has no winning strategy in the $k$-Porada game

$kP(bX,bX\setminus X)$**

Recall that a space $X$ is projectively Menger provided every separable metrizable continuous image of $X$ is Menger. The following result was established by Bonanzinga et al. (see [2, Theorem 6]).

Theorem 4.2**.**

(Bonanzinga, Cammaroto, Matveev) The following conditions are equivalent:

1. (A)

$X$* is projectively Menger,* 2. (B)

For every sequence $(\mathscr{U}_{n})_{n\in\omega}$ of countable covers of $X$ by cozero-sets, there is a sequence $(\mathscr{V}_{n})_{n\in\omega}$ such that for every $n$, $\mathscr{V}_{n}$ is a finite subfamily of $\mathscr{U}_{n}$ and the family $\bigcup_{n\in\omega}\mathscr{V}_{n}$ covers $X$.

This suggests the following counterpart of Theorem 4.1.

Theorem 4.3**.**

The following two conditions are equivalent:

1. (A)

$X$* has the projective Menger property* 2. (B)

Player I has no winning strategy in the $z$-Porada game

$zP(\beta X,\beta X\setminus X)$**

Proof.

Assume that $X$ is projectively Menger. Let $\sigma$ be a strategy for player I in the $z$-Porada game $zP(\beta X,\beta X\setminus X)$. We need to show that there is a play

[TABLE]

where player I applies her strategy and fails, i.e. $\emptyset\neq\bigcap_{n\in\omega}{V_{n}}\subseteq\beta X\setminus X$.

By induction on $|s|$, for $s\in\omega^{<\omega}$, we construct an open subset $V_{s}$ of $\beta X$ and a zero-set $K_{s}$ in $\beta X\setminus X$ such that:

1. (i)

For every $s\in\omega^{<\omega}$, the tuple $\left(V_{s|0},V_{s|1},\ldots,V_{s}\right)$ is admissible. 2. (ii)

If $\sigma(V_{s|0},V_{s|1},\ldots,V_{s})=(K,U)$, then $K_{s}=K$, for every $s\in\omega^{<\omega}$. 3. (iii)

$\operatorname{cl}_{\beta X}(V_{s\frown(n+1)})\subseteq V_{s\frown n}$, for every $s\in\omega^{<\omega}$ and $n\in\omega$ 4. (iv)

$K_{s}=\bigcap_{n\in\omega}V_{s\frown n}$ for every $s\in\omega^{<\omega}$ 5. (v)

$\operatorname{cl}_{\beta X}(V_{s\frown n})\subseteq V_{s}$, for every $s\in\omega^{<\omega}$ and $n\in\omega$.

We have $\sigma(\emptyset)=(L,W)$, for some $L\in\mathcal{Z}(\beta X,\beta X\setminus X)$ and an open subset $W$ of $\beta X$ satisfying $L\subseteq W$.

Set $V_{\emptyset}=W$. Fix $m\in\omega$ and let $p\in\omega^{<\omega}$ be a sequence of length $m$. Suppose that the sets $V_{s}$ are constructed for all $s$ satisfying $|s|\leq m$ and the sets $K_{s}$ are constructed for all $s$ satisfying $|s|<m$, in such a way that the conditions (i)–(v) are satisfied. We will define $K_{p}$ and $V_{p\frown n}$, for all $n\in\omega$. Since (i) holds for $p$, the pair

[TABLE]

is well defined and $K\subseteq U\subseteq V_{p}$. We set $K_{p}=K$. Now, the set $K_{p}$ is a zero-subset of $\beta X$ so we can write

[TABLE]

We set $V_{p\frown n}=V^{\prime}_{n}$. This finishes the inductive construction.

To simplify notation, let us denote

[TABLE]

For every $s\in Q$, fix a continuous map $f_{s}:\beta X\to[0,1]$ witnessing $K_{s}$ being zero-subset of $\beta X$, i.e.

[TABLE]

Let $f:\beta X\to[0,1]^{Q}$ be the diagonal map given by the family $\{f_{s}:s\in Q\}$, i.e.

[TABLE]

Let $M=f(X)$ and $bM=f(\beta X)$. Since, $Q$ is countable, the space $bM$ is metrizable.

Note that

[TABLE]

True, take $a\in f(K_{s})$. We have $f(K_{s})=\{y\in bM\subseteq[0,1]^{Q}:y(s)=0\}$, by (2), so $a(s)=0$. On the other hand, if $a\in M$, then $a=f(x)$ for some $x\in X$ (because $M=f(X)$). But $K_{s}\cap X=\emptyset$, so $f(x)(s)=a(s)\neq 0$, by (2); a contradiction.

