## Relating games of Menger, countable fan tightness, and selective separability

#### The University of South Alabama

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

### Abstract

The speaker will adapt some techniques of Arhangelskii, Barman, and Dow to relate the existence of winning limited information strategies in several selection games.

## 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.
• Each separable Frechet space is $$SS$$. (Barman Dow 2011)

## 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 $$SS^{+mark}$$.

Idea: look at related games where strategies cannot always be improved to Markov.

## Countable Fan Tightness

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:

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

## Countable (Dense) Fan Tightness Game

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:

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

These equivalencies also hold for Markov strategies.

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

## $$\Omega$$-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 $$n<\omega$$ (folklore?)

## $$\Omega$$-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

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):

#### Theorem

$$X$$ is $$\Omega M$$ if and only if $$C_p(X)$$ is $$CFT$$ if and only if $$C_p(x)$$ is $$CDFT$$.

#### Idea of proof (Sakai?)

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 $$F\subseteq\mathbf x^{-1}[(-\frac{1}{2^n},\frac{1}{2^n})]\in\mathcal U_B$$. $$\Box$$

• If $$X$$ is $$\sigma$$-compact and metrizable, then $$C_p(X)$$ is $$CFT^{+}$$.

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:

#### Theorem

$$X$$ is $$\sigma$$-compact if and only if $$X$$ is $$\Omega M^{+mark}$$.

#### Proof

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. $$\Box$$

## $$\Omega M$$ and $$CFT$$

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

• $$X$$ is $$\Omega M$$ iff $$C_p(X)$$ is $$C(D)FT$$
• $$X$$ is $$\Omega M^{+}$$ iff $$C_p(X)$$ is $$C(D)FT^{+}$$
• $$X$$ is $$\Omega M^{+mark}$$ iff $$C_p(X)$$ is $$C(D)FT^{+mark}$$

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

• Take the one-point Lindelofication $$\omega_1\cup\{\infty\}$$; this space isn't $$\sigma$$-compact and therefore not $$\Omega M^{+mark}$$. However, there exists a winning strategy, showing $$\Omega M^+$$.
• It follows $$C_p(\omega_1\cup\{\infty\})$$ is $$CFT^+$$ but not $$CFT^{+mark}$$.

Unfortunately, $$C_p(\omega_1\cup\{\infty\})$$ isn't separable.

• Let $$X$$ be non-$$\sigma$$-compact but $$\Omega M^+$$. Could $$C_p(X)$$ be separable? This would be a $$SS^+$$, $$\neg SS^{+mark}$$ space.

### References

• A.V. Arhangelskii, Hurewicz spaces, analytic sets and fan tightness of function spaces, Soviet Math. Dokl. 33 (1986) 396–399.
• Doyel Barman, Alan Dow, Selective separability and SS+. Topology Proc. 37 (2011), 181–204.
• Sakai
• Scheepers

# Thank you!

## Questions?

Slides available at Clontz.org.