THE DESCRIPTIVE COMPLEXITY OF CONNECTEDNESS IN POLISH SPACES

GABRIEL DEBS AND JEAN SAINT RAYMOND

Abstract. We investigate the descriptive complexity of connectedness (pathwise connectedness, local connectedness) of Polish spaces, and prove that even in the frame of finite dimensional euclidean spaces this complexity can be as high as possible, and much beyond the first projective classes Σ_{1}^{1} and Π_{1}^{1}. In particular we prove that several of these notions are Π_{2}^{1}-complete.

1. Introduction

This work is a continuation of [2] where was investigated the maximal possible complexity of the set of all zero-dimensional closed subsets of a given Polish space. In the present work we study the analog question for connectedness which is a kind of strong negation of zero-dimension. In fact we shall investigate this question for several variation of the notion of connectedness. More precisely we shall compute the maximal possible complexity, Polish space X, of the following sets :

C(X): the set of all closed connected subsets of X,
Č(X): the set of all closed subsets of X whose complement is connected,
C_{path}(X): the set of all closed pathwise connected subsets of X,
C_{loc}(X): the set of all closed locally connected subsets of X.

It turns out that this complexity depends on whether the ground space X is compact or merely Polish, as shown by the following table which gives upper bounds for the complexity of these sets:

C(X)  Č(X)  C_{path}(X) C_{loc}(X)

X compact  Π_{1}^{0}  Π_{1}^{1}  Π_{2}^{1}  Π_{3}^{0}

X Polish  Π_{2}^{1}  Π_{2}^{1}  Π_{2}^{1}  Ǎ(Π_{1}^{1})

2010 Mathematics Subject Classification. Primary 03E15, 28A05, 54D05; Secondary 54H05.

Key words and phrases. Connectedness, local connectedness, Effros Borel strucure, projective sets.

Note that in the compact case the complexity is computed in the compact space K(X) of all compact subsets of X, while in the Polish case the complexity is relatively to the canonical Borel structure of the space F(X) of all closed subsets of X (see Section 2). These bounds are obtained by a straightforward expansion of definitions, except the bound Ǎ(Π_{1}^{1}) (see Section 3 for the notation) which relies on some specific analysis and properties of Polish locally connected spaces.

In this work we prove that all these bounds are optimal. More precisely we prove:

a) There exists a Polish space X ⊂ ℝ^{3} with C(X) Π_{2}^{1}-complete.

b) There exists a Polish space X ⊂ ℝ^{3} with Č(X) Π_{2}^{1}-complete.

c) There exists a compact space X ⊂ ℝ^{3} with Č(X) Π_{1}^{1}-complete.

d) There exists a compact space X ⊂ ℝ^{4} with C_{path}(X) Π_{2}^{1}-complete.

e) There exists a Polish space X ⊂ ℝ^{3} with C_{loc}(X) Ǎ(Π_{1}^{1})-complete.

Observe that since any Polish space can be embedded as a closed subset of the Polish space ℝ^{ω} it follows from the previous results that the sets C(ℝ^{ω}) and C_{path}(ℝ^{ω}) are Π_{2}^{1}-complete, while the set C_{loc}(ℝ^{ω}) is Ǎ(Π_{1}^{1})-complete. Note however that as we shall see (Proposition 7.5) the set Č(ℝ^{ω}) is actually Borel.

To finish we list next some open questions.

1) There is a priori no topological reason which prevents a 2-dimensional version of the previous results by imposing that X ⊂ ℝ^{2}. However the constructions used in the paper, which necessitate some geometrical freedom in the surrounding euclidean space, are not adaptable to provide such a requirement.

2) These results give the maximal complexity of the considered sets. They do not describe the whole spectrum of possible complexity of these sets for an arbitrary Polish space X.

3) One can try to release the restriction on the space X of being Polish. In particular one can consider the same questions for an analytic space X. We recall that in this case the canonical Borel structure on F(X) is isomorphic to the Borel structure of a Σ_{1}^{1} subset of 2^{ω} and considering descriptive complexity of subsets of F(X) is meaningful. Then resuming the same computation than in the Polish case one can check that for an analytic space X the sets C(X) and Č(X) are still Π_{2}^{1}, while the natural complexity upper bound of the sets C_{path}(X) and C_{loc}(X) jumps up to Π_{4}^{1}; and it is not clear whether in the two latter cases this bound is optimal.

4) Finally note that one can investigate similarly the descriptive complexity of various topological notions other than connectedness. We mention in particular the case of the set F_{0}(X) of all closed zero-dimensional subsets of a Polish space X. It is not difficult to check that F_{0}(X) is a Σ_{2}^{1} set, and it follows from the results of [2] that the maximal complexity of this set is at least ⅁Σ_{2}^{0} (see [7] or [2] for the definition of this class of Δ_{2}^{1} sets). But the maximal complexity of the set F_{0}(X) is unknown.

2. Descriptive Notions and Notations

Along the paper we work in the frame of Polish spaces and we assume the reader to be familiar with basic notions and results from Descriptive Set Theory in this setting, for example as presented in [5] or [7]. We recall that the notion of complexity of a set A is a priori relative to some specified space X containing A. However when the surrounding space X is not specified the complexity is to be understood in the maximal sense, that is the complexity relatively to any Polish space X in which A can be embedded.

We shall mainly consider the classical pointclasses of Borel and projective sets, namely:

Σ_{ξ}^{0}: the additive Baire class of rank ξ

Π_{ξ}^{0}: the multiplicative Baire class of rank ξ (so Π_{2}^{0} = G_{δ})

Σ_{1}^{1}: the class of analytic sets.

Π_{1}^{1}: the class of co-analytic sets.

Δ_{1}^{1}: the class of Borel sets.

Σ_{2}^{1}: the class of projections of co-analytic sets, also denoted by P CA.

Π_{2}^{1}: the class of complements of P CA sets, also denoted by CP CA.

Δ_{2}^{1} := Σ_{2}^{1} ∩ Π_{2}^{1}

Also given any pointclass Γ we denote by Γ̌ its dual class, so for any Polish space X, Γ̌(X) = {X \ A : A ∈ Γ}; and by A(Γ) the class of all sets obtained by operation A applied to a Souslin scheme of sets from Γ. It is well known that A(Σ_{1}^{1}) = Σ_{1}^{1} and we shall consider in the sequel the class A(Π_{1}^{1}) and its dual class Ǎ(Π_{1}^{1}) which are both subclasses of the class of C-sets, hence of the class Δ_{2}^{1} (see [1] or [2] for more details).

The basic tool for computing complexity is the notion of reduction. We recall that a mapping φ : X → Y is said to reduce the set A ⊂ X to the set B ⊂ Y if A = φ^{−1}(B). If Λ is a class of functions, A ⊂ X is said to be Λ-reducible to B ⊂ Y if there exists φ : X → Y in Λ which reduces A to B; and when Λ is the class of continuous functions we simply say that A ⊂ X is reducible to B ⊂ Y. Note that in writing A ⊂ X we mean a pair (A, X) such that A ⊂ X, and similarly for B ⊂ Y; and reducibility is actually a relation between such pairs. However when there is no ambiguity on the surrounding spaces X and Y, we shall simply say that the set A is reducible to the set B. As a matter of fact in many instances, if we impose on the spaces X and Y to be Polish, the notion of reducibility is absolute in the sense that it does not depend on the specific Polish spaces in which the sets A and B are embedded.

Given a pointclass Γ and a Polish space Y, the set B ⊂ Y is said to be Γ-hard if any set A ⊂ 2^{ω} in Γ is reducible to B; if moreover B is in Γ then the set B is said to be Γ-complete.

From the proof of the main result of [3] we have the following:

Theorem 2.1. (Kechris [3]) Let Γ be a pointclass containing the class Π_{1}^{1} and closed by Δ_{1}^{1} substitutions, Y be a Polish space and B ⊂ Y. If any set A ⊂ 2^{ω} in Γ is Δ_{1}^{1}-reducible to B then B is Γ-hard.

Note that since any uncountable Polish space is Δ_{1}^{1}-isomorphic to 2^{ω}, it follows from Theorem 2.1 that for a projective class Γ, being Γ-complete for a subset B of some Polish space Y is absolute and does not depend on Y.

3. Topologies and Borel Structures on F(X)

3.1. The topological space K(X)

For any space X we denote by K(X) the space of all compact subsets of X equipped with the Vietoris topology that is the topology generated by the sets of the form:

V^{−} = {K ∈ K(X) : K ⊂ V } and V^{+} = {K ∈ K(X) : K ∩ V ≠ ∅}

where V is an arbitrary open subset of X. We recall that if the space X is zerodimensional (compact, Polish) then so is the space K(X) itself; also if X is perfect then so is K^{∗}(X) = K(X) \ {∅}. In particular K^{∗}(2^{ω}) is homeomorphic to 2^{ω}.

3.2. The Borel structure of the space F(X)

If the space X is not compact there is no canonical topology on the space F(X) of all closed subsets of X. In this case we will always equip F(X) with its Effros structure, that is the Borel structure generated by the sets of the form V^{+} = {F ∈ F(X) : F ∩ V ≠ ∅}. If X is compact we recover then the Borel structure generated by the canonical topology on K(X) = F(X). It is also well known that if X is Polish then this canonical Borel structure on F(X) is standard, that is can be generated by some Polish topology. One way of constructing such a topology is to embed the Polish space X in some compact space X̂ and then to identify any set F ∈ F(X) with its closure \overline{F} in X̂. The set F(X) is then identified to a Π_{2}^{0} subset of K(X̂), namely:

F(X) ≈ \{K ∈ K(X̂) : \overline{K ∩ X} = K\} .

This defines on F(X) a Polish topology which is said to be admissible and one can easily check that the Borel structure defined by an admissible topology is indeed generated by the sets of the form V^{+}. Nevertheless, when X is not compact we shall never consider any topological notion when working in F(X) which will be equipped with its Borel structure solely. Note however that all Borel notions and beyond such as Σ_{1}^{1}, Π_{1}^{1}, Σ_{2}^{1}, Π_{2}^{1}, ⋯ are meaningful for F(X).

3.3. We state next the main properties of the space F(X) when X is Polish, which we will use freely in the sequel:

a) The set {(F, G) ∈ F(X) × F(X) : F ⊂ G} is Δ_{1}^{1}.

b) The set {(F, K) ∈ F(X) × K(X) : F ∩ K ≠ ∅} is Δ_{1}^{1}.

c) The set {(F, G) ∈ F(X) × F(X) : F ∩ G ≠ ∅} is Σ_{1}^{1}.

d) If A is a Σ_{1}^{1} subset of X then the set {F ∈ F(X) : F ∩ A ≠ ∅} is Σ_{1}^{1}.

e) Let F^{∗}(X) = F(X) \ {∅}; then there exists a sequence (c_{k})_{k∈ω} of Borel functions c_{k} : F^{∗}(X) → X such that for all F ∈ F^{∗}(X), c_{k}(F) ∈ F and the set {c_{k}(F) : k ∈ ω} is dense in F.

