*Proof of Theorem 6.1.* Let $X$ be a compact Polish space. We prove that some copy of $X$ in $\mathcal{V}(Q)$ does not enumerate $A$. It implies the result, because any name of a copy of $X$ in $\mathcal{V}(Q)$ computes a presentation of $X$.

The space $\mathcal{I}(Q) = \{\phi : Q \to Q \text{ continuous one-to-one}\}$, with the topology induced by the metric $\rho(\phi, \psi) = \sup_x d(\phi(x), \psi(x))$ is Polish (Corollary 1.3.11 in [44]). For any compact set $K \subseteq Q$ and any non-c.e. set $A \subseteq \mathbb{N}$, we prove that the set $\{\phi \in \mathcal{I}(Q) : A \leq_e \phi(K)\}$ is meager in $\mathcal{I}(Q)$, which implies the existence of a copy $K'$ of $K$ such that $A \not\leq_e K'$ ($K'$ is seen as an element of the space $\mathcal{V}(Q)$). It is done by showing that for each Turing machine $M$, the set $\{\phi \in \mathcal{I}(Q) : M(\phi(K)) = A\}$ is nowhere dense in $\mathcal{I}(Q)$.

Let $\phi \in \mathcal{I}(Q)$ be such that $M(\phi(K)) = A$. For $\epsilon > 0$, we prove that there exists $\psi \in \mathcal{I}(Q)$ such that $\rho(\phi, \psi) < \epsilon$ and such that $\psi(K) \in F_{M,A}$ (the set $F_{M,A}$ was defined before Lemma 6.3). It implies the result, because for every $\psi'$ sufficiently close to $\psi$, one also has $\psi' \in F_{M,A}$ as $F_{M,A}$ is open.

Lemma 6.6 provides a computably overt set $A$ containing $\phi(K)$, in which the set of $\epsilon$-deformations of $\phi(K)$ is dense. By Lemma 6.3, $F_{M,A}$ intersects $A$. As $F_{M,A}$ is open, there exists an $\epsilon$-deformation of $\phi(K)$ in $F_{M,A}$. Let $f$ be the corresponding $\epsilon$-perturbation, and $\psi = f \circ \phi$. One has $\rho(\psi, \phi) < \epsilon$ and $\psi(K) \in F_{M,A}$. $\square$

# 7 Comparing compact and Polish presentations

Let $X$ be a compact Polish space. In the proofs we frequently used the fact that the jump of any Polish presentation of $X$ computes a compact presentation of $X$, which is stated precisely in Lemma 2.1. Here we investigate whether it can compute more. Of course, it always computes $0'$, and we show that if $X$ is perfect, then it does not compute more in general: every compact presentation of $X$, paired with $0'$, computes the jump of a Polish presentation of $X$. However, we will see after that it is no more true for non-perfect spaces and we give a counter-example.

**Theorem 7.1.** Let $X$ be a compact perfect Polish space. If $X$ has a $\mathbf{d}$-computable compact presentation, then there exists $\mathbf{e}$ such that $X$ has an $\mathbf{e}$-computable Polish presentation and $\mathbf{e}' \leq \mathbf{d} \oplus 0'$.

**Corollary 7.2.** Let $X$ be a compact perfect Polish space. One has

$$ \{\mathbf{e}' : \mathbf{e} \in \mathrm{Sp}(X)\} = \{\mathbf{d} \oplus 0' : \mathbf{d} \in \mathrm{Sp}_c(X)\}. $$

*Proof.* If $\mathbf{e} \in \mathrm{Sp}(X)$, then let $\mathbf{d} = \mathbf{e}'$. One has $\mathbf{d} \in \mathrm{Sp}_c(X)$ by Lemma 2.1, and $\mathbf{e}' = \mathbf{d} \oplus 0'$.

