Meager-Nowhere dense games, by Marion Scheepers

Topology Atlas Document # zaab-03.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 3 (1996), 19-20.

Meager-Nowhere dense games
by

Marion Scheepers

(Boise State University, Boise, Idaho, U.S.A.)

These games, originally defined by Marion Scheepers, are an analysis of the classical "zig-zag" argument which is used to prove such things as: the union of countably many finite sets is countable; the union of countably many first category sets is first category, and so on. These examples can all be described in the language of topology, using the notions of nowhere dense -- and meager (i.e., first category) sets. The basic idea is that two players, ONE and TWO, play an inning per positive integer. In each inning ONE first chooses a meager set O(n) and TWO respond with a nowhere dense set T(n); TWO wins if the union of TWO's nowhere dense sets covers the union of ONE's meager sets. The standard "zig-zag" argument shows that TWO has a winning strategy -- but this strategy requires perfect memory from TWO. Does TWO have a winning strategy which depends on less than perfect memory? Many possible restrictions on TWO's memory, and restrictions on the actions of the two players have been investigated from this point of view and there are many open problems in this regard. Here is a sample of three:

1. The Coding Strategy Conjecture: ONE is restricted to playing so that for each n, O(n) is a subset of or equal to O(n+1); TWO wins if the union of the T(n)'s covers the union of the O(n)'s. TWO's memory is such that when choosing T(n+1), TWO knows only O(n+1) and T(n). The conjecture is that TWO has a winning strategy which depends on only this memory. (Known: The Generalized Continuum Hypothesis implies the Coding Strategy Conjecture. Evidence suggests that the consistency of the existence of some large cardinal numbers may imply the consistency of the negation of the Coding Strategy Conjecture.)

2. The Three-Tactic Conjecture: ONE is restricted to playing so that for each n, O(n) is a proper subset of O(n+1). The winning condition is as before. Fix a positive integer k. TWO's memory is such that when choosing T(n), TWO remembers no more than the most recent k moves of ONE, and none of the previous moves of TWO. A strategy of TWO depending on only this information is said to be a k-tactic. The conjecture is that if TWO has a winning k-tactic, then TWO has a winning 3-tactic. (Known: There are many examples where there is no k for which TWO has a winning k-tactic. TWO has a winning 1-tactic if, and only if, every meager set is nowhere dense. There are examples where TWO has a winning 2-tactic, but not a winning 1-tactic. There are examples where TWO doesn't have a winning 2-tactic, and it is undecidable whether TWO has a winning 3-tactic (the real line is such an example). In all known examples where it could be shown that TWO has a winning k-tactic, it could be shown that TWO has a winning 3-tactic; this and a few general but not all-encompassing theorems suggest the truth of the conjecture.)

3. The Countable-Finite Conjecture: Look at the game described in the Three Tactic Conjecture, where now ONE chooses countable sets and TWO chooses finite sets. The conjecture is that TWO has a winning 2-tactic. (Known: If the underlying set from which they are choosing these subsets has cardinality less than aleph - omega, then TWO has a winning 2-tactic. It is consistent that for any underlying set TWO has a winning 2-tactic -- even in the presence of very large cardinal numbers.)

For a more extensive list of problems and for references to the relevant literature, contact the author by e-mail.

Received by the editors: December 19, 1995.