Given a subbase $𝒮$ of a space $X$, the game $PO(𝒮,X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$-th move, a point $x_n\in X$ and a set $U_n\in 𝒮$ such that $x_n\in U_n$. The game stops after the moves $\{x_n,U_n:n\in ø\}$ have been made and the player $P$ wins if ${\bigcup }_{n\in ø}{U}_n=X$; otherwise $O$ is the winner. Since $PO(𝒮,X)$ is an evident modification of the well-known point-open game $PO\left(X\right)$, the primary line of research is to describe the relationship between $PO\left(X\right)$ and $PO(𝒮,X)$ for a given subbase $𝒮$. It turns out that, for any subbase $𝒮$, the player $P$ has a winning strategy...

We characterize spaces with $\sigma$-$n$-linked bases as specially embedded subspaces of separable spaces, and derive some corollaries, such as the $𝐜$-productivity of the property of having a $\sigma$-linked base.

If Nonempty has a winning strategy against Empty in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows Nonempty to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits Nonempty to consider the previous move by Empty. We show that Nonempty has a stationary winning strategy for every second-countable T₁ Choquet space. More generally, Nonempty has a stationary winning strategy for...