Using the families $\{K_{s}:s\in Q\}$ and $\{V_{s}:s\in Q\}$, we will recursively construct a strategy $\tau$ for player I in the $k$-Porada game $kP(bM,bM\setminus M)$. We will make sure that if $(W_{0},\ldots,W_{m-1})$ is an admissible $m$-tuple for the strategy $\tau$ and if $\tau(W_{0},\ldots,W_{m-1})=(C_{m},U_{m})$, then there is $s\in Q$ of length $m$, such that

1. (vi)

$C_{m}=f(K_{s})$ 2. (vii)

$V_{s|i}\subseteq f^{-1}(W_{i-1})$, for every $1\leq i\leq m$.

Let $\tau(\emptyset)=(f(K_{\emptyset}),bM)$ and put $C_{0}=f(K_{\emptyset})$, $U_{0}=bM$. Fix $n\geq 0$ and suppose that $\tau$ is defined for all admissible $n$-tuples $(W_{0},\ldots W_{n-1})$ (if $n=0$, then a tuple $(W_{0},\ldots W_{n-1})$ is empty) in such a way that conditions (vi)-(vii) hold. Suppose that

[TABLE]

By our inductive assumption, the pair $(C_{n},U_{n})=\tau(W_{0},\ldots,W_{n-1})$ is well defined and (vi) holds, i.e. $C_{n}=f(K_{s})$ for some $s\in\omega^{n}$. By (4), we have

[TABLE]

Thus,

[TABLE]

According to (ii), there is an open set $U_{s}\supseteq K_{s}$ such that

[TABLE]

Using (iii), (iv) and (5), by compactness we can find $k\in\omega$ such that

[TABLE]

We set

[TABLE]

This finishes the construction of $\tau$. Note that $\eqref{zawieranie}$ guarantees that (vii) holds.

By our assumption, the space $X$ is projectively Menger so $M$ must be Menger being a separable metrizable continuous image of $X$. Hence, by Theorem 4.1, the strategy $\tau$ is not winning. It follows that there exists a play

[TABLE]

in which player I applies her strategy and fails, i.e.

[TABLE]

By condition (vii), the above play generates an infinite sequence $s\in\omega^{\omega}$ such that

[TABLE]

According to (i) and (ii), for every $n$, the tuple $(V_{s|0},\ldots,V_{s|n})$ is admissible and $K_{s|n}$ is the compact set in the pair $\sigma(V_{s|0},\ldots,V_{s|n})$. This means that

[TABLE]

is a play in the game $zP(\beta X,\beta X\setminus X)$. We claim that player II wins this run of the game.

Indeed, we have $\bigcap_{n\in\omega}V_{s|n}=\bigcap_{n\in\omega}\operatorname{cl}_{\beta X}(V_{s|n})$, by (v). So this intersection must be nonempty by compactness. Moreover, $\bigcap_{n\in\omega}V_{s|n}\subseteq\beta X\setminus X$, by (7) and (8) and the fact that $f(X)=M$. This finishes the proof of $(A)\Rightarrow(B)$.

To prove the converse, assume that player I has no winning strategy in the game $zP(\beta X,\beta X\setminus X)$. Striving for a contradiction, suppose that $X$ is not projectively Menger. Consequently, $X$ maps continuously onto a separable metric non-Menger space $M$. Let $f:X\to M$ be a continuous surjection. Since $M$ is a separable metric space, it has a metric compactification $bM$. Let $\widetilde{f}:\beta X\to bM$ be the continuous extension of $f$. The space $M$ is not Menger, so by Theorem 4.1, player I has a winning strategy $\sigma$ in the $k$-Porada game $kP(bM,bM\setminus M)$.

Using $\sigma$, we will inductively construct a winning strategy $\tau$ of player I in the game $zP(\beta X,\beta X\setminus X)$ in such a way that for every $n\geq 0$, the following condition is satisfied:

1. (viii)

If $(V^{\prime}_{0},\ldots,V^{\prime}_{n-1})$ is an admissible $n$-tuple in the game $zP(\beta X,\beta X\setminus X)$ (i.e. for the strategy $\tau$), then there is an admissible $n$-tuple $(V_{0},\ldots,V_{n-1})$ in the game $kP(bM,bM\setminus M)$ (i.e. for the strategy $\sigma$). And if $\sigma(V_{0},\ldots,V_{n-1})=(K_{n},U_{n})$, then $\tau(V^{\prime}_{0},\ldots,V^{\prime}_{n-1})=\left(\widetilde{f}^{-1}(K_{n}),\widetilde{f}^{-1}(U_{n})\right)$.

