Selection Properties and Games

The selection property \(S_{fin}(\mathcal A,\mathcal B)\) states that given \(A_n\in\mathcal A\) for \(n<\omega\), there exist \(B_n\in[A_n]^{<\omega}\) such that \(\bigcup_{n<\omega}B_n\in \mathcal B\).

The selection game \(G_{fin}(\mathcal A,\mathcal B)\) proceeds analogously, with player \(\mathcal I\) choosing \(A_n\) and player \(\mathcal{II}\) choosing \(B_n\) each round of the game. \(\mathcal{II}\) wins in the case that \(\bigcup_{n<\omega}B_n\in \mathcal B\).

Introduced by Scheepers in various papers entitled "Combinatorics of open covers".

Note \(\mathcal{II}\uparrow G_{fin}(\mathcal A,\mathcal B)\) implies \(S_{fin}(\mathcal A,\mathcal B)\), but the converse need not hold.

Selective Separability

A space is selectively separable or \(SS\) if \(S_{fin}(\mathcal D,\mathcal D)\) holds, where \(\mathcal D\) denotes the dense subsets of the space.

• All \(SS\) spaces are separable.
• A space \(X\) is \(SS\) if and only if the set \(I\) of isolated points is countable and \(X\setminus I\) is \(SS\).
• Each separable Frechet space is selectively separable.

Selective Separability Game

Let \(G_{fin}(\mathcal D,\mathcal D)\) denote the selective separability game, and \(SS^+\) denote \(\mathcal{II}\uparrow G_{fin}(\mathcal D,\mathcal D)\).

• Barman/Dow gave an example of a space which is \(SS\) but not \(SS^+\).

Markov strategies

A Markov strategy for a game only considers the round number and latest move of the opponent.

Let \(SS^{+mark}\) denote \(\mathcal{II}\uparrow_{mark} G_{fin}(\mathcal D,\mathcal D)\).

• Gruenhage has asked if all \(SS^+\) spaces are \(SS^{+mark}\).
• Barman/Dow showed that all countable \(SS^+\) spaces are \(SS^{+mark}\).

Countable Fan Tightness

A space has countable (dense) fan tightness at a point \(x\) if \(S_{fin}(\mathcal B_x,\mathcal B_x)\) holds, where \(\mathcal B\) denotes the (dense) sets which have \(x\) as a limit point.

A space has countable (dense) fan tightness if it has countable (dense) fan tightness at every point. Denote this with \(C(D)FT\).

Barman/Dow showed that the following are equivalent:

• \(X\) is \(SS\)
• \(X\) is separable and \(CDFT\)
• \(X\) has countable dense fan-tightness at each point of some countable dense subset.

Countable Fan Tightness Game

We may similarly consider \(G_{fin}(\mathcal B_x,\mathcal B_x)\). If \(\mathcal{II}\uparrow G_{fin}(\mathcal B_x,\mathcal B_x)\) then we write \(C(D)FT^{+}\)

Similarly, the following are equivalent:

• \(X\) is \(SS^+\)
• \(X\) is separable and \(CDFT^{+}\)
• \(\mathcal{II}\uparrow G_{fin}(\mathcal B_x,\mathcal B_x)\) for each point in some countable dense subset of \(X\)

The above also holds for Markov strategies.

Note: \(CF(D)T\)-type properties are hereditary.

\(\Omega\)-Menger property

An \(\omega\)-cover of a space is an open cover for which every finite set is contained in some member of the cover.

A space is \(\Omega\)-Menger or \(\Omega M\) when \(S_{fin}(\Omega,\Omega)\) holds, where \(\Omega\) denotes the \(\omega\)-covers of \(X\).

• \(X\) is \(\Omega\)-Menger if and only if \(X^n\) is Menger for \(n<\omega\)

\(\Omega\)-Menger game

We may similarly consider \(G_{fin}(\Omega,\Omega)\). If \(\mathcal{II}\uparrow G_{fin}(\Omega,\Omega)\) then we write \(\Omega M^{+}\).

Bringing it together

Let \(C(X)\) be the subspace of \(\mathbb R^X\) consisting of continuous functions. (For this reason, we only consider completely regular spaces.)

Arhangel'skii proved the following:

Idea of proof

There's a natural corespondance between \(\omega\) covers of completely regular spaces \(X\) and subsets of \(C(X)\) containing \(\mathbf 0\) as a limit point.

Given an \(\omega\) cover \(\mathcal U\), we may assume it consists of co-zero sets. For \(U\in\mathcal U\), let \(\mathbf x_U\in C(X)\) satisfy \(U=\mathbf x_U^{-1}[(-1,1)]\). It follows \( \mathbf 0 \in cl(\{\frac{1}{2^n}\mathbf x_U:U\in\mathcal U,n<\omega\}) \).

Likewise, given \(\mathbf 0\in cl(B)\), let \( \mathcal U_B = \{ \mathbf x^{-1}[(-\frac{1}{2^n},\frac{1}{2^n})] : \mathbf x\in B, n<\omega \} \); it follows \(\mathcal U_B\) is an \(\omega\) cover.

Barman/Dow made a similar observation:

• If \(X\) is \(\sigma\)-compact, then \(C(X)\) is \(CFT^{+mark}\).

Since \(CFT^{+mark}\) implies \(CDFT^{+mark}\), any separable subspace of such a \(C(X)\) must be \(SS^{+mark}\).

We get a direct comparison with Arhangel'skii's result by observing the following:

So if \(X=\bigcup_{n<\omega}X^n\) for \(X_n\) compact and increasing, let \(\sigma(\mathcal U,n)\) be an \(n\) cover of \(\mathcal U\) for \(X_n\). It follows that \(\bigcup_{n<\omega}\sigma(\mathcal U,n)\) is an \(\omega\) cover.

Alternately, if given a Markov strategy, let \(X_n = \bigcap_{\mathcal U\in\Omega}\bigcup\sigma(\mathcal U,n)\). Since any open cover may be closed under finite unions to obtain an \(\omega\) cover, it's not hard to see that \(X_n\) is relatively compact, and therefore \(\overline{X_n}\) is compact by regularity of \(X\). Finally, it's not hard to show \(\sigma\) isn't winning if \(X\not=\bigcup_{n<\omega}X_n\). \(\Box\)

\(\Omega M\) vs. \(CFT\)

As it turns out, the idea of Arhangel'skii's original result yields all of the following:

• \(X\) is \(\Omega M\) iff \(C(X)\) is \(CFT\)
• \(X\) is \(\Omega M^{+}\) iff \(C(X)\) is \(CFT^{+}\)
• \(X\) is \(\Omega M^{+mark}\) iff \(C(X)\) is \(CFT^{+mark}\)

So it's easy to find an example of a space which is \(CFT^+\) but not \(CFT^{+mark}\).

• Take the one-point Lindelofication \(\omega_1^\dagger\); this space isn't \(\sigma\)-compact and therefore not \(\Omega M^{+mark}\), but it can be shown that it is \(\Omega M^+\).
• It follows \(C(\omega_1^\dagger)\) is \(CFT^+\) but not \(CFT^{+mark}\).

So if \(C(\omega_1^\dagger)\) was separable, we'd have a \(SS^+\), \(\neg SS^{+mark}\) space... but it's not even ccc.

So searching for \(\Omega M^+\), \(\neg\Omega M^{+mark}\) spaces \(X\) yields \(CFT^+\), \(\neg CFT^{+mark}\) spaces \(C(X)\), but finding an example where \(C(X)\) is separable may not be possible.