2016 August 3rd
Presentation for the 31st Summer Conference on Topology and its Applications special session on Topology and Foundations at The University of Leicester
The speaker will adapt some techniques of Arhangelskii, Barman, and Dow to relate the existence of winning limited information strategies in several selection 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.
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.
Let \(G_{fin}(\mathcal D,\mathcal D)\) denote the selective separability game, and \(SS^+\) denote \(\mathcal{II}\uparrow G_{fin}(\mathcal D,\mathcal D)\).
A Markov strategy for a game only considers the round number and latest move of the opponent.
Let \(SS^{+mark}\) denote \(\mathcal{II}\uparrow_{mark} G_{fin}(\mathcal D,\mathcal D)\).
Idea: look at related games where strategies cannot always be improved to Markov.
A space has countable fan tightness at a point \(x\) if \(S_{fin}(\mathcal B_x,\mathcal B_x)\) holds, where \(\mathcal B_x\) denotes the sets which have \(x\) as a limit point.
A space has countable dense fan tightness at a point \(x\) if \(S_{fin}(\mathcal D,\mathcal B_x)\) holds.
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 (2011) showed that the following are equivalent:
We may similarly consider game versions of these properties.
If \(\mathcal{II}\uparrow G_{fin}(\mathcal B_x,\mathcal B_x)\) for each point \(x\) then we write \(CFT^{+}\).
If \(\mathcal{II}\uparrow G_{fin}(\mathcal D,\mathcal B_x)\) for each point \(x\) then we write \(CDFT^{+}\).
Slightly modifying the proof of the previous result, the following are equivalent:
These equivalencies also hold for Markov strategies.
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 \(X\).
We may similarly consider \(G_{fin}(\Omega,\Omega)\). If \(\mathcal{II}\uparrow G_{fin}(\Omega,\Omega)\) then we write \(\Omega M^{+}\).
Assume all spaces from here on are completely regular.
Let \(C_p(X)\) be the subspace of \(\mathbb R^X\) consisting of continuous functions.
Arhangel'skii proved the following (1986):
\(X\) is \(\Omega M\) if and only if \(C_p(X)\) is \(CFT\) if and only if \(C_p(x)\) is \(CDFT\).
There's a natural corespondence between members of \(\Omega\) (\(\omega\)-covers of \(X\)), members of \(\mathcal D\) (dense subsets of \(C_p(X)\)), and members of \(\mathcal B_{\mathbf 0}\) (subsets of \(C_p(X)\) containing \(\mathbf 0\) as a limit point).
For \(\mathcal U\in\Omega\), let \( D = \{ \mathbf x\in C_p(X) : \mathbf x[X\setminus U]=\{1\} \text{ for some } U\in\mathcal U \} \).
Any neighborhood in \(C_p(X)\) may only restrict some finite subset \(F\subseteq X\). Since \(F\) is contained in some \(U\in\mathcal U\), this witnesses a member of \(D\) guaranteed to be in that neighborhood. So, \(D\in\mathcal D\).
Likewise, given \(B\in\mathcal B_{\mathbf 0}\), let \( \mathcal U_B = \{ \mathbf x^{-1}[(-\frac{1}{2^n},\frac{1}{2^n})] : \mathbf x\in B \} \).
For each finite set \(F\subseteq X\), the neighborhood of \(\mathbf 0\) restricting \(F\) to \((-\frac{1}{2^n},\frac{1}{2^n})\) contains some \(\mathbf x\in B\). Thus \(F\subseteq\mathbf x^{-1}[(-\frac{1}{2^n},\frac{1}{2^n})]\in\mathcal U_B\). \(\Box\)
Barman/Dow made a similar observation:
Digging into the proof, they really showed: If \(X\) is \(\sigma\)-compact, then \(C_p(X)\) is \(CFT^{+mark}\).
As a result, any separable subspace of such a \(C_p(X)\) must be \(SS^{+mark}\).
We may get a more direct correspondance with Arhangel'skii's result by proving the following:
\(X\) is \(\sigma\)-compact if and only if \(X\) is \(\Omega M^{+mark}\).
First note that \(X\) is compact if and only if for each \(\omega\) cover of the space and \(n<\omega\), there exists a finite subcollection of the \(\omega\) cover for which each subset of \(X\) of size \(n\) is contained in some member of the subcollection. (Hint: consider \(X^n\).)
So if \(X=\bigcup_{n<\omega}X^n\) for \(X_n\) compact and increasing, we may define a Markov strategy by letting \(\sigma(\mathcal U,n)\) such a finite subcollection of \(\mathcal U\) for \(X_n\). It follows that \(\bigcup_{n<\omega}\sigma(\mathcal U,n)\) is an \(\omega\) cover of \(X\).
Conversely, if given a Markov strategy \(\sigma\), let \(X_n = \bigcap_{\mathcal U\in\Omega}\bigcup\sigma(\mathcal U,n)\). For any open cover of \(X\), let \(\mathcal U\) be its closure under finite unions, an \(\omega\)-cover of \(X\). By the definition of \(X_n\), \(\sigma(\mathcal U,n)\) is a finite subcover of \(X_n\). Thus \(X_n\) is relatively compact to \(X\), and \(\overline{X_n}\) is compact by the regularity of \(X\).
Finally, it's routine to show that if \(X\not=\bigcup_{n<\omega}X_n\), then \(\sigma\) wasn't a winning strategy. \(\Box\)
As it turns out, the idea of Arhangel'skii's original result yields all of the following:
So it's easy to find an example of a space which is \(CFT^+\) but not \(CFT^{+mark}\).
Unfortunately, \(C_p(\omega_1\cup\{\infty\})\) isn't separable.
Slides available at Clontz.org.