We relate the property of discrete selectivity and its corresponding game, both recently introduced by V.V.Tkachuck, to a variety of selection principles and point picking games. In particular we show that player II can win the discrete selection game on C p (X) if and only if player II can win a variant of the point open game on X. We also show that the existence of limited information strategies in the discrete selection game on C p (X) for either player are equivalent to other well-known topological properties.In the course of studying the strong domination of function spaces by second countable spaces and countable spaces, G. Sanchez and Tkachuk isolated the topological property of discrete selectivity[1] [2]. A space is discretely selective if for every sequence {U n : n ∈ ω} of non-empty open subsets of the space, there are points x n ∈ U n so that {x n : n ∈ ω} is closed discrete. In subsequent work, Tkachuk showed that for T 3.5 -spaces, C p (X) is discretely selective if and only if X is uncountable.Discrete selectivity naturally generates a game, in which player I plays open sets, player II responds with points from those open sets, and player II wins if the points form a closed discrete set. Tkachuk explored what happens when player I has a winning strategy for this game, showing that the existence of a winning strategy for player I in this game on C p (X) is equivalent to player I having a winning strategy for Gruenhage's W -game on C p (X, 0) and is also equivalent to player I having a winning strategy for the pointopen game on X [3]. Tkachuk also showed that if player II has a winning strategy in the point-open game on X, then player II has a winning strategy in the discrete selection game on C p (X). Tkachuk hypothesized that the implication partially reverses for player II (considering ω-covers), and posed this problem as an open question. All of the strategies Tkachuk worked with were perfect information strategies.By considering limited information strategies and other topological games, we were able to answer
By adapting techniques of Arhangel'skii, Barman, and Dow, we may equate the existence of perfect-information, Markov, and tactical strategies between two interesting selection games. These results shed some light on Gruenhage's question asking whether all strategically selectively separable spaces are Markov selectively separable.
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