Leírás
Speaker: Tamas Csernák.
The title of his talk: The point separation game in topological spaces
Abstract:
Assume that X is a topological space, Y ⊂ X, and β is an ordinal.
We define the set membership game Gβ (Y, X) and the point separating game Gβ (X)
as follows.
The gameplay is the same for both games; the difference lies in how the winner
is determined. There are two players: the Seeker and the Hider. The gameplay
consists of β-many turns. In the αth turn, the Seeker first chooses an open sets
U(α) ∈ τ_X , and then the Hider selects i(α) ∈ 2.
Let B(U,I) be the intersection of {U(alpha)" : i(alpha)=1} - union{U(alpha):i(alpha)=0}.
In the game Gβ (Y, X), the Seeker wins iff
B (U, i) ⊂ Y or B (U, i) ∩ Y = ∅.
In the game Gβ (X), the Seeker wins iff
| B (U, i)| ≤ 1.
We write S ↑ Gβ (Y, X) (or H ↑ Gβ (Y, X), respectively) iff the Seeker (or the
Hider, respectively) has a winning strategy in the game Gβ (Y, X).
We write S ↑ Gβ (X) (or H ↑ Gβ (X), respectively) iff the Seeker (or the Hider,
respectively) has a winning strategy in the game Gβ (X).
We investigate properties of these games.