In fact if we endow F(X) with some admissible topology then in a) and b) “Δ_{1}^{1}” can be replaced by “Π_{1}^{0}” and in e) “Borel” can be replaced by “first Baire class”.

We recall that a compact set valued mapping Ψ : Σ → K(T) is said to be u.s.c. if for any open subset W of T, the set Ψ^{−1}(W^{−}) = {α ∈ Σ : Ψ(α) ⊂ W } is an open subset of Σ. Note that if T is compact this is equivalent to say that the set {(α, x) ∈ Σ × T : x ∈ Ψ(α)} is closed in Σ × T.

The following result will be of constant use in the sequel.

Lemma 3.4. Let Σ be a Polish space, T be a compact space, and Ψ : Σ → K(T) be an u.s.c. mapping. Let S be a Π_{2}^{0} subset of T and Φ : Σ → F(S) be the mapping defined by Φ(α) = S ∩ Ψ(α).

If for all α ∈ S, Φ(α) is dense in Ψ(α), then the mapping Φ is Borel.

Proof. Let V be any open subset of S. Fix W an open subset of T such that V = S ∩ W, write W = ⋃_{n} F_{n} with all the F_{n}’s closed in T and set W_{n} = T \ F_{n}. It follows then from the density condition that

Φ(α) ∩ V ≠ ∅ ⟺ Φ(α) ∩ W ≠ ∅ ⟺ Ψ(α) ∩ W ≠ ∅
⟺ ∃n, Ψ(α) ∩ F_{n} ≠ ∅ ⟺ ∃n, Ψ(α) ⊄ W_{n}.

Hence Φ^{−1}(V^{+}) = ⋃_{n}(Σ \ Ψ^{−1}(W_{n}^{−})) is a Σ_{2}^{0} set, which proves that Φ is Borel.

□

3.5. A family of projective complete sets.

Let Δ ≈ 2^{ω} be a copy of the Cantor space and Q ⊂ Δ be a countable dense subset of Δ, we define inductively for n ≥ 1 a pair (ℙ_{n}(Q, Δ) , ℚ_{n}(Q, Δ)) = (ℙ_{n}, ℚ_{n}) of subsets of K^{∗}(Δ^{n}) ≈ K^{∗}(2^{ω}) ≈ 2^{ω} by:

\{ \begin{matrix}ℙ_{1} = K^{∗}(Q) \\ ℙ_{n} = \{K ∈ K(Δ^{n}) : ∀α ∈ Δ, K(α) ∈ ℚ_{n−1}\} for n ≥ 2 \\ ℚ_{n} = K^{∗}(Δ^{n}) \ ℙ_{n} \end{matrix}

Hence for K ∈ K^{∗}(Δ^{n}), if n ≥ 1 is odd:

K ∈ ℙ_{n} ⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ, ⋯, ∃α_{n−1} ∈ Δ, K(α_{1}, ⋯, α_{n−1}) ⊂ Q ,

and if n is even:

K ∈ ℙ_{n} ⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ, ⋯, ∀α_{n−1} ∈ Δ, K(α_{1}, ⋯, α_{n−1}) ⊄ Q .

It follows then by a straightforward induction that the set ℙ_{n} is Π_{n}^{1} and the set ℚ_{n} is Σ_{n}^{1}. It is also well known that the set ℙ_{1} is Π_{1}^{1}-complete.

Lemma 3.6. For all n ≥ 1, ℙ_{n} is Π_{n}^{1}-complete.

Proof. If B is any Π_{n}^{1} subset of 2^{ω} then by definition we can find a sequence (A_{k}, B_{k})_{1≤k≤n} satisfying for all k:

(i) B_{n} = B, (ii) A_{k} ⊂ 2^{ω} × Δ^{n−k} is Σ_{k}^{1} and B_{k} = (2^{ω} × Δ^{n−k}) \ A_{k} is Π_{k}^{1},

(iii) A_{n−k} ⊂ 2^{ω} × Δ^{k} is the projection on 2^{ω} × Δ^{k} of B_{n−k−1} ⊂ (2^{ω} × Δ^{k}) × Δ. Hence A_{1} ⊂ C := 2^{ω} × Δ^{n−1} is Σ_{1}^{1} and setting P = Δ \ Q ≈ ω^{ω} we can find a closed subset F of C × P such that A_{1} is the projection of F on C. Then if H denotes the closure in C × Δ = 2^{ω} × Δ^{n} of the set F, the mapping ε ↦ H(ε) from 2^{ω} to K(Δ^{n}) is u.s.c. hence by Lemma 3.4 Φ is Borel, and for all ε ∈ 2^{ω}:

– if n is odd we have:

ε ∈ B_{n} ⟺ ∀α_{1} ∈ Δ, (ε, α_{1}) ∈ A_{n−1}
⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ : (ε, α_{1}, α_{2}) ∈ B_{n−2}

⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ, ⋯, ∃α_{n−1} ∈ Δ, ∀α_{n} ∈ P, (ε, α_{1}, ⋯, α_{n}) ∉ F ⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ, ⋯, ∃α_{n−1} ∈ Δ, ∀α_{n} ∈ P, (ε, α_{1}, ⋯, α_{n}) ∉ H

⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ ⋯, ∃α_{n−1} ∈ Δ, H(ε, α_{1}, ⋯, α_{n−1}) ⊂ Q
⟺ H(ε) ∈ ℙ_{n}

– similarly if n is even:

ε ∈ B_{n} ⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ, ⋯, ∀α_{n−1} ∈ Δ, ∃α_{n} ∈ P : (ε, α_{1}, ⋯, α_{n}) ∈ H

⟺ ∀α_{1} ∈ Δ, ∃α_{2} ∈ Δ, ⋯, ∀α_{n−1} ∈ Δ, H(ε, α_{1}, ⋯, α_{n−1}) ⊄ Q ⟺ H(ε) ∈ ℙ_{n}

This proves that any Π_{n}^{1} subset of 2^{ω} is Δ_{1}^{1}-reducible to ℙ_{n} hence by Theorem 2.1 the set ℙ_{n} is Π_{n}^{1}-complete. □

4. Connectedness

We recall that C(X) denotes the set of all closed connected subsets of X. It is well known that if X is compact then C(X) is a closed, hence compact, subset of K(X).

Proposition 4.1. If X is a Polish space then C(X) is a Π_{2}^{1} subset of F(X).

Proof. By definition of C(X) for any F ∈ F(X) we have:

F ∉ C(X) ⟺ ∃ F_{0}, F_{1} ∈ F(X) \ {∅, F } : F_{0} ∩ F_{1} = ∅ and F ⊂ F_{0} ∪ F_{1} .

Since on F(X) the relation “F′ ≠ F ” is Borel and both relations “F_{0} ∩ F_{1} = ∅ ” and “F ⊂ F_{0} ∪ F_{1}” are Π_{1}^{1} it follows that F(X) \ C(X) is a Σ_{2}^{1} set, hence C(X) is a Π_{2}^{1} set. □

Theorem 4.2. There exists a Polish space X ⊂ 𝕀^{3} for which the set C(X) is Π_{2}^{1}-complete.

Proof. Fix Δ ≈ 2^{ω} such that \{\frac{1}{4}, \frac{3}{4}\} ⊂ Δ ⊂ [\frac{1}{4}, \frac{3}{4}] and Q a dense countable subset of Δ; then setting P = Δ \ Q we define:

X = {(x, y, z) ∈ 𝕀^{3} : x ∉ Δ or (y ∈ P and z = 0)}

which is clearly a Π_{2}^{0} subset of 𝕀^{3} hence a Polish space.

For any set A ⊂ 𝕀^{2} we set Ã = A × {0} ⊂ 𝕀^{3}. Let Ψ : K^{∗}(Δ^{2}) → K(𝕀^{3}) be the mapping defined for all K ∈ K^{∗}(Δ^{2}) by:

Ψ(K) = \{u = (x, y, z) ∈ 𝕀^{3} : z + d(x, Δ) ≥ \frac{1}{2}d(u, K̃)\}

where d denotes the standard metric on 𝕀 as well as the ℓ^{1} metric on 𝕀^{3}:

d((x, y, z), (x′, y′, z′)) = |x − x′| + |y − y′| + |z − z′| ,

and observe that K̃ ⊂ Ψ(K). Finally we define Φ : K^{∗}(Δ^{2}) → F(X) by:

Φ(K) = X ∩ Ψ(K).

Lemma 4.3. The mapping Φ is Borel.

Proof. We apply Lemma 3.4: observe first that since the set {(K, u) : u ∈ Ψ(K)} is a closed subset of K^{(}Δ^{2}) × 𝕀^{3} then the mapping Ψ is u.s.c. and we now check that for all K ∈ K^{∗}(Δ^{2}), Φ(K) is dense in Ψ(K).

Let U be an open subset of 𝕀^{3} such that U ∩ Ψ(K) ≠ ∅; we have to show that U ∩ Φ(K) ≠ ∅. Fix u = (x, y, z) ∈ U ∩ Ψ(K); we have to find some

u′ = (x′, y′, z′) ∈ U ∩ Φ(K). For all t ∈ 𝕀 and ε > 0 let J_{t}^{ε} =]t − \frac{ε}{2}, t + \frac{ε}{2}[∩𝕀; we

may suppose that U = J_{x}^{ε} × J_{y}^{ε} × J_{z}^{ε} for some ε > 0.

If x ∉ Δ then u ∈ Φ(K) and we can take u′ = u. If not then d(x, Δ) = 0 and d((x, y), K) + z = d(u, K̃) ≤ 2z hence d((x, y), K) ≤ z; and we distinguish three cases:

– If z < 1 pick x′ ∈ J_{x}^{ε} \ Δ, let δ < d(x′, Δ) such that z + δ < 1 and pick z′ ∈]z, z +δ[; then u′ = (x′, y, z′) ∈ U ∩X, d(u′, u) = |x′ − x|+|z′ − z| ≤ 2d(x′, Δ) and d(u′, K̃) ≤ d(u, K̃) + 2d(x′, Δ) ≤ 2z + 2d(x′, Δ) ≤ 2z′ + 2d(x′, Δ); hence u′ ∈ Φ(K).

– If d((x, y), K) < z = 1: let δ < ε such that d((x, y), K) < z − δ and pick x′ ∈ J_{x}^{δ} \ Δ; then u′ = (x′, y, z) ∈ U ∩ X and d(u′, u) = |x′ − x| ≤ δ hence d(u′, K̃) ≤ d(u, K̃) + δ = d((x, y), K) + z + δ ≤ 2z so u′ ∈ Φ(K).