If $\mathbf{d} \in \mathrm{Sp}_c(X)$, then there exists $\mathbf{e}_0 \in \mathrm{Sp}(X)$ such that $\mathbf{e}'_0 \leq \mathbf{d} \oplus 0'$ by Theorem 7.1. Friedberg's jump inversion theorem relative to $\mathbf{e}_0$ states that if $\mathbf{a} \geq \mathbf{e}'_0$, then there exists $\mathbf{f}$ such that $\mathbf{a} = (\mathbf{e}_0 \oplus \mathbf{f})'$. We apply it to $\mathbf{a} = \mathbf{d} \oplus 0'$, and let $\mathbf{e} = \mathbf{e}_0 \oplus \mathbf{f}$. One has $\mathbf{d} \oplus 0' = \mathbf{e}'$ and $\mathbf{e} \in \mathrm{Sp}(X)$ as $\mathbf{e} \geq \mathbf{e}_0 \in \mathrm{Sp}(X)$. $\square$

**Reformulation.** Again, we use an overtness argument to reformulate the problem.
*Proof of Theorem 6.1.* Let $X$ be a compact Polish space. We prove that some copy of $X$ in $\mathcal{V}(Q)$ does not enumerate $A$. It implies the result, because any name of a copy of $X$ in $\mathcal{V}(Q)$ computes a presentation of $X$.

The space $\mathcal{I}(Q) = \{\phi : Q \to Q \text{ continuous one-to-one}\}$, with the topology induced by the metric $\rho(\phi, \psi) = \sup_x d(\phi(x), \psi(x))$ is Polish (Corollary 1.3.11 in [44]). For any compact set $K \subseteq Q$ and any non-c.e. set $A \subseteq \mathbb{N}$, we prove that the set $\{\phi \in \mathcal{I}(Q) : A \leq_e \phi(K)\}$ is meager in $\mathcal{I}(Q)$, which implies the existence of a copy $K'$ of $K$ such that $A \not\leq_e K'$ ($K'$ is seen as an element of the space $\mathcal{V}(Q)$). It is done by showing that for each Turing machine $M$, the set $\{\phi \in \mathcal{I}(Q) : M(\phi(K)) = A\}$ is nowhere dense in $\mathcal{I}(Q)$.

Let $\phi \in \mathcal{I}(Q)$ be such that $M(\phi(K)) = A$. For $\epsilon > 0$, we prove that there exists $\psi \in \mathcal{I}(Q)$ such that $\rho(\phi, \psi) < \epsilon$ and such that $\psi(K) \in F_{M,A}$ (the set $F_{M,A}$ was defined before Lemma 6.3). It implies the result, because for every $\psi'$ sufficiently close to $\psi$, one also has $\psi' \in F_{M,A}$ as $F_{M,A}$ is open.

Lemma 6.6 provides a computably overt set $A$ containing $\phi(K)$, in which the set of $\epsilon$-deformations of $\phi(K)$ is dense. By Lemma 6.3, $F_{M,A}$ intersects $A$. As $F_{M,A}$ is open, there exists an $\epsilon$-deformation of $\phi(K)$ in $F_{M,A}$. Let $f$ be the corresponding $\epsilon$-perturbation, and $\psi = f \circ \phi$. One has $\rho(\psi, \phi) < \epsilon$ and $\psi(K) \in F_{M,A}$. $\square$

# 7 Comparing compact and Polish presentations

Let $X$ be a compact Polish space. In the proofs we frequently used the fact that the jump of any Polish presentation of $X$ computes a compact presentation of $X$, which is stated precisely in Lemma 2.1. Here we investigate whether it can compute more. Of course, it always computes $0'$, and we show that if $X$ is perfect, then it does not compute more in general: every compact presentation of $X$, paired with $0'$, computes the jump of a Polish presentation of $X$. However, we will see after that it is no more true for non-perfect spaces and we give a counter-example.

**Theorem 7.1.** Let $X$ be a compact perfect Polish space. If $X$ has a $d$-computable compact presentation, then there exists $e$ such that $X$ has an $e$-computable Polish presentation and $e' \leq d \oplus 0'$.

**Corollary 7.2.** Let $X$ be a compact perfect Polish space. One has

$$ \{e' : e \in \mathrm{Sp}(X)\} = \{d \oplus 0' : d \in \mathrm{Sp}_c(X)\}. $$

*Proof.* If $e \in \mathrm{Sp}(X)$, then let $d = e'$. One has $d \in \mathrm{Sp}_c(X)$ by Lemma 2.1, and $e' = d \oplus 0'$.

