Title: Theory / Unimportant / Math / Logic / Noélie
Tags: hasSeenConjDef

Definition: \(X^{<\omega}\setminus\emptyset\)
Page: 12
Chapter: Notation
Variables: \(X\) a set
Notation: \(\mathrm{Seq}(X)\)

Name: A pruned tree
Definition: A tree without terminal node
Page: 12
Chapter: Notation
Variables: \(X\) a set

Name: The set of infinite branches of \(T\)
Page: 12
Chapter: Notation
Variables: \(X\) a set
\(T\) a tree on \(X\)
Notation: \([T]\)

Name: The cardinality of the continuum
Definition: \(2^{\aleph_0}\)
Page: 12
Chapter: Notation
Variables: \(X\) a set
\(T\) a tree on \(X\)
Notation: \(\mathfrak c\)

Definition: the set of open subsets of \(X\) if \(i=0\)
Page: 12
Chapter: Notation
Variables: \(X\) a topological set
Notation: \(\Sigma_i^1(X)\)
Definition2: the first projection of a set \(B\in\Pi_{i-1}^1(X\times \omega^\omega)\) if \(i>0\)
Conjdef: either

Definition: The set of complements of elements of \(\Sigma_i^1(X)\)
Page: 12
Chapter: Notation
Variables: \(X\) a topological set
Notation: \(\Delta_i^1(X)\)

Name: The class of projective subsets of \(X\)
Definition: \(\bigcup_{n\in\omega}\Sigma_n^i(X)\)
Page: 12
Chapter: Notation
Variables: \(X\) a topological set

Name: A suitable class of subsets of \(X\)
Definition: contains the class of Borel sets
Page: 12
Chapter: Notation
Variables: \(X\) a topological set
Definition2: is stable under finite unions and intersections
Conjdef: A class of subsets of \(X\) which
Definition3: is stable under Borel inverse images
Assuming: \(X\) Polish

Name: A game
Definition: \(X\) a set of possible moves
Page: 15
Chapter: I
Section: 1
Notation: \(\mathcal G(T,\mathcal X)\)
Definition2: a nonempty tree \(T\subseteq X^{<\omega}\) without terminal nodes, called the rules
Conjdef: A tuple with
Definition3: a set \(\mathcal X\subseteq[T]\) the target set

Name: A winning strategy of \(\mathcal G(T,\mathcal X)\) for I
Definition: A strategy such that I wins, whatever II plays
Page: 15
Chapter: I
Section: 1
Variables: \(X\) a set of possible moves
\(T\) a tree on \(X\), set of rules
\(\mathcal X\) a target set

Name: A strategy of \(\mathcal G(T,\mathcal X)\) for I
Definition: A function from the elements \(s\in T\) of even length to an element \(x\in X\) such that \(s\frown x\in T\)
Page: 15
Chapter: I
Section: 1
Variables: \(X\) a set of possible moves
\(T\) a tree on \(X\), set of rules
\(\mathcal X\) a target set

Name: An outcome
Definition: A mapping from \([T]\) to some set
Page: 15
Chapter: I
Section: 1
Variables: \(X\) a set of possible moves
\(T\) a tree on \(X\), set of rules
Denoted by: \(F\)

Name: A strategy in \(\mathcal G(T,F)\) to reach \(\mathcal Y\)
Definition: A strategy in the game \(\mathcal G(T,F^{-1}(\mathcal Y))\)
Page: 15
Chapter: I
Section: 1
Variables: \(X\) a set of possible moves
\(T\) a tree on \(X\), set of rules
\(Y\) a set
\(F\) an outcome function to \(Y\)
\(\mathcal Y\subseteq Y\)

Name: A determined game
Definition: A game such that one of the player has a winning strategy
Page: 15
Chapter: I
Section: 1

Name: A game on \(X\)
Definition: A game whose rule tree is \(X^{<\omega}\)
Page: 15
Chapter: I
Section: 1
Variables: \(X\) a set

Definition: The assumption «every \(\Gamma\)-subset of \(\omega^\omega\) is determined»
Page: 16
Chapter: I
Section: 1
Variables: \(\Gamma\) a class of Polish space
Notation: \(\mathrm{Det}_\omega(\Gamma)\)

Definition: The assumption «every \(\Gamma\)-subset of \(\mathbb R^\omega\) is determined, when \(\mathbb R\) is endowed with its usual Polish topology, and \(\mathbb R^\omega\) with the product topology»
Page: 16
Chapter: I
Section: 1
Variables: \(\Gamma\) a class of Polish space
Notation: \(\mathrm{Det}_{\mathbb R}(\Gamma)\)