– If d((x, y), K) = z = 1: take (x_{0}, y_{0}) ∈ K such that d((x, y), (x_{0}, y_{0})) = 1;

then since |y − y_{0}| ≤ \frac{3}{4} we have |x − x_{0}| ≥ \frac{1}{4} and since x, x_{0} ∈ Δ we can find x′ ∈]x − ε, x + ε[ \Δ such that |x′ − x_{0}| < |x − x_{0}| and then u′ = (x′, y, 1) ∈ U ∩ Φ(K). □

Let ℙ_{2} = ℙ_{2}(Q, Δ) (see Section 3.5).

Lemma 4.4. Let (x, y) ∈ (𝕀 \ Δ) × 𝕀 and set for all t ∈ 𝕀, u_{t} = (x, y, t).

a) If u_{z} ∈ Φ(K) then u_{t} ∈ Φ(K) for all t ≥ z.

b) If K ∈ ℙ_{2} then there exists z ∈ 𝕀 such that u_{z} ∈ Φ(K); in particular u_{1} ∈ Φ(K).

Proof. Observe first that since x ∉ Δ then for all t ∈ 𝕀, u_{t} ∈ X \ K̃ hence

u_{t} ∈ Φ(K) ⟺ δ(t) := t + d(x, Δ) − \frac{1}{2}d(u_{t}, Δ) ≥ 0.

a) If u_{z} ∈ Φ(K) then for all t ≥ z:

δ(t) − δ(z) ≥ (t − z) − \frac{1}{2}d(u_{t}, u_{z}) = (t − z) − \frac{1}{2}(t − z) ≥ 0

hence δ(t) ≥ δ(z) ≥ 0 and u_{t} ∈ Φ(K).

b) Fix a ∈ Δ such that d(x, Δ) = |x − a|. Since K ∈ ℙ_{2} then there exists b ∈ P such that (a, b, 0) ∈ K̃. Then for all z ≥ 1 − d(x, Δ):

d(u_{z}, K̃) ≤ |x − a| + |y − b| + z ≤ d(x, Δ) + 1 + z ≤ 2 d(x, Δ) + 2z

hence δ(z) ≥ 0; so u_{z} ∈ Φ(K) and it follows from part a) that u_{1} ∈ Φ(K). □

Lemma 4.5. If K ∉ ℙ_{2} then Φ(K) ∉ C(X).

Proof. Observe first that for any K ∈ K^{∗}(Δ^{2}) and any ε ∈ {0, 1}, v_{ε} = (ε, 0, 1) is a member of Φ(K). Indeed if w_{ε} = (a, b, 0) ∈ K̃ is such that d(v_{ε}, K̃) = d(v_{ε}, w_{ε}) then

d(v_{ε}, K̃) = |ε − a| + |b| + 1 ≤ \frac{3}{4} + \frac{3}{4} + 1 = 2(\frac{1}{4} + 1) = 2(d(ε, Δ) + 1).

If K ∉ ℙ_{2} then by definition of ℙ_{2} there exists some α ∈ Δ such that K(α) ⊂ Q and it follows then from the definition of X that Φ(K)(α) = ∅. Hence the sets V_{0} = {(x, y, z) ∈ Φ(K) : x < α} and V_{1} = {(x, y, z) ∈ Φ(K) : x > α} form an open partition of Φ(K) with v_{ε} ∈ V_{ε}, which shows that Φ(K) is not connected. □

Lemma 4.6. If K ∈ ℙ_{2} then Φ(K) ∈ C(X).

Proof. Fix K ∈ ℙ_{2}. Let J denote the set of all connected components of 𝕀 \ Δ and set for all J ∈ J:

Φ_{J}(K) = Φ(K) ∩ (J × 𝕀^{2}).

\underline{Claim 1}: Φ_{J}(K) is pathwise connected.

Proof. Set C_{J} = J × 𝕀 × {1}. Since K ∈ ℙ_{2} then by part b) of Lemma 4.4, C_{J} ⊂ Φ_{J}(K) and since it is convex the set C_{J} is pathwise connected. Moreover it follows from part a) of Lemma 4.4 that any element u = (x, y, z) ∈ Φ_{J}(K) can be connected in Φ_{J}(K) by a linear path to the element u_{1} = (x, y, 1) ∈ C_{J}. Hence Φ_{J}(K) is pathwise connected. ⋄

Suppose now that (V_{0}, V_{1}) is a partition of Φ(K) into two clopen subsets and that V_{0} contains v_{0} = (0, 0, 1) and set for ε = 0, 1, J_{ε} = {J ∈ J : Φ_{J}(K) ⊂ V_{ε}}. It follows then from Claim 1 that J = J_{0} ∪ J_{1}.

\underline{Claim 2}: If u = (a, b, 0) ∈ K̃ with a ∈ \overline{⋃ J_{ε}} then u ∈ V_{ε}.

Proof. Fix a sequence (J_{k})_{k} in J_{ε} with for each k some x_{k} ∈ J_{k} such that a = lim_{k} x_{k}, and let z_{k} = |x_{k} − a|. Since x_{k} ∉ Δ then u_{k} = (x_{k}, b, z_{k}) ∈ X. Moreover since u ∈ K̃ then u ∈ Φ(K) and

d(u_{k}, K̃) ≤ d(u_{k}, u) = |x_{k} − a| + z_{k} = 2z_{k} ≤ 2(z_{k} + d(x_{k}, Δ))

hence u_{k} ∈ Φ_{J_{k}}(K) ⊂ V_{ε}; and since V_{ε} is a closed subset of Φ(K) then u = lim_{k} u_{k} ∈ V_{ε}. ⋄

\underline{Claim 3}: J = J_{0}.

Proof. For otherwise J_{1} ≠ ∅; let a = inf(⋃ J_{1}). Since by assumption v_{0} ∈ V_{0}

then [0, \frac{1}{4}[∈ J_{0} and inf J ∈ Δ for every J ∈ J_{1}, hence a ∈ Δ. And since K ∈ ℙ_{2},

we can find some b ∈ P such that u = (a, b, 0) ∈ K̃ ∩ X. Then fixing for all k some I_{k} ∈ J with I_{k}∩]a − 2^{−k}, a[≠ ∅, we have that I_{k} ∈ J_{0}. Also we can find for all k some J_{k} ∈ J_{1} with J_{k}∩]a, a + 2^{−k}[≠ ∅. It follows then from Claim 2 that u ∈ V_{0} ∩ V_{1} which is a contradiction. ⋄

It follows from Claim 3 that ((𝕀 \ Δ) × 𝕀^{2}) ∩ Φ(K) ⊂ V_{0}. Moreover since (Δ × 𝕀^{2}) ∩ Φ(K) ⊂ K̃ then it follows from Claim 2 that Φ(K) ⊂ V_{0}. This proves that Φ(K) is connected and finishes the proof of Lemma 4.6. □

So by Lemma 4.3, Lemma 4.5 and Lemma 4.6 Φ : K^{∗}(Δ^{2}) → F(X) is a Borel mapping which reduces the complete Π_{2}^{1} set ℙ_{2} to C(X). Hence by Theorem 2.1 the set C(X) is Π_{2}^{1}-complete. □

5. Pathwise Connectedness

We recall that C_{path}(X) denotes the set of all closed pathwise connected subsets of X.

Proposition 5.1. If X is a Polish space then C_{path}(X) is a Π_{2}^{1} subset of F(X).

Proof. Fix a compatible complete metric d on X and equip the space Π(X) of all continuous paths γ : 𝕀 → X with the Polish topology of uniform convergence.

Then for any set F ∈ F(X) we have:

F ∈ C_{path}(X) ⟺ \{ \begin{matrix}∀x, y ∈ F, ∃ γ ∈ Π(X) : γ(0) = x, \\ γ(1) = y and ∀t ∈ ℚ, γ(t) ∈ F \end{matrix}

where ℚ denotes the set of rational numbers in 𝕀.

Since the relations “x ∈ F ” and “γ(t) = x” determine relatively Borel, hence Σ_{1}^{1}, subsets of X × F(X) and Π(X) × 𝕀 × X then C_{path}(X) is clearly a Π_{2}^{1} subset of F(X). □

As we shall see the previous complexity bound is optimal even when X is compact. The proof of this relies on some general constructions of pathwise connected spaces that we present first.

Definition 5.2. Given any element a in some set X we set 𝕀_{a} = 𝕀 × {a}. A set A ⊂ 𝕀 × X is said to be pinned at a if A ⊃ 𝕀_{a}.

We denote by π_{0} and π_{1} the projection mappings from 𝕀 × X onto 𝕀 and X respectively.

Proposition 5.3. Let A ⊂ 𝕀 × X be a pinned set at a ∈ X and suppose that π_{0}(A \ 𝕀_{a}) is of empty interior in 𝕀. Then A is pathwise connected if and only if for all ε ∈ 𝕀 the section A(ε) is pathwise connected.

Proof. Suppose that all the sections of A are pathwise connected and let (α, x) and (β, y) be two elements of A. Then (α, x) can be joined to (α, a) by a path in {α} × A(α) ⊂ A, and (β, a) can be joined to (β, y) by a path in {β} × A(β) ⊂ A. Since (α, a) can trivially be joined to (β, a) by an affine path in 𝕀_{a} then by concatenation, (α, x) can be joined to (β, y) by a path in A.

Conversely suppose that A is pathwise connected . Let x and y be two elements of some section A(ε), and let γ be a path in A joining (ε, x) to (ε, y).

– If γ(𝕀) ∩ 𝕀_{a} = ∅ then π_{0}(γ(𝕀)) is a connected subset of 𝕀 with empty interior, hence it is a singleton, and since γ(0) = (ε, x) then necessarily π_{0}(γ(𝕀)) = {ε}. It follows that π_{1} ∘ γ is a path in A(ε) joining x to y.

– If γ(𝕀) ∩ 𝕀_{a} ≠ ∅ let u = min{t ∈ 𝕀 : γ(t) ∈ 𝕀_{a}} then γ([0, u[) ∩ 𝕀_{a} = ∅ and as in the previous case we necessarily have π_{0} ∘ γ(t) = ε for all t < u hence by continuity γ(u) = (ε, a) and it follows that γ_{0} = π_{1}∘γ_{|[0,u]} is a path in A(ε) joining x to a. Similarly if v = max{t ∈ 𝕀 : γ(t) ∈ 𝕀_{a}} then π_{0} ∘ γ(t) = ε for all t ≥ v and γ_{1} = π_{1} ∘ γ_{|[v,1]} is a path in A(ε) joining x to a. Then by the concatenation of γ_{0} and γ_{1} one gets a path in A(ε) joining x to y. □

Proposition 5.4. Let A, B ⊂ 𝕀 × X be two pinned sets at a and b respectively, and suppose that E = π_{0}(A \ 𝕀_{a}) ∪ π_{0}(B \ 𝕀_{b}) is of empty interior in 𝕀.

If A and B are pathwise connected then C = A ∪ B is pathwise connected if and only if for some ε ∈ 𝕀 the section C(ε) is pathwise connected.