Let $\sigma(\emptyset)=(K_{0},U_{0})$. Denote

[TABLE]

Since $\widetilde{f}$ is surjective, the sets $K^{\prime}_{0}$ and $U^{\prime}_{0}$ are nonempty. Moreover, $K^{\prime}_{0}$ being closed $G_{\delta}$-subset of $\beta X$, is a zero-set in $\beta X$. In addition, $K^{\prime}_{0}\subseteq\beta X\setminus X$. This is because $\widetilde{f}(X)=f(X)=M$ and $K_{0}$ misses $M$. Hence, the pair

[TABLE]

is a legal move of player I in the game $zP(\beta X,\beta X\setminus X)$ and the condition (viii) holds for $n=0$.

Fix $n\geq 0$ and suppose that $\tau$ is defined for all admissible $m$-tuples $(V_{0},\ldots,V_{m-1})$, where $m\leq n$, and the condition (viii) holds for all such tuples. Let $(V^{\prime}_{0},\ldots,V^{\prime}_{n})$ be an arbitrary admissible $(n+1)$-tuple in $zP(\beta X,\beta X\setminus X)$ and let $\tau(V^{\prime}_{0},\ldots,V^{\prime}_{n-1})$ be the last move of player I (it is well defined by the inductive assumption). According to (viii), there is an admissible $n$-tuple $(V_{0},\ldots,V_{n-1})$ in the game $kP(bM,bM\setminus M)$ such that if $\sigma(V_{0},\ldots,V_{n-1})=(K_{n},U_{n})$, then

[TABLE]

The $(n+1)$-tuple $(V^{\prime}_{0},\ldots,V^{\prime}_{n})$ is admissible which means that $V^{\prime}_{n}$ is an open subset of $\beta X$ with

[TABLE]

Let

[TABLE]

By Lemma 2.1, the set $V_{n}$ is open in $bM$ and $K_{n}\subseteq V_{n}\subseteq U_{n}$. This means that the tuple $(V_{0},\ldots,V_{n-1},V_{n})$ is admissible in the game $kP(bM,bM\setminus M)$. Let

[TABLE]

Denote

[TABLE]

As for $K^{\prime}_{0}$ and $U^{\prime}_{0}$ we argue that these sets are nonempty, $K^{\prime}_{n+1}$ is a zero-set in $\beta X$, $U^{\prime}_{n+1}$ is open in $\beta X$, and $K^{\prime}_{n+1}\subseteq\beta X\setminus X$. Since $U_{n+1}\subseteq V_{n}=\widetilde{f}^{\#}(V^{\prime}_{n})$ we must have $U^{\prime}_{n+1}\subseteq V^{\prime}_{n}$ (cf. Lemma 2.1). Therefore, the pair

[TABLE]

is a well-defined $(n+1)$-st move of player I in the game $zP(\beta X,\beta X\setminus X)$. This finishes the induction.

Let us prove that the strategy $\tau$ that we have just constructed, is winning for player I. To this end, consider an arbitrary play $P$:

[TABLE]

in the game $zP(\beta X,\beta X\setminus X)$, where player I applies the strategy $\tau$. We need to show that either

[TABLE]

By (viii), the play $P$ induces a play

[TABLE]

in the game $kP(bM,bM\setminus M)$ such that, for every $n\in\omega$ we have:

[TABLE]

By our assumption, the strategy $\sigma$ is winning for player I, so either

[TABLE]

or

[TABLE]

According to (9), we have

[TABLE]

So, if (10) holds, then $\bigcap_{n\in\omega}V^{\prime}_{n}=\emptyset$, which means that player I wins in the game $zP(\beta X,\beta X\setminus X)$. Suppose that (11) holds and pick $y\in M\cap\bigcap_{n\in\omega}U_{n}$. Since the map $f:X\to M$ is surjective, there is $x\in X$ with $f(x)=\widetilde{f}(x)=y$, whence

[TABLE]

In particular the latter set is nonempty which means that player I wins in this case too. ∎

Acknowledgements

The research of the first author was supported by Fundación Séneca - ACyT Región de Murcia project 21955/PI/22, Agencia Estatal de Investigación (Government of Spain) and ERDF project PID2021-122126NB-C32 and European Union - NextGenerationEU funds through María Zambrano fellowship.