Residuality of Families of F_{σ} Sets

Shingo SAITO

（斎藤新悟） University College London

http://www.ucl.ac.uk/~ucahssa/

1 Residuality of Families of Closed Sets

Work in I := [0, 1]. K := {K ⊂ I | K: closed}. d: Hausdorff metric on K.

(d(K, ∅) := 1 for nonempty K ∈ K.) d defines residuality on K. \underline{e.g.}

{K ∈ K | K: null}: residual in K.

We say typical closed sets are null.

2 Residuality of Families of F_{σ} Sets

2.1 How about giving F_{σ} a topology? Give F_{σ} a topology via the surjection

K^{ℕ} ⟶  F_{σ}
∈  ∈
(K_{n}) ⟼ ⋃_{n=1}^{∞} K_{n}

It turns out that F ⊂ F_{σ}: residual ⟺ {F ∈ F_{σ} | F: dense} ⊂ F This residuality is not useful.

2.2 Another approach \underline{Def} F ⊂ F_{σ} is residual if

{(K_{n}) ∈ K^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}

is residual in K^{ℕ}.

Another natural surjection:

K_{↗}^{ℕ} := {(K_{n}) ∈ K^{ℕ} | K_{1} ⊂ K_{2} ⊂ ⋯ }

K_{↗}^{ℕ} ⟶ F_{σ}; (K_{n}) ⟼ ⋃_{n=1}^{∞} K_{n}

\underline{Def} F ⊂ F_{σ} is ↗-residual if

{(K_{n}) ∈ K_{↗}^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}

is residual in K_{↗}^{ℕ}.

\underline{Main Thm} F ⊂ F_{σ} is residual iff it is ↗-residual.

2.3 Evidences that this is the definition

\underline{Thm} For a σ-ideal I on I, T.F.A.E.:

(1) I ∩ F_{σ} is residual in F_{σ};

(2) I ∩ K is residual in K.

\underline{Cor} Typical F_{σ} sets are null.

Relations with Functions C(I) := {f: I ⟶ ℝ | f: continuous}

with the sup norm ∥·∥

\underline{Def} x ∈ I is a knot point of f ∈ C(I) if

D^{+}f(x) := lim sup_{y↘x} \frac{f(y) − f(x)}{y − x} = ∞,

D_{+}f(x) = −∞, D^{−}f(x) = ∞, D_{−}f(x) = −∞.

For f ∈ C(I),

N(f) := {x ∈ I | x is not a knot point of f}
∈ F_{σ}.

\underline{Thm} (Preiss and Zajíček (unpublished)) For a σ-ideal I on I, T.F.A.E.:

(1) N(f) ∈ I for typical f ∈ C(I);

(2) I ∩ K is residual in K.

\underline{Thm} For F ⊂ F_{σ}, T.F.A.E.:

(1) N(f) ∈ F for typical f ∈ C(I);

(2) F is residual in F_{σ}.

3 Proof of Main Thm

\underline{Main Thm} For F ⊂ F_{σ}, T.F.A.E.:

(1) {(K_{n}) ∈ K^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}: residual in K^{ℕ};

(2) {(K_{n}) ∈ K_{↗}^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}: residual in K_{↗}^{ℕ}.

3.1 Banach-Mazur game

X: metric space, S ⊂ X. Players 1 ◯ and 2 ◯ alternately choose a ball. They must choose a subset of the ball chosen in the previous turn.

\begin{matrix}B_{1} \\ ↑ \\ 1 & ◯ \end{matrix} ⊃ \begin{matrix}B_{2} \\ ↑ \\ 2 & ◯ \end{matrix} ⊃ \begin{matrix}B_{3} \\ ↑ \\ 1 & ◯ \end{matrix} ⊃ \begin{matrix}B_{4} \\ ↑ \\ 2 & ◯ \end{matrix} ⊃ ⋯

Player 2 ◯ wins iff ⋂_{n=1}^{∞} B_{n} ⊂ S.

\underline{Fact} Player 2 ◯ has a winning strategy iff S is residual.

3.2 Plan for the proof

We look at ↗-residual ⟹ residual.

F ⊂ F_{σ}: ↗-residual.

\underline{K_{↗}^{ℕ}-game}

• X = K_{↗}^{ℕ}.

• S = {(K_{n}) ∈ K_{↗}^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}.