Proof. If C(ε) is pathwise connected for some ε then we can find a path in C joining (ε, a) ∈ A to (ε, b) ∈ B, and since A and B are pathwise connected then clearly C is pathwise connected too.

Conversely if C is pathwise connected pick any α ∈ 𝕀, let γ be a path in C joining (α, a) to (α, b), consider

u = max{t ∈ 𝕀 : γ(t) ∈ 𝕀_{a}} and v = min{t ∈ [u, 1] : γ(t) ∈ 𝕀_{b}}

and set γ(u) = (α′, a) and γ(v) = (α′′, b).

– If u = v then α′ = α′′ and a = b and since by assumption π_{0}(C \ 𝕀_{a}) = E is of empty interior in 𝕀 then the set C is pinned at a and by Proposition 5.3, C(ε) is pathwise connected for all ε.

– If u ≠ v then γ(]u, v[) ∩ (𝕀_{a} ∪ 𝕀_{b}) = ∅ hence π_{0}(γ(]u, v[)) ⊂ E is of empty interior in 𝕀, hence as in Proposition 5.3, π_{0}(γ(]u, v[)) is a singleton say {ε}. It follows by continuity that π_{0}(γ([u, v])) = {ε} hence α′ = α′′ = ε and π_{1} ∘ γ_{|[u,v]} is a path in C(ε) joining a to b, and since by Proposition 5.3, A(ε) and B(ε) are pathwise connected, then C(ε) is pathwise connected. □

Remark 5.5. Let X be a compact space and suppose that Φ : 2^{ω} → K(X) is a continuous mapping reducing some Π_{2}^{1} subset A of 2^{ω} to the set C_{path}(X). Then since C_{path}(X) is a subset of the compact space C(X) any element from \overline{A} \ A is necessarily mapped by Φ to an element in C(X) \ C_{path}(X). Hence Theorem5.6 below provides implicitly compact sets which are connected but not pathwise connected. As a matter of fact the construction of the space X in this proof will

make use of the classical result that if G is the graph of the function x ↦ sin \frac{1}{x} then its closure \overline{G} = G ∪ (\{0\} × [−1, 1]) is connected but not pathwise connected. More precisely, we shall use the fact that no element from {0} × [−1, 1] can be joined to an element from G by a path in \overline{G}.

Theorem 5.6. The set C_{path}(𝕀^{4}) is Π_{2}^{1}-complete.

Proof. The construction will actually be achieved in the space 𝕀^{2} × 𝕁^{2} ≈ 𝕀^{4} where 𝕁 = [−1, 1].

Let ε ∈ 2^{ω} that we view as the characteristic function of some set M_{ε} ⊂ ω and set N_{ε} = {0} ∪ M_{ε}. Let ν_{ε} ∈ ω^{≤ω} \ {∅} be the increasing enumeration of N_{ε} and set:

H_{ε}^{+} = ⋃_{j∈dom (ν_{ε})}\{(x, y) ∈]2^{−ν_{ε}(j+1)}, 2^{−ν_{ε}(j)}] × 𝕁 : y = 2^{−ν_{ε}(j)} sin \frac{π}{x}\} ⊂]0, 1] × 𝕁

where by convention 2^{−ν(i)} = 0 if i ∉ dom (ν). We also set:

H_{ε}^{−} = H^{−} = ([−1, 0] × {0}) ∪ ({0} × 𝕁) ⊂ [−1, 0] × 𝕁

and

H_{ε} = H_{ε}^{−} ∪ H_{ε}^{+} ⊂ 𝕁^{2} .

Observe that for all ε, H_{ε} is non empty and compact. More precisely: – if N_{ε} is infinite then \overline{H_{ε}^{+}} = H_{ε}^{+} ∪ \{(0, 0)\},

– if N_{ε} is finite and k = max N_{ε} then \overline{H_{ε}^{+}} = H_{ε}^{+} ∪ (\{0\} × 2^{−k}𝕁).

Let:

P = {ε ∈ 2^{ω} : ∀m, ∃n ≥ m, ε(n) = 1} ,
Q = {ε ∈ 2^{ω} : ∃m, ∀n ≥ m, ε(n) = 0} .

Lemma 5.7. a) For all ε, H_{ε}^{−} and H_{ε}^{+} are pathwise connected.

b) If ε ∈ P then H_{ε} is pathwise connected.

c) If ε ∈ Q then H_{ε} is not pathwise connected.

Proof. a) For all ε, H_{ε}^{−} = H^{−} is clearly pathwise connected and H_{ε}^{+} is the graph of a continuous function f_{ε} : ]0, 1] → 𝕁 hence also pathwise connected.

b) If ε ∈ P then lim_{x→0} f_{ε}(x) = 0 and \overline{H_{ε}^{+}} is also the graph of a continuous function f̃_{ε} : 𝕀 → 𝕁 with f̃_{ε}(0) = 0, hence \overline{H_{ε}^{+}} is pathwise connected. Then writing H_{ε} = H^{−} ∪\overline{H_{ε}^{+}} each of both sets in this union is pathwise connected and contains the element (0, 0). It follows that H_{ε} is pathwise connected too.

c) If ε ∈ Q and k = max(N_{ε}) then G_{ε} = H_{ε} ∩ (]0, 2^{−k}] × 𝕁) is the graph of the function x ↦ 2^{−k} sin(\frac{π}{x}) on ]0, 2^{−k}] and \overline{G_{ε}} = G_{ε} ∪ ([0, 2^{−k}] × 2^{−k}𝕁). And as

pointed out in Remark 5.5, no element from \overline{G_{ε}} \ G_{ε} can be joined to an element from G_{ε} by a path in \overline{G_{ε}}, hence by a path in H_{ε} since \overline{G_{ε}} is a retract of H_{ε}. □

From now on we identify 2^{ω} to a compact set Δ ⊂ 𝕀 with endpoints 0 and 1 and set ℙ_{1} = ℙ_{1}(Q, Δ) and ℙ_{2} = ℙ_{2}(Q, Δ).

Let a = (−1, 0) and b = (1, 0) viewed as elements of 𝕁^{2} so 𝕀_{a} ∪ 𝕀_{b} ⊂ 𝕀 × 𝕁^{2}. Observe that for ε ∈ 2^{ω} we have a ∈ H_{ε}^{−} and b ∈ H_{ε}^{+}. We then define the mapping Ψ : K(Δ) → K(𝕀 × 𝕁^{2}) by

Ψ(L) = 𝕀_{a} ∪ 𝕀_{b} ∪ ⋃_{ε∈L}{ε} × H_{ε} .

Lemma 5.8. Ψ(L) ∈ C_{path}(𝕀 × 𝕁^{2}) if and only if L ∉ ℙ_{1}.

Proof. Set Ψ^{−}(L) = 𝕀_{a} ∪ (Ψ(L) ∩ ([−1, 0] × 𝕁^{2})) and Ψ^{+}(L) = 𝕀_{b} ∪ (Ψ(L) ∩ (]0, 1] × 𝕁^{2})). Then the subsets Ψ^{−}(L) and Ψ^{+}(L) of 𝕀 × 𝕁^{2} are pinned at a and b respectively, and by Lemma 5.7 for all ε ∈ L the sets Ψ^{−}(L)(ε) = {a}∪H_{ε}^{−} = H_{ε}^{−} and Ψ^{+}(L)(ε) = {b}∪H_{ε}^{+} = H_{ε}^{+} are pathwise connected hence by Proposition 5.3, Ψ^{−}(L) and Ψ^{+}(L) are pathwise connected. Since Ψ(L) = Ψ^{−}(L) ∪ Ψ^{+}(L) and π_{0}(Ψ^{−}(L) \ 𝕀_{a}) ∪ π_{0}(Ψ^{+}(L) \ 𝕀_{b}) ⊂ L is of empty interior in 𝕀, it follows then from Proposition 5.4 and Lemma 5.7 that:

Ψ(L) ∈ C_{path}(𝕀 × 𝕁^{2}) ⟺ ∃ε ∈ 𝕀 : Ψ(L)(ε) ∈ C_{path}(𝕁^{2})
⟺ ∃ε ∈ L : H_{ε} ∈ C_{path}(𝕁^{2})
⟺ L ∩ ℙ ≠ ∅
⟺ L ∉ ℙ_{1} .  □

Let c = (0, b) ∈ 𝕀 × 𝕁^{2} so 𝕀_{c} ⊂ 𝕀^{2} × 𝕁^{2} and observe that for all L ∈ K(Δ), c ∈ Ψ(L). We then define Φ : K^{∗}(Δ^{2}) → K(𝕀^{2} × 𝕁^{2}) by

Φ(K) = 𝕀_{c} ∪ ⋃_{α∈Δ}{α} × Ψ(K(α)).

Lemma 5.9. Φ(K) ∈ C_{path}(𝕀^{2} × 𝕁^{2}) if and only if K ∈ ℙ_{2}.

Proof. The set Φ(K) ⊂ 𝕀 × (𝕀 × 𝕁^{2}) is pinned at c. It follows then from Proposition 5.3 and Lemma 5.8 that:

Φ(K) ∈ C_{path}(𝕀^{2} × 𝕁^{2}) ⟺ ∀α ∈ 𝕀, Φ(K)(α) ∈ C_{path}(𝕀 × 𝕁^{2})
⟺ ∀α ∈ Δ, Ψ(K(α)) ∈ C_{path}(𝕀 × 𝕁^{2})
⟺ ∀α ∈ Δ, α ∉ ℙ_{1}
⟺ K ∈ ℙ_{2}.  □

To finish the proof of Theorem 5.6 observe that the set {(K, z) : z ∈ Φ(K)} is a closed subset of K^{∗}(Δ^{2}) × (𝕀^{2} × 𝕁^{2}), so the mapping Φ is u.s.c. hence Borel, and by Lemma 5.8 and Lemma 5.9, Φ reduces the Π_{2}^{1}-complete set ℙ_{2} to C_{path}(𝕀^{2}×𝕁^{2}). Hence by Theorem 2.1 the set C_{path}(𝕀^{2} × 𝕁^{2}) is Π_{2}^{1}-complete. □

6. Local Connectedness

We recall that C_{loc}(X) denotes the set of all closed locally connected subsets of X.

We also recall that a topological space is said to be locally connected if any element has a fundamental system of connected (not necessarily open) neighbourhoods. But a basic result asserts that in a locally connected space any element has in fact a fundamental system of connected open neighbourhoods, equivalently that the connected components of any open set are all open.

Finally we recall that A(Π_{1}^{1}) denotes the class of all sets obtained by the Souslin A-operation from Π_{1}^{1} sets, and Ǎ(Π_{1}^{1}) its dual class.

Proposition 6.1. If X is a compact space then C_{loc}(X) is a Π_{3}^{0} subset of K(X).

