In this paper we study connections between topological games such as Rothberger, Menger, and compact-open games, and we relate these games to properties involving covers by ${{G}_{\delta }}$ subsets. The results include the following: (1) If TWO has a winning strategy in theMenger game on a regular space $X$, then $X$ is an Alster space. (2) If TWO has a winning strategy in the Rothberger game on a topological space $X$, then the ${{G}_{\delta }}$-topology on $X$ is Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.

We show that there exists a locally compact separable metrizable space $L$ such that $C_{0}(L)$, the Banach space of all continuous complex-valued functions vanishing at infinity with the supremum norm, is almost transitive. Due to a result of Greim and Rajagopalan [3], this implies the existence of a locally compact Hausdorff space $\tilde L$ such that $C_{0}(\tilde L)$ is transitive, disproving a conjecture of Wood [9]. We totally owe our construction to a topological characterization due to Sánches [8].

It is proved that for every Hausdorff space ℝ and for every Hausdorff (regular or Moore) space X, there exists a Hausdorff (regular or Moore, respectively) space S containing X as a closed subspace and having the following properties:

• la) Every continuous map of S into ℝ is constant.

• b) For every point x of S and every open neighbourhood U of x there exists an open neighbourhood V of x, V ⊆ U such that every continuous map of V into ℝ is constant.

• 2) Every continuous map f of S into S (f ≠ identity on S) is constant.