• Player 2 ◯ knows a winning strategy.

\underline{K^{ℕ}-game}

• X = K^{ℕ}.

• S = {(K_{n}) ∈ K^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}.

• Player 2 ◯ is looking for a winning strategy.

2 ◯ knows ; strategy ↙ transfer

↓

K^{ℕ}-game  K_{↗}^{ℕ}-game
1 ◯:  B_{1}  ⟶  B̃_{1} ↷_{strategy}
2 ◯:  B̃_{2}  ⟵  B_{2}
1 ◯:  B_{3}  ⟶  B̃_{3} ↷_{strategy}
2 ◯:  B̃_{4}  ⟵  B_{4}
⋮  ⋮  ⋮

3.3 Transfers What matters is only the centres (not the radii). We may assume that the centres are

sequences of disjoint finite sets in K^{ℕ}-game, sequences of finite sets in K_{↗}^{ℕ}-game.

How about the maps

K^{ℕ} ⟶ K_{↗}^{ℕ}; (K_{n}) ⟼ (K_{1}, K_{1} ∪ K_{2}, …),
K_{↗}^{ℕ} ⟶ K^{ℕ}; (K_{n}) ⟼ (K_{1}, K_{2} \ K_{1}, …)?

They don’t work.

Why don’t they work?

K^{ℕ}-game  K_{↗}^{ℕ}-game
1 ◯: ( \begin{matrix}K_{1} : • \\ K_{2} :  • \\ K_{3} :  • \end{matrix} ⟶ ( \begin{matrix}K_{1} : • \\ K_{2} : • • \\ K_{3} : • • • \end{matrix}

2 ◯: ( \begin{matrix}K_{1} : • \\ K_{2} : ∘ • \\ K_{3} :  ∘ • \end{matrix} ⟵ ( \begin{matrix}K_{1} : • \\ K_{2} : •∘ • \\ K_{3} : •∘ •∘ • \end{matrix}

Throw these points ∘ away:

K^{ℕ}-game  K_{↗}^{ℕ}-game
1 ◯: ( \begin{matrix}K_{1} : • \\ K_{2} :  • \\ K_{3} :  • \end{matrix} ⟶ ( \begin{matrix}K_{1} : • \\ K_{2} : • • \\ K_{3} : • • • \end{matrix}

2 ◯: ( \begin{matrix}K_{1} : • \\ K_{2} :  • \\ K_{3} :  • \\ K_{4} : ∘ ∘ \end{matrix} ⟵ ( \begin{matrix}K_{1} : • \\ K_{2} : •∘ • \\ K_{3} : •∘ •∘ • \end{matrix}

Next step:

K^{ℕ}-game  K_{↗}^{ℕ}-game

2 ◯: ( \begin{matrix}K_{1} : • \\ K_{2} :  • \\ K_{3} :  • \\ K_{4} : • • \end{matrix} ⟵ ( \begin{matrix}K_{1} : • \\ K_{2} : •• • \\ K_{3} : •• •• • \end{matrix}

1 ◯: ( \begin{matrix}K_{1} : • \\ K_{2} :  • \\ K_{3} :  •∘ \\ K_{4} : •∘ •∘ \end{matrix} ⟶ ( \begin{matrix}K_{1} : • \\ K_{2} : ••∘ • \\ K_{3} : ••∘ ••∘ •∘ \end{matrix}

3.4 What remains to be proved

Suppose (intersection of balls in K^{ℕ}-game) = {(P_{n})}, (intersection of balls in K_{↗}^{ℕ}-game) = {(Q_{n})}.

(P_{n}), (Q_{n}): limit of centres. We know ⋃_{n=1}^{∞} Q_{n} ∈ F. We want ⋃_{n=1}^{∞} P_{n} ∈ F, so shall prove ⋃_{n=1}^{∞} P_{n} = ⋃_{n=1}^{∞} Q_{n}.

\underline{Problem}

K^{ℕ}-game  K_{↗}^{ℕ}-game

( K_{1} : ; K_{2} : •^{1} ; ⋮ ; K_{10} : •^{2} ; ⋮ ; K_{73} : •^{3} ⟷^{( \begin{matrix}K_{1} : \\ K_{2} : •^{1} •^{2}•^{3} \end{matrix}}

The limit must belong to P_{1}.