If $d \in \mathrm{Sp}_c(X)$, then there exists $e_0 \in \mathrm{Sp}(X)$ such that $e'_0 \leq d \oplus 0'$ by Theorem 7.1. Friedberg's jump inversion theorem relative to $e_0$ states that if $a \geq e'_0$, then there exists $f$ such that $a = (e_0 \oplus f)'$. We apply it to $a = d \oplus 0'$, and let $e = e_0 \oplus f$. One has $d \oplus 0' = e'$ and $e \in \mathrm{Sp}(X)$ as $e \geq e_0 \in \mathrm{Sp}(X)$. $\square$

**Reformulation.** Again, we use an overtness argument to reformulate the problem.
As we have already seen, the degrees of Polish presentations of $X$ coincide with the degrees of copies of $X$ as elements of $\mathcal{V}(Q)$. In the same way, we show that the jumps of these degrees are exactly the jumps of the copies of $X$ in $\mathcal{V}(Q)$.

Again, we abstract away from $\mathcal{V}(Q)$ to make the results easier to read. Let $S$ be a countably-based space with a fixed index basis $(B_i)_{i \in \mathbb{N}}$ that is closed under finite intersections. Let $(U_i)_{i \in \mathbb{N}}$ be the canonical enumeration of the effective open subsets of $S$ defined by $U_i = \bigcup_{j \in W_i} B_j$, where $W_i$ is the $i$th c.e. subset of $\mathbb{N}$. We will apply the next result to $S = \mathcal{V}(Q)$.

**Definition 7.3.** The jump of $x \in S$ is the set $J(x) = \{i \in \mathbb{N} : x \in U_i\}$.

**Lemma 7.4.** If $x$ is a point of an effective countably-based space $S$, the Turing degree of $J(x)$ is the least element of $\{\mathbf{d}' : \mathbf{d} \text{ computes } x\}$.

*Proof.* Let $\delta_S : \mathbb{N}^\mathbb{N} \to S$ be the standard representation of $S$, mapping $p$ to $x$ if $\{i : \exists n, p(n) = i + 1\} = \{i : x \in B_i\}$, in which case we say that $p$ is a name of $x$. The function $\delta_S$ is computable and effectively open: the image of an effective open set is an effective open set, uniformly. Observe that $\mathbf{d}$ computes $x$ if and only if $\mathbf{d}$ computes some name of $x$.

As $\delta_S$ is computable, the preimages of effective open sets are effectively open, so if $p$ is a name of $x$ then $p'$ computes $J(x)$.

Conversely, we show that $J(x)$ computes $p'$ for some name $p$ of $x$. Let $(U_n)_{n \in \mathbb{N}}$ be the canonical enumeration of the effective open subsets of $\mathbb{N}^\mathbb{N}$. The set $\delta_S^{-1}(x)$ is $\Pi_2^0$ relative to $J(x)$: indeed, one has $p \in \delta_S^{-1}(x)$ iff $\forall i, [x \in B_i \iff \exists n, p(n) = i + 1]$, which is a $\Pi_2^0$ formula relative to $J(x)$. Let then $V_n$ be uniformly $J(x)$-effective open sets such that $\delta_S^{-1}(x) = \bigcap_n V_n$. Moreover, if $W \subseteq \mathbb{N}^\mathbb{N}$ is an effective open set, given by an index, then we can decide using $J(x)$ whether $W$ intersects $\delta_S^{-1}(x)$, because it is equivalent to $x \in \delta_S(W)$, which is an effective open set for which we have an index.

At stage $s$, we have produced a finite prefix $p_s$ of $p$, such that $[p_s]$ is contained in $V_{s-1}$ and intersects $\delta_S^{-1}(x)$. Using $J(x)$, we can decide whether $[p_s] \cap U_s$ intersects $\delta_S^{-1}(x)$. If it does, then we extend $p_s$ to $p_{s+1}$ so that $[p_{s+1}]$ is contained in $U_s \cap V_s$ and intersects $\delta_S^{-1}(x)$, and we declare that $p \in U_s$. If it does not, then we simply extend $p_s$ to $p_{s+1}$ so that $[p_{s+1}]$ is contained in $V_s$ and intersects $\delta_S^{-1}(x)$, and we declare that $p \notin U_s$.

