Let S be a topological property of sequences (such as, for example, "to contain a convergent subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses a non-empty open subet U_n of X, and Player B responds by selecting a point x_n in U_n. Player B wins the game if the sequence (x_n) satisfies property S in X; otherwise, Player A wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness.

中文翻译：

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由开放点游戏定义的紧凑性属性

令 S 是序列的拓扑属性（例如，“包含收敛子序列”或“具有累积点”）。我们在拓扑空间 X 上引入以下开点博弈 OP(X,S)。 在第 n 步，玩家 A 选择 X 的一个非空的开子集 U_n，玩家 B 通过在其中选择一个点 x_n 作为响应联合国。如果序列 (x_n) 满足 X 中的属性 S，则玩家 B 赢得游戏；否则，玩家 A 获胜。OP(X,S) 在 OP(X,S) 中（不）存在规则或固定获胜策略，这定义了基础空间 X 的新紧凑性属性。我们彻底研究了这些属性并构建了区分其中一半的示例，对于任意属性 S夹在顺序紧致性和可数紧致性之间。