Proof. Fix a countable basis (U_{j})_{j∈ω} of X and let F ∈ F(X). If F is locally connected then for all ε > 0 and all x ∈ F there exists an open set V in X such that x ∈ V, diam V < ε and V ∩ F is connected; then F′ = \overline{V ∩ F} is a closed connected subset of X such that diam F′ ≤ ε and if j is such that x ∈ U_{j} ⊂ V then U_{j} ∩ F ⊂ F′. Hence if F is locally connected then

(⋆) \{ \begin{matrix}∀n, ∀x ∈ F, ∃ j ∈ ω : x ∈ U_{j}, ∃ F′ ∈ C(X) : \\ diam F′ ≤ 2^{−n} and U_{j} ∩ F ⊂ F′ \end{matrix}

Conversely if (⋆) holds then for all ε > 0, any element of F admits a connected neighbourhood of diameter < ε and so F is locally connected.

Then it follows from the compactness of F that condition (⋆) is equivalent to

(⋆⋆) \{ \begin{matrix}∀n, ∃ J finite ⊂ ω : F ⊂ ⋃_{j∈J} U_{j} and ∀j ∈ J, \\ ∃ F′ ∈ C(X) : diam F′ ≤ 2^{−n} and U_{j} ∩ F ⊂ F′ \end{matrix}

Hence:

C_{loc}(X) = ⋂_{n} ⋃_{{J finite \atop J⊂ω}} ⋂_{j∈J} L_{j,J,n}

where L_{j,J,n} is the subset of K(X) defined by the second line of (⋆⋆), and which is closed as the projection of a closed subset of the compact space K(X)×C(X). □

Theorem 6.2. The set C_{loc}(𝕀^{2}) is Π_{3}^{0}-complete.

Proof. For any a, b ∈ ℝ^{2} let [a, b] denote the affine segment joining a to b. Let S = [(0, 0) , (0, 1)] and given any α ∈ 𝕀^{ω} let S_{n}(α) = [(0, 0) , (2^{−n}, α(n))] and K(α) = S ∪ ⋃_{n} S_{n}(α). Then clearly K(α) is a compact connected subset of 𝕀^{2}.

If lim_{n} α(n) = 0 then the open subset K(α) \ {(0, 0)} of K(α) is locally homeomorphic to 𝕀, hence it is locally connected. Moreover for all s, t > 0 the open neigbourhood V_{s,t} = K(α)∩[0, s[×[0, t[ of {(0, 0)} in K(α) is itself homeomorphic to K(α), so V_{s,t} is connected. Hence K(α) is locally connected.

If lim sup_{n} α(n) = δ > 0 then any connected neigbourhood of (0, δ) in K(α) contains the whole segment [(0, 0) , (0, δ)]. Hence K(α) is not locally connected.

Since for all n the mapping Φ_{n} : α ↦ S ∪ ⋃_{p≤n} S_{p}(α) is clearly continuous it follows that the uniform limit Φ : α ↦ K(α) is continuous too, and by the previous observations Φ reduces the set A = {α ∈ 𝕀^{ω} : lim_{n} α(n) = 0} to the set C_{loc}(𝕀^{2}), and it is well known that the set A is Π_{3}^{0}-complete. □

Theorem 6.3. If X is a Polish space then C_{loc}(X) is a Ǎ(Π_{1}^{1}) subset of F(X).

Proof. Fix a sequence (c_{k})_{k∈ω} of Borel functions c_{k} : F^{∗}(X) → X such that for all F ∈ F^{∗}(X), c_{k}(F) ∈ F and the set {c_{k}(F) : k ∈ ω} is dense in F (see Section 3). Then given any F ∈ F^{∗}(X) and any V open subset of X we define for all k:

C_{k}(F, V) = \overline{\{x : ∃K ∈ C(X) compact : \{x, c_{k}(F)\} ⊂ K ⊂ F ∩ V \}}

which is clearly a closed connected subset of F containing c_{k}(F).

\underline{Claim 1}: The set A_{k} = \{(x, F) ∈ X × F^{∗}(X) : x ∈ C_{k}(F, V)\} is Σ_{1}^{1}.

Proof. Let X̂ be any compactification of X and Û_{i} be open subsets of X̂ such that U_{i} = X ∩ Û_{i}; then

x ∈ C_{k}(F, V) ⟺ ∀i, x ∉ U_{i} or ∃K ∈ C(X̂) : \{ \begin{matrix}K ∩ Û_{i} ≠ ∅, K ⊂ F ∩ V, \\ x ∈ K, c_{k}(F) ∈ K \end{matrix}

Then observe that Û_{i} being open and V being a Π_{2}^{0} subset of the compact space X̂ the conditions “K ∈ C(X̂), K ∩ Û_{i} ≠ ∅, K ⊂ F ∩ V ” define a Π_{2}^{0} subset of the compact space K(X̂). Also since c_{k} is Borel then the condition “c_{k}(F) ∈ K” determines a Borel, hence a Σ_{1}^{1}, subset of F^{∗}(X) × K(X̂). It follows that A_{k} is Σ_{1}^{1}. ⋄

\underline{Claim 2}: For any open set U ⊂ V, the set \{F ∈ F^{∗}(X) : F ∩ U ⊂ C_{k}(F, V)\} is Σ_{1}^{1}.

Proof. Let J = \{j ∈ ω : \overline{U_{j}} ⊂ U\}; then U = ⋃_{j∈J} U_{j} and

F ∩ U = ⋃_{j∈J} F ∩ U_{j} ⊂ ⋃_{j∈J} \overline{F ∩ U_{j}} ⊂ ⋃_{j∈J} F ∩ \overline{U_{j}} = F ∩ U

hence, since {c_{i}(F) : i ∈ ω} ∩ U_{j} is dense in F ∩ U_{j}:

F ∩ U ⊂ C_{k}(F, V) ⟺ ∀j ∈ J, \overline{F ∩ U_{j}} ⊂ C_{k}(F, V)

⟺ ∀j ∈ J, ∀i ∈ ω, c_{i}(F) ∉ U_{j} or c_{i}(F) ∈ C_{k}(F, V) .

The conclusion follows then from Claim 1. ⋄

Consider the tree

T = \{s ∈ ω^{<ω} : ∀m < |s| , diam(U_{s(m)}) ≤ 2^{−m} and \overline{U_{s(m)}} ⊂ U_{s(m−1)}\}.

Set E_{∅} = F^{∗}(X) and for all s ∈ T ∩ ω^{n+1} let

E_{s} = {F ∈ F^{∗}(X) : F ∩ U_{s(n)} ≠ ∅ and ∀k, F ∩ U_{s(n)} ⊄ C_{k}(F, U_{s(0)})}

\underline{Claim 3}: F^{∗}(X) \ C_{loc}(X) = A((E_{s})_{s∈T} ).

Proof. If F is not locally connected we can find some U_{i_{0}} and x ∈ F ∩ U_{i_{0}} such that F ∩U_{i_{0}} does not contain any connected neigbourhood of x in F. We can also find an infinite branch σ of T such that σ(0) = i_{0} and x ∈ U_{σ(n)} for all n. Then for all k, since C_{k}(F, U_{i_{0}}) is a connected subset of F ∩ U_{σ(0)}, then for no n we can have F ∩ U_{σ(n)} ⊂ C_{k}(F, U_{i_{0}}). Hence F ∈ E_{σ_{|n}} for all n, and so F ∈ A((E_{s})_{s∈T} ).

Conversely if for some infinite branch σ of T, we have F ∈ E_{σ_{|n}} for all n then F is not locally connected. For otherwise let {x} = ⋂_{n} U_{σ(n)} and let C be the connected component of x in F ∩ U_{σ(0)}. Since F is assumed to be locally connected then C is open in F and we can find some m and then some k such that c_{k}(F) ∈ F ∩ U_{σ(m)} ⊂ C. Moreover C is a connected and locally connected Polish space, and it is well known that such a space is actually pathwise connected. Hence any element in C can be joined to c_{k}(F) by a path in C ⊂ F ∩ U_{σ(0)}. It follows that x ∈ F ∩ U_{σ(m)} ⊂ C_{k}(F, U_{σ(0)}) which contradicts that F ∈ E_{σ_{|m+1}}. ⋄

To finish the proof of Theorem 6.1 observe that by Claim 2 each E_{s} is a Π_{1}^{1} set hence by Claim 3 the set C_{loc}(X) is in the class Ǎ(Π_{1}^{1}). □

Theorem 6.4. There exists a Polish space X ⊂ 𝕀^{3} for which the set C_{loc}(X) is Ǎ(Π_{1}^{1})-complete.

Proof. The space X will be of the form:

X = ⋃_{s∈ω^{<ω}} X_{s}

where each X_{s} for s ≠ ∅, is contained in a 2-dimensional affine space and is affinely isomorphic to a fixed Polish space Y ⊂ 𝕁 × 𝕁 of the plane. The proof is rather technical and will be split in three parts:

a) Definition and study of the space Y.

b) Definition of the space X.

c) Construction for any given A(Π_{1}^{1}) set A ⊂ 2^{ω} of a Borel mapping Φ : 2^{ω} → F^{∗}(X) which reduces A to F(X) \ C_{loc}(X).

a) \underline{Definition of the space Y}: We fix \{−\frac{1}{2}, \frac{1}{2}\} ⊂ Δ ⊂ [−\frac{1}{2}, \frac{1}{2}] a copy of 2^{ω} such that Q := Δ ∩ ℚ is dense in Δ, and set ℙ_{1} = ℙ_{1}(Q, Δ) ⊂ K^{∗}(Δ).

Let a = (0, 1) , a^{∗} = (0, −1) , b = (1, 0) , b^{∗} = (−1, 0) and let Σ ⊂ 𝕁 × 𝕁 be the full square of vertices a , a^{∗} , b , b^{∗}, that is the convex hull of the set {a, a^{∗}, b, b^{∗}}. We point out here that despite the formal symmetry between a, a^{∗} and b, b^{∗} the central role in the construction will be played by a, a^{∗}. We then define

Y = Σ \ (Q × {0})

which is clearly a Π_{2}^{0} subset of 𝕁 × 𝕁.

In this part of the proof by square we shall mean a subsquare of Σ whose edges are parallel to the edges of Σ. Hence if we consider the convex cone

C = {(x, y) ∈ ℝ^{2} : y ≥ |x|}

then a square is a set of the form (v+C)∩(w−C) where (v, w) is a pair of elements of Σ with same first coordinate. Note that any element u ∈ Σ admits a basis of neighbourhoods in Σ constituted by squares with vertices v, w ∈ Σ \ (𝕁 × {0}).

For any E ⊂ ℝ^{2} we set:

E^{0} = E ∩ (ℝ × {0}) ; E^{+} = E ∩ (ℝ×]0, +∞[) ; E^{−} = E ∩ (ℝ×] − ∞, 0[) .

so

E = E^{0} ∪ E^{+} ∪ E^{−}

Lemma 6.5. There exists a Borel mapping Ψ : K^{∗}(Δ) → F(Y) such that for all L ∈ K^{∗}(Δ),

a) Ψ(L) is locally connected; moreover for any convex set W containing a (resp. a^{∗}) the set W^{+} ∩ Ψ(L) (resp. W^{−} ∩ Ψ(L)) is connected.

b) if L ∉ ℙ_{1} then Ψ(L) ∈ C(Y).

c) if L ∈ ℙ_{1} then a and a^{∗} are separated by a clopen set, hence Ψ(L) ∉ C(Y).