In the limit, we obtain some $p \in \bigcap_s V_s$ so $p$ is a name of $x$. For each $s$, we have decided along the construction whether $p \in U_s$, so we have computed $p'$. $\square$

*Proof of Theorem 7.1.* Let $X \in \mathcal{K}(Q)$ be **d**-computably compact and perfect. Using **d** and **o'** as oracles, we progressively compute a copy $K$ of $X$, together with its jump $J(K)$, seeing $K$ as a point of $\mathcal{V}(Q)$. Let $(U_n)_{n \in \mathbb{N}}$ be an effective enumeration of the effective open subsets of $\mathcal{V}(Q)$. For each $n$, we need to decide as long as we build $K$, whether $K \in \mathcal{U}_n$.

We start from some $\epsilon_0 > 0$ and some basic $\epsilon_0/2$-ball $B_0$ in the Hausdorff metric, containing $X$. $B_0$ is a computably overt set in $\mathcal{V}(Q)$, so we can decide using $\mathbf{o'}$ whether it intersects $\mathcal{U}_0$. There are two cases:

(1) Assume that $B_0$ intersects $\mathcal{U}_0$. As in Lemma 6.6 there exists an $\epsilon_0$-perturbation $f_0$
As we have already seen, the degrees of Polish presentations of $X$ coincide with the degrees of copies of $X$ as elements of $\mathcal{V}(Q)$. In the same way, we show that the jumps of these degrees are exactly the jumps of the copies of $X$ in $\mathcal{V}(Q)$.

Again, we abstract away from $\mathcal{V}(Q)$ to make the results easier to read. Let $S$ be a countably-based space with a fixed index basis $(B_i)_{i \in \mathbb{N}}$ that is closed under finite intersections. Let $(U_i)_{i \in \mathbb{N}}$ be the canonical enumeration of the effective open subsets of $S$ defined by $U_i = \bigcup_{j \in W_i} B_j$, where $W_i$ is the $i$th c.e. subset of $\mathbb{N}$. We will apply the next result to $S = \mathcal{V}(Q)$.

**Definition 7.3.** The jump of $x \in S$ is the set $J(x) = \{i \in \mathbb{N} : x \in U_i\}$.

**Lemma 7.4.** If $x$ is a point of an effective countably-based space $S$, the Turing degree of $J(x)$ is the least element of $\{\mathbf{d}' : \mathbf{d} \text{ computes } x\}$.

*Proof.* Let $\delta_S : \mathbb{N}^\mathbb{N} \to S$ be the standard representation of $S$, mapping $p$ to $x$ if $\{i : \exists n, p(n) = i + 1\} = \{i : x \in B_i\}$, in which case we say that $p$ is a name of $x$. The function $\delta_S$ is computable and effectively open: the image of an effective open set is an effective open set, uniformly. Observe that $\mathbf{d}$ computes $x$ if and only if $\mathbf{d}$ computes some name of $x$.

As $\delta_S$ is computable, the preimages of effective open sets are effectively open, so if $p$ is a name of $x$ then $p'$ computes $J(x)$.

Conversely, we show that $J(x)$ computes $p'$ for some name $p$ of $x$. Let $(U_n)_{n \in \mathbb{N}}$ be the canonical enumeration of the effective open subsets of $\mathbb{N}^\mathbb{N}$. The set $\delta_S^{-1}(x)$ is $\Pi_2^0$ relative to $J(x)$: indeed, one has $p \in \delta_S^{-1}(x)$ iff $\forall i, [x \in B_i \iff \exists n, p(n) = i + 1]$, which is a $\Pi_2^0$ formula relative to $J(x)$. Let then $V_n$ be uniformly $J(x)$-effective open sets such that $\delta_S^{-1}(x) = \bigcap_n V_n$. Moreover, if $W \subseteq \mathbb{N}^\mathbb{N}$ is an effective open set, given by an index, then we can decide using $J(x)$ whether $W$ intersects $\delta_S^{-1}(x)$, because it is equivalent to $x \in \delta_S(W)$, which is an effective open set for which we have an index.

