Choquet博弈中的稳定策略和收敛策略
Stationary and convergent strategies in Choquet games
摘要：
如果在Choquet博弈中，NONEMPTY在一个空间上对EMPTY有获胜策略，则该空间被称为Choquet空间。这样的获胜策略允许NONEMPTY在进行每一步新的移动之前考虑之前移动的整个有限历史;静态策略只允许NONEMPTY考虑EMPTY的前一步移动。我们证明了NONEMPTY对每一个第二可数的T1 Choquet空间都有一个平稳的获胜策略。更一般地说，NONEMPTY对于任何具有开有限基的T1 Choquet空间都有一个固定的获胜策略。我们还研究了Choquet博弈的收敛策略，证明了以下结果。(1)T1空间X是完全度量空间的开像当且仅当NONEMPTY在X上的Choquet博弈中有收敛的获胜策略。(2)一个T1空间X是一个度量空间的紧开象当且仅当X是亚紧的，并且在X上的Choquet博弈中NONEMPTY有平稳收敛策略。(3)一个T1空间X是一个完全度量空间的紧开象当且仅当X是亚紧的，并且在X上的Choquet博弈中NONEMPTY有一个平稳收敛的获胜策略。
英文摘要：
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 any T1 Choquet space with an open-finite basis. We also study convergent strategies for the Choquet game, proving the following results. (1) A T1 space X is the open image of a complete metric space if and only if NONEMPTY has a convergent winning strategy in the Choquet game on X. (2) A T1 space X is the compact open image of a metric space if and only if X is metacompact and NONEMPTY has a stationary convergent strategy in the Choquet game on X. (3) A T1 space X is the compact open image of a complete metric space if and only if X is metacompact and NONEMPTY has a stationary convergent winning strategy in the Choquet game on X.
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