Proof. Let Ψ̂ : K^{∗}(Δ) → K(Σ) and Ψ : K^{∗}(Δ) → F(Y) defined for all L ∈ K^{∗}(Δ) by:

Ψ̂(L) = {(x, y) ∈ Σ : |y| ≥ d(x, L)} ⊃ L × {0}

where d is the standard metric on 𝕀, and

Ψ(L) = Ψ̂(L) ∩ Y .

The mapping Ψ̂ is clearly u.s.c. and for all L ∈ K^{∗}(Δ), Ψ(L) is dense in Ψ̂(L) hence by Lemma 3.4 the mapping Ψ is Borel.

We then set for ε = 0, +, − :

Ψ^{ε}(L) = Ψ(L) ∩ Y^{ε}

and for all α ∈ 𝕁 :

C_{α} = {(x, y) ∈ ℝ^{2} : y ≥ |x − α|} = (α, 0) + C
C′_{α} = {(x, y) ∈ ℝ^{2} : y ≤ − |x − α|} = (α, 0) − C.

Hence

Ψ^{+}(L) = ⋃_{α∈L} C_{α} ∩ Σ^{+} and Ψ^{−}(L) = ⋃_{α∈L} C′_{α} ∩ Σ^{−} .

The inclusions from right to left are obvious since for all x, α, d(x, L) ≤ |x − α|, and from left to right since for all x, d(x, L) = |x − α| for some α.

a) Let u ∈ Ψ(L) and ε > 0. Consider a square V = (v + C) ∩ (w − C) such that u ∈ V with diam(V) < ε. By symmetry we may suppose that u ∉ Ψ^{−}(L) and w ∈ Σ^{+}.

Since the cones C_{α} and w − C have parallel edges then V ∩ C_{α} ≠ ∅ if and only w ∈ C_{α}, and since w ∈ Σ^{+} then

V ∩ Ψ^{+}(L) = ⋃{V ∩ C_{α} ∩ Σ^{+} : w ∈ C_{α}} .

Note that each V ∩ C_{α} ∩ Σ^{+} is a convex, hence connected, set and since in the formula above all the sets V ∩C_{α}∩Σ^{+} have w as a common element then V ∩Ψ^{+}(L) is connected. We now distinguish two cases:

– If u ∈ Ψ^{+}(L) then we can choose V small enough so that V ∩ Ψ(L) = V ∩ Ψ^{+}(L) and we are done.

– If u ∈ Ψ^{0}(L) then (V ∩ Ψ^{+}(L)) ∪ (V ∩ Ψ^{0}(L)) = \overline{V ∩ Ψ^{+}(L)}^{Y} is connected, and we necessarily have that v ∈ Σ^{−} hence (V ∩Ψ^{−}(L))∪(V ∩Ψ^{0}(L)) is connected too, and since V ∩ Ψ^{0}(L) ≠ ∅ it follows that V ∩ Ψ(L) is connected.

Finally if W is any convex set containing a then for any x ∈ Z = W^{+} ∩ Ψ(L) the segment [a, x] is contained in Z hence Z is connected. ⋄

b) Recall that Ψ^{+}(L) = ⋃_{α∈L} C_{α}∩Σ^{+} and the sets C_{α}∩Σ^{+} are all convex hence connected; and since a = (0, 1) ∈ C_{α} ∩ Σ^{+} for all α then Ψ^{+}(L) is connected. Similarly Ψ^{−}(L) is connected. Then as above Ψ^{+}(L) ∪ Ψ^{0}(L) = \overline{Ψ^{+}(L)}^{Y} and Ψ^{−}(L) ∪ Ψ^{0}(L) = \overline{Ψ^{−}(L)}^{Y} are connected. Hence if L ∉ ℙ_{1} then Ψ^{0}(L) ≠ ∅ and Ψ(L) = Ψ^{0}(L) ∪ Ψ^{+}(L) ∪ Ψ^{0}(L) is connected.

c) If L ∈ ℙ_{1} then Ψ^{0}(L) = ∅ and then (Ψ^{+}(L), Ψ^{−}(L)) form an non trivial open covering of Ψ(L) separating a and a^{∗}. ⋄

This finishes the proof of Lemma 6.5. □

\underline{Some variations of the space Y}:

We recall that a = (0, 1) , a^{∗} = (0, −1) , b = (1, 0) , b^{∗} = (−1, 0) and Σ ⊂ 𝕁 × 𝕁 is the full square of vertices a , a^{∗} , b , b^{∗}.

For any pair (u^{∗}, u) of distinct elements in ℝ^{3} and any ε > 0 let h : ℝ^{2} → ℝ^{3} be the unique affine mapping such that

h(a) = u, h(a^{∗}) = u^{∗}, h(b) = \frac{u + u^{∗}}{2} + (0, 0, \frac{ε}{2}), h(b^{∗}) = \frac{u + u^{∗}}{2} − (0, 0, \frac{ε}{2}) .

Set Σ^{(u^{∗},u,ε)} = h(Σ) , Σ_{+}^{(u^{∗},u,ε)} = h(Σ^{+}) , Σ_{−}^{(u^{∗},u,ε)} = h(Σ^{−}), Y^{(u^{∗},u,ε)} = h(Y) and let Ψ^{(u^{∗},u,ε)} : K^{∗}(Δ) → F(Y^{(u^{∗},u)}) be the mapping defined by:

Ψ^{(u^{∗},u,ε)}(L) = h(Ψ(L))

where Ψ is the mapping given by Lemma 6.5.

b) \underline{Definition of the space X}:

For any s ∈ ω^{≤ω} let s^{+} be the element of (ω \ {0})^{≤ω} of the same length such that s^{+}(i) = s(i) + 1 for all i < |s^{+}| = |s|, let χ(s) ∈ 𝕀 be the real whose expansion into continued fraction is given by s^{+} and let I_{s} denote the closed interval of endpoints χ(s) and χ(s^{⌢}⟨0⟩). We recall that:

(i) χ(∅) = 0, χ(⟨n⟩) = \frac{1}{n + 1} and χ(s^{⌢}⟨n, 0⟩) = χ(s^{⌢}⟨n + 1⟩);

(ii) χ(s) is a rational number if and only if the sequence s is finite; (iii) for all s ≺ t, χ(t) ∈ I_{t} ⊂ I_{s}; (iv) I_{s^{⌢}⟨n⟩} ∩ I_{s^{⌢}⟨n+1⟩} = {χ(s^{⌢}⟨n⟩)} and I_{s^{⌢}⟨n⟩} ∩ I_{s^{⌢}⟨m⟩} = ∅ if |m − n| > 1;

(v) lim sup_{n→∞} I_{s^{⌢}⟨n⟩} = {χ(s)} .

For any s ∈ ω^{<ω} we set ν(s) = |s| + ∑_{k<|s|} s(k) and let:

ε_{s} = 2^{−ν(s)}  ;  a_{s} = (χ(s), ε_{s}, 2^{−|s|})

and if s ≠ ∅ we set s^{∗} = s_{||s|−1} and let

Σ_{s} = Σ^{(a_{s^{∗}},a_{s},ε_{s})} ; Σ_{s}^{+} = Σ_{+}^{(a_{s^{∗}},a_{s},ε_{s})} ; Σ_{s}^{−} = Σ_{−}^{(a_{s^{∗}},a_{s},ε_{s})}
X_{s} = Y^{(a_{s^{∗}},a_{s},ε_{s})} ; N_{s} = I_{s^{∗}} × [0, ε_{s^{∗}}] × [0, 2^{−|s^{∗}|}]
Ψ_{s} = Ψ^{(a_{s^{∗}},a_{s},ε_{s})} : K^{∗}(Δ) → F(X_{s}) .

Observe that if χ(s) = χ(t) and |s| = |t| then s = t, so a_{s} ≠ a_{t} whenever s ≠ t.

Finally for any set E ⊂ ℝ we set Ẽ := E × {0} × {0} and define

X_{∅} = ℙ̃ and X = ⋃_{s∈ω^{<ω}} X_{s}.

One can then easily check that for all s:

(i) for all t ⪰ s, a_{t} ∈ X_{t} ⊂ Σ_{t} ⊂ N_{t} ⊂ N_{s}; (ii) for all t ≠ s, Σ_{s} ∩ Σ_{t} ≠ ∅ if and only if Σ_{s} ∩ Σ_{t} = {a_{r}}

with: r = s = t^{∗} or r = t = s^{∗} or r = s^{∗} = t^{∗}; (iii) lim_{n→∞} a_{s^{⌢}⟨n⟩} = â_{s} := (χ(s), 0, 2^{−|s|−1}); (iv) lim sup_{n→∞} Σ_{s^{⌢}⟨n⟩} = lim inf_{n→∞} X_{s^{⌢}⟨n⟩} = Σ̂_{s} is the line segment [â_{s}, a_{s}];

(v) lim sup_{n→∞} ⋃_{t≻s^{⌢}⟨n⟩} Σ_{t} = Σ̃_{s} := {(χ(s), 0, z) : 0 ≤ z ≤ 2^{−|s|−1}}; (vi) Σ̂_{s} ∩ X = {a_{s}} and Σ̃_{s} ∩ X = ∅.

Lemma 6.6. X is a Polish space.

Proof. Observe first that X_{∅} = ℙ̃ = X ∩𝕀̃ is closed in X. Consider now a sequence (u_{k})_{k} in X converging to u = (x, y, z) with for all k, u_{k} ∈ X_{s_{k}} for some s_{k} ≠ ∅. Then up to the extraction of some subsequence we may suppose that:

– either there exists s ∈ ω^{<ω} such that s_{k} = s for all k, and then u ∈ X_{s};

– or there exists s ∈ ω^{<ω} such for all k, s_{k} ⊃ s^{⌢}⟨n_{k}⟩ with lim_{k→∞} n_{k} = ∞, and then by conditions (iv) and (v), u ∈ Σ̂_{s} ∪ Σ̃_{s}, hence by property (vi) either u ∉ X or u = a_{s};

– or there exists an infinite sequence σ ∈ ω^{ω} such for all k, s_{k} ⊃ σ_{|ℓ_{k}} with lim_{k→∞} ℓ_{k} = ∞. It follows then from property (i) that u = lim u_{k} = (χ(σ), 0, 0) ∈ P̃.

Hence if \overline{X} denotes the closure of X in 𝕀^{3} then

\overline{X} \ X = ⋃_{s}(Σ̃_{s} ∪ Σ̂_{s} \ \{a_{s}\})

which is a countable union of compact sets, hence X is a Π_{2}^{0} subset of 𝕀^{3}. □

c) \underline{Construction of the reduction mappings}:

We now fix an arbitrary regular Souslin system (A_{s})_{s∈ω^{<ω}} of Π_{1}^{1}-sets and let A = A((A_{s})_{s∈ω^{<ω}}) the result of the Souslin operation on this system. Since ℙ_{1} is Π_{1}^{1}-complete we can fix for all σ ∈ ω^{<ω}, a continuous mapping ψ_{s} : 2^{ω} → K^{∗}(Δ) which reduces A_{s} to ℙ_{1}, and set φ_{s} = Ψ_{s}∘ψ_{s} for s ≠ ∅. So by Lemma 6.5, φ_{s} : 2^{ω} → F^{∗}(X_{s}) is a Borel mapping satisfying for all α:

a) φ_{s}(α) is locally connected and for any convex set W containing a_{s} the set W ∪ Σ_{s}^{+} ∩ φ_{s}(α) is connected (and similarly for a_{s^{∗}} and Σ_{s}^{−}).

b) if α ∉ A_{s} then φ_{s}(α) ∈ C(X_{s}).

c) if α ∈ A_{s} then a_{s} and a_{s}^{∗} are separated by a clopen set, hence φ_{s}(α) ∉ C(X_{s}). We finally define for all α ∈ 2^{ω}

Φ(α) = P̃ ∪ ⋃_{{s∈ω^{<ω} \atop s≠∅}} φ_{s}(α) .

Lemma 6.7. Φ(α) ∈ F^{∗}(X).

Proof. Consider as in the proof of Lemma 6.6 a sequence (u_{k})_{k} in Φ(α) converging to some u ∈ X. Again since P̃ is closed in Φ(α) we may suppose that for all k, u_{k} ∈ φ_{s_{k}}(α) for some s_{k} ≠ ∅. Then repeating the arguments of Lemma 6.6 we are led to three alternatives. In the first alternative we are reduced to the case where s_{k} = s for all k, and then u_{k} ∈ φ_{s}(α) for all k, hence u ∈ φ_{s}(α) ⊂ Φ(α). In the two other alternatives we have u ∈ (Σ̃_{s} ∪ Σ̂_{s}) ∩ X hence u = a_{s}, or u ∈ P̃, and in both cases u ∈ Φ(α). □

Lemma 6.8. Φ(α) is locally connected at any element of Φ(α) \ P̃ .

Proof. Observe that by property (iv) the set F := {a_{s} : s ∈ ω^{<ω}} is a closed subset of Φ(α), and it follows from properties (iv), (v) and (vi) that for all s in S^{∗} = ω^{<ω} \ {∅} the set Σ_{s} \ {a_{s}, a_{s^{∗}}} is open in X hence U_{s} := φ_{s}(α) \ {a_{s}, a_{s^{∗}}} is an open subset of Φ(α), and by the very choice of φ_{s} the set U_{s} is locally connected. It follows that U := Φ(α) \ (P̃ ∪ F) = ⋃_{s∈S^{∗}} U_{s} is a locally connected open subset of Φ(α). So Φ(α) is locally connected at any element of U.

If u = a_{s} ∈ F for some s ∈ ω^{<ω} then the set

S = {r ∈ ω^{<ω} : r ≠ s and ν(r) ≤ 1 + ν(s)}

is finite and u ∉ Σ̂_{r} for all r ∈ S. It follows that the open neighbourhood V = \{(x, y, z) ∈ 𝕀^{2} × ℝ : y > \frac{ε_{s}}{2} \} of u meets only finitely many segments [a_{r}, â_{r}]

with r ∈ S and that u does not belong to F_{u} = \overline{⋃\{Σ_{r} : r^{∗} ∈ S, r ≠ s}\}. So one can find a convex neighbourhood W ⊂ V of u such that W ∩ F_{u} = ∅ and that

(x, y, z) ∈ W ⟹ \frac{3ε_{s}}{4} < y < \frac{5ε_{s}}{4} whence Σ_{t} ∩ W ≠ ∅ ⟹ u ∈ Σ_{t}, that is t = s or

t^{∗} = s and moreover Ψ_{s}^{−1}(W) ⊂ Σ^{+} and Ψ_{s^{⌢}⟨n⟩}^{−1}(W) ⊂ Σ^{−}.

Hence W ∩ Φ(α) = ⋃{W ∩ φ_{t}(α) : t = s or t^{∗} = s} and it follows from property a) stated above that W ∩ φ_{t}(α) is connected and contains u for such a t, which proves that W ∩ Φ(α) is connected, hence that Φ(α) is locally connected at u. □

Lemma 6.9. If α ∉ A then Φ(α) ∈ C_{loc}(X).

Proof. By Lemma 6.8 we only need to prove that Φ(α) is locally connected at any element of P̃. So let ε > 0 and u ∈ P̃; note that u = (χ(σ), 0, 0) for some unique σ ∈ ω^{ω}. Since α ∉ A there exists some s ≺ σ such that α ∉ A_{s} and if |s| is big enough then diam (X ∩ N_{s}) < ε.

Then for all t ∈ ω^{<ω} such that s ≺ t^{∗}, α ∉ A_{t} hence φ_{t}(α) is connected. It follows by induction that for all p ≥ |s| the set ⋃_{s⪯t,|t|≤p} φ_{t}(α) is connected, hence

Φ(α) ∩ N_{s} = \overline{⋃_{s⪯t} φ_{t}(α)}^{X} is connected. □

Lemma 6.10. If α ∈ A then Φ(α) ∉ C_{loc}(X).

Proof. Fix σ ∈ ω^{ω} such that α ∈ A_{s} for all s ≺ σ. We shall prove that the element u = (χ(σ), 0, 0) ∈ P̃ has no connected neighbourhood in Φ(α).

So let V be any neighbourhood of u in Φ(α). We can find t ≺ σ such that N_{t} ⊂ V hence φ_{t}(α) ⊂ V. Since α ∈ A_{t} there exists a clopen partition (W, W^{∗}) in φ_{t}(α) such that a_{t} ∈ W and a_{t}^{∗} ∈ W^{∗}. Then setting U = W ∪ \overline{⋃_{s≻t} φ_{s}(α)} and U^{∗} = W^{∗} ∪ \overline{⋃_{s⪰̸t} φ_{s}(α)} we have U ∩ U^{∗} ∩ P̃ = ∅ hence (U, U^{∗}) is a partition of Φ(α) into two relatively closed subsets, with a_{t} ∈ U ∩ V and a_{t}^{∗} ∈ U^{∗} ∩ V, which proves that V is not connected. □

Observe that for any open set V, Φ^{−1}(V^{+}) = ⋃_{s∈ω^{<ω}} φ_{s}^{−1}(V^{+}) and since each mapping φ_{s} is Borel then the mapping Φ is Borel too. Also by Lemma 6.9 and Lemma 6.10 Φ reduces the set 2^{ω} \ A to the set C_{loc}(X). Hence by Theorem 2.1 the set C_{loc}(X) is Ǎ(Π_{1}^{1})-complete. □

7. Connectedness of open subsets

Our goal here is to study the complexity of the set of all open connected subsets of a given space X that we identify to the set

Č(X) = {F ∈ F(X) : X \ F is connected }.

Proposition 7.1. For any Polish space X there exists a Borel mapping δ : F(X) → F(X × ℝ) which reduces the set Č(X) to the set C(X × ℝ).

Proof. Set J =]0, +∞[ ≈ ℝ and fix a compatible distance d < 1 on X. For all F ∈ F^{∗}(X) let δ(F) be the graph of the the restriction of the function d(·, F) to U = X \ F, so:

δ(F) = {(x, t) ∈ X × J : d(x, F) = t}

and set δ(∅) = X × {1}. Then for all F ∈ F(X), δ(F) is a closed subset of X × J, which is homeomorphic to U, hence the mapping δ : F(X) → F^{∗}(X × J) reduces the set Č(X) to the set C(X × J).

To see that δ is Borel, fix a countable basis (U_{i})_{i∈ω} of open sets in X. Since X is Polish we can also fix a sequence (c_{k})_{k∈ω} of Borel functions c_{k} : F^{∗}(X) → X such that for all F ∈ F^{∗}(X), c_{k}(F) ∈ F and the set {c_{k}(F) : k ∈ ω} is dense in F (see Section 3). In particular if a_{j} = c_{j}(X) then {a_{j} : j ∈ ω} is dense in X. Then for any basic open set V ×]s, t[ in X × J and F ≠ ∅, δ(F) ∩ (V ×]s, t[) ≠ ∅ if and only if:

∃(n, i, j) : a_{j} ∈ U_{i} ⊂ V, U_{i} ∩ F = ∅ and \{ \begin{matrix}∀k, d(a_{j}, c_{k}(F)) ≥ s + 2^{−n} \\ ∃k, d(a_{j}, c_{k}(F)) < t \end{matrix}

which is a Borel condition on F. □

Let X be a Polish space and Γ be any projective class. Then by Proposition 7.1 if C(X×ℝ) is in Γ then Č(X) is in Γ, and if Č(X) is Γ-hard then C(X×ℝ) is Γ-hard. Note however that the converse is false: if X = ℝ^{ω} ≈ X ×ℝ then since any Polish space can be embedded as a closed subset in ℝ^{ω}, it follows from Theorem 4.2 that the set C(ℝ^{ω}) is Π_{2}^{1}-complete while as we shall see in Proposition 7.5 the set Č(ℝ^{ω}) is Borel.

We recall that if the space X is compact then C(X) is a compact subset of K(X).

Proposition 7.2. If X is a compact space then Č(X) is a Π_{1}^{1} subset of K(X).

Proof. As in the proof of Proposition 4.1 we can write dually:

F ∉ Č(X) ⟺ ∃ F_{0}, F_{1} ∈ F(X) \ {∅, F } : F_{0} ∪ F_{1} = X and F ⊃ F_{0} ∩ F_{1}

and since X is compact then the relation “F_{0} ∪ F_{1} = X ” is compact and the relation “F′ ≠ F ” is open hence σ-compact, while the relation “F ⊃ F_{0} ∩ F_{1}” is G_{δ} and it follows that the right hand side defines a Σ_{1}^{1} set hence Č(X) is Π_{1}^{1}. □

Theorem 7.3. There exists a compact (connected) space X ⊂ 𝕀^{2} for which the set Č(X) is Π_{1}^{1}-complete.

Proof. Fix an enumeration (q_{n})_{n≥1} of ℚ and consider the Kuratowski-Sierpinski function f : ℙ → 𝕀 (see [6]) defined by

f(x) = ∑_{n=1}^{∞} 2^{−n} sin(\frac{1}{x − q_{n}}).

It is clear that the function f is continuous on ℙ but has no continuous extension to 𝕀. More precisely for all q ∈ ℚ the set of all cluster values of f at q is a non trivial interval J_{q}. Note that the graph G := Gr(f) ≈ ℙ is not connected. However as shown in [6] (see also [2]) if f̃ : 𝕀 → 𝕀 is any extension of f with f̃(q) ∈ J_{q} for all q ∈ ℚ then Gr(f̃) is connected. In particular the closure of G in 𝕀^{2}:

X := \overline{G} = G ∪ ⋃_{q∈ℚ} J_{q}

is a compact connected space. It follows that if G ⊂ H ⊂ \overline{G} and for all q ∈ ℚ, H(q) ≠ ∅ then H is connected.

We also need the following result from [2]. (Observe that the set \overline{F} of Lemma 9.7 in [2] is the graph of a function g with the properties stated in Lemma 7.4.)

Lemma 7.4. There exists a perfect zero-dimensional compact set Δ ⊂ 𝕀 such that Q = Δ ∩ ℚ is dense in Δ and f_{|Δ∩ℙ} admits a continuous extension g : Δ → 𝕀.

So let (Q, Δ) as in Lemma 7.4 and consider the mapping Φ : K^{∗}(Δ) → K(X) defined by

Φ(K) = Gr(g_{|K}).

Then Φ is clearly continuous and we now check that Φ reduces the Π_{1}^{1}-complete set ℙ_{1} = ℙ_{1}(Q, Δ) to Č(X). So let K ∈ K^{∗}(Δ) and set U = X \ Φ(K):

– If K ∈ ℙ_{1} then K ⊂ ℚ; then for all q ∈ ℚ, U(q) = J_{q} \ {g(q)} if q ∈ K, and U(q) = J_{q} if not. Hence for all q ∈ ℚ, U(q) ≠ ∅ so U is connected, that is Φ(K) ∈ Č(X).

– If K ∉ ℙ_{1} then K contains some irrational number α and then \overline{G}(α) = {f(α)} = {g(α)} hence U(α) = ∅ and we can write U = (U ∩ [0, α[×𝕀) ∪ (U ∩]α, 1] × 𝕀) which proves that U is not connected, that is Φ(K) ∉ Č(X). □

We recall (Proposition 4.1) that if the space X is Polish then C(X) is a Π_{2}^{1} subset of F(X).

Proposition 7.5. a) If X is a Polish space then Č(X) is a Π_{2}^{1} subset of F(X).

b) If moreover X is locally connected then Č(X) is a Δ_{1}^{1} subset of F(X), more precisely a Π_{2}^{0} subset for any admissible topology on F(X).

Proof. a) Again for F ∈ F(X) we have:

F ∉ Č(X) ⟺ ∃ F_{0}, F_{1} ∈ F(X) \ {∅, F } : F_{0} ∪ F_{1} = X and F ⊃ F_{0} ∩ F_{1}

and by the same arguments than in as in the proof of Proposition 4.1 one can check that the right hand side defines a Σ_{2}^{1} set hence Č(X) is a Π_{2}^{1} set.

b) Suppose that X is locally connected. Fix a countable basis (U_{i})_{i∈ω} of open sets in X. Let U be any open subset of X, since U is also a Polish locally connected space, then U is connected if and only if it is pathwise connected. Hence for any F ∈ F(X), F ∈ Č(X) if and only if:

(⋆) \{ \begin{matrix}∀i, j, F ∩ U_{i} ≠ ∅ or F ∩ U_{j} ≠ ∅ or ∃K ⊂ X such that: K ∩ F = ∅, \\ K ∩ U_{i} ≠ ∅, K ∩ U_{j} ≠ ∅, and K is compact and connected \end{matrix}

Note that if U ⊂ X is open and K ⊂ X is compact then for any admissible topology on F(X) the sets

U^{+} = {F ∈ F(X) : F ∩ U ≠ ∅} and K^{−} = {F ∈ F(X) : F ∩ K = ∅}

are open subsets of F(X). Hence if E_{i,j} denotes the set of all compact subsets of X satisfying the conditions in the second line of (⋆) then

Č(X) = ⋂_{i,j}(U_{i}^{+} ∪ U_{j}^{+} ∪ ⋃_{K∈E_{i,j}} K^{−}).

Hence Č(X) is a Π_{2}^{0} subset of F(X). □

Theorem 7.6. There exists a Polish space X ⊂ 𝕀^{3} for which the set Č(X) is Π_{2}^{1}-complete.

The proof is in some sense a dual version of the proof of Theorem 4.2 and for this we need the following result which can be considered as a kind of dual version of Lemma 3.5 for n = 2.

Lemma 7.7. There exists a Π_{2}^{0} set G ⊂ Δ × Δ and a continuous mapping α ↦ K_{α} from Δ to K^{∗}(Δ) such that the set

Q_{2} = {α ∈ Δ : ∃γ ∈ K_{α}, G(γ) = {0}}

is Σ_{2}^{1}-complete.

Proof. For simplicity we shall assume that Δ = 2^{ω} and fix a homeomorphism ı : (α, β) ↦ ⟨α, β⟩ from 2^{ω}×2^{ω} onto 2^{ω}. Let Q = {ε ∈ 2^{ω} : ∃n, ∀m ≥ n, ε(m) = 0} and P = 2^{ω} \ Q ≈ ω^{ω}; we denote by 0 the element of Q such that 0(n) = 0 for all n.

Let A be any Σ_{2}^{1} subset of 2^{ω}, fix a Π_{1}^{1} set B ⊂ 2^{ω} × 2^{ω} such that A is the projection of B on the first factor, and consider B̃ = ı(B) ⊂ 2^{ω}. Since the Σ_{1}^{1} set 2^{ω} \ B̃ is the projection of some closed subset of 2^{ω} × P we can find a compact set H of 2^{ω} × 2^{ω} such that 2^{ω} \ B̃ is the projection of H ∩ (2^{ω} × P).

Let

G = {(γ, ε) ∈ 2^{ω} × 2^{ω} : ε = 0 or (γ, ε) ∈ H ∩ (2^{ω} × P)}

and for all α ∈ 2^{ω},

K_{α} = ı({α} × 2^{ω}).

Then clearly G is a Π_{2}^{0} set and the mapping φ : α ↦ K_{α} is continuous from Δ to K^{∗}(Δ); and we now check that φ reduces A to Q_{2}.

If α ∈ A fix β such that (α, β) ∈ B; then setting γ = ı(α, β) we have γ ∈ K_{α} and since γ ∈ B̃ then there is no ε ∈ P such that (γ, ε) ∈ H hence G(γ) = {0}; so α ∈ Q_{2}.

Conversely if α ∈ Q_{2} fix γ ∈ K_{α} such that G(γ) = {0}; for all ε ∈ P, since ε ≠ 0, then (γ, ε) ∉ H ∩ (2^{ω} × P), hence γ ∈ B̃. So γ = ı(α, β) with (α, β) ∈ B, which proves that α ∈ A. □

\underline{Proof of Theorem 7.6.} Fix Δ ≈ 2^{ω} such that \{\frac{1}{4}, \frac{3}{4}\} ⊂ Δ ⊂ [\frac{1}{4}, \frac{3}{4}], G a Π_{2}^{0} subset of Δ × Δ, α ↦ K_{α} a continuous mapping from Δ to K^{∗}(Δ), and Q_{2} as in Lemma 7.7 and let

X = {(x, y, z) ∈ 𝕀^{3} : x ∉ Δ or ((x, y) ∈ G and z = 0)}.

which is clearly a Π_{2}^{0} subset of 𝕀^{3}. We then define mappings Ψ : Δ → K^{∗}(𝕀^{3}) and Φ : Δ → F(X) by:

Ψ(α) = {(x, y, z) ∈ 𝕀^{3} : z + d(x, K_{α}) ≤ d(x, Δ) + y}
Φ(α) = X ∩ Ψ(α) .

Lemma 7.8. Φ is Borel.

Proof. Following the proof and notation of Theorem 4.2 we apply Lemma 3.4. Again the mapping Ψ is u.s.c. and we now show that for all α, Φ(α) is dense in Ψ(α). Suppose that u = (x, y, z) ∈ U ∩ Ψ(K) with U = J_{x}^{ε} × J_{y}^{ε} × J_{z}^{ε} for some ε > 0. If x ∉ Δ then u ∈ U ∩ Ψ(K) and we are done. If not then z + d(x, K_{α}) ≤ y

and d(x, K_{α}) ≤ diam(Δ) = \frac{1}{2}, and we distinguish three cases:

– If z > 0 let δ < min(ε, z), pick x′ ∈ J_{x}^{δ} \ Δ and z − ε < z′ < z − δ then z′ ∈ J_{z}^{ε} and z′ + d(x′, K_{α}) ≤ z − δ + d(x, K_{α}) + δ ≤ y hence u′ = (x′, y, z′) ∈ U ∩ Φ(K).

– If y < 1 let δ = min(ε, 1−y), pick x′ ∈ J_{x}^{δ}\Δ and y+δ < y′ < y+ε then y′ ∈ J_{y}^{ε} and z +d(x′, K_{α}) ≤ z +d(x, K_{α})+δ ≤ y+δ ≤ y′ hence u′ = (x′, y′, z) ∈ U ∩Φ(K).

– If z = 0 and y = 1 let δ < min(ε, \frac{1}{2}) pick x′ ∈ J_{x}^{δ} \Δ then since d(x, K_{α}) ≤ \frac{1}{2}, z + d(x′, K_{α}) ≤ d(x, K_{α}) + δ ≤ 1 = y hence u′ = (x′, 1, 0) ∈ U ∩ Φ(K). □

Lemma 7.9. Φ reduces Q_{2} to F^{∗}(X) \ Č(X).

Proof. Let α ∈ Δ and set U = X \ Φ(α); then

U = ((G \ K_{α}) × {0}) ∪ {(x, y, z) ∈ 𝕀^{3} : x ∉ Δ and z + d(x, K_{α}) > d(x, Δ) + y}.

If α ∈ Q_{2} then by definition there exists some γ ∈ K_{α} such that G(γ) = {0}. It follows that U ∩ {(x, y, z) : x = γ} = ∅, and setting U_{0} = U ∩ {(x, y, z) : x < γ} and U_{1} = U ∩ {(x, y, z) : x > γ} the pair (U_{0}, U_{1}) is a clopen partition of U with v_{0} = (0, 0, 1) ∈ U_{0} and v_{1} = (1, 0, 1) ∈ U_{1}. Hence U is not connected.

Conversely if α ∉ Q_{2} one first proves in the lines of the proof of Theorem 4.2 that for any connected component J of 𝕀 \ Δ the set U_{J}(α) = U ∩ (J × 𝕀^{2}) is pathwise connected. Then given any partition clopen (V_{0}, V_{1}) of U one proves in the same lines that if v_{0} = (0, 0, 1) ∈ V_{0} then for all J, U_{J}(α) ⊂ V_{0}, which proves that U is connected. □

This finishes the proof of Theorem 7.6. □

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Gabriel Debs, Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France

Jean Saint Raymond, Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France