At stage $s$, we have produced a finite prefix $p_s$ of $p$, such that $[p_s]$ is contained in $V_{s-1}$ and intersects $\delta_S^{-1}(x)$. Using $J(x)$, we can decide whether $[p_s] \cap U_s$ intersects $\delta_S^{-1}(x)$. If it does, then we extend $p_s$ to $p_{s+1}$ so that $[p_{s+1}]$ is contained in $U_s \cap V_s$ and intersects $\delta_S^{-1}(x)$, and we declare that $p \in U_s$. If it does not, then we simply extend $p_s$ to $p_{s+1}$ so that $[p_{s+1}]$ is contained in $V_s$ and intersects $\delta_S^{-1}(x)$, and we declare that $p \notin U_s$.

In the limit, we obtain some $p \in \bigcap_s V_s$ so $p$ is a name of $x$. For each $s$, we have decided along the construction whether $p \in U_s$, so we have computed $p'$. $\square$

*Proof of Theorem 7.1.* Let $X \in \mathcal{K}(Q)$ be $\mathbf{d}$-computably compact and perfect. Using $\mathbf{d}$ and $\mathbf{0}'$ as oracles, we progressively compute a copy $K$ of $X$, together with its jump $J(K)$, seeing $K$ as a point of $\mathcal{V}(Q)$. Let $(U_n)_{n \in \mathbb{N}}$ be an effective enumeration of the effective open subsets of $\mathcal{V}(Q)$. For each $n$, we need to decide as long as we build $K$, whether $K \in \mathcal{U}_n$.

We start from some $\epsilon_0 > 0$ and some basic $\epsilon_0/2$-ball $B_0$ in the Hausdorff metric, containing $X$. $B_0$ is a computably overt set in $\mathcal{V}(Q)$, so we can decide using $\mathbf{0}'$ whether it intersects $\mathcal{U}_0$. There are two cases:

(1) Assume that $B_0$ intersects $\mathcal{U}_0$. As in Lemma 6.6 there exists an $\epsilon_0$-perturbation $f_0$
mapping $X$ to $X_0 \in \mathcal{U}_0$ (as $X$ is perfect, the computably overt set given by Lemma 6.6 can be replaced by the $\epsilon_0/2$-ball $\mathcal{B}_0$).

We now show that one can compute such an $f_0$. The space $\mathcal{P}_{\epsilon_0}$ of $\epsilon_0$-perturbations is a computable Polish space, the function $\Phi : \mathcal{P}_{\epsilon_0} \to \mathcal{V}(Q)$ mapping $f$ to $f(X)$ is $\mathbf{d}$-computable, so $\Phi^{-1}(\mathcal{U}_0)$ is a $\mathbf{d}$-effective open set, therefore one can $\mathbf{d}$-computably find some $f_0$ there.

We now pick a ball $\mathcal{B}_1$ of some radius $\epsilon_1$ (to be chosen, see below) around $X_0$, whose closure is contained in $\mathcal{B}_0 \cap \mathcal{U}_0$ and in which we are going to stay forever, so that in the limit, the copy $K$ of $X$ belongs to $\mathcal{U}_0$. We declare that $K \in \mathcal{U}_0$.

(2) Now assume that $\mathcal{B}_0$ does not intersect $\mathcal{U}_0$. We do nothing and proceed by picking a ball $\mathcal{B}_1$ of some radius $\epsilon_1$ around $X$ whose closure is contained in $\mathcal{B}_0$. In the sequel, we stay forever in this ball so in the limit, $K \in \mathcal{B}_0$ hence $K$ does not belong to $\mathcal{U}_0$. We declare that $K \notin \mathcal{U}_0$.

In both cases, we have decided whether the set $K$ belongs to $\mathcal{U}_0$. We now iterate this process with $\mathcal{U}_1, \mathcal{U}_2$, etc., choosing $\epsilon_1, \epsilon_2, \dots$ smaller and smaller so that the composition of the $\epsilon_i$-perturbations converges to an injective function (the Inductive Convergence Criterion [44, Theorem 1.6.2]) tells us that we can always choose the next $\epsilon_i$ sufficiently small to ensure that the limit is an injective, and moreover $\epsilon_i$ can be chosen in a computable way), and taking the closure of $\mathcal{B}_{n+1}$ contained in $\mathcal{B}_n$. In the limit, we have built a copy $K$ of $X$ and computed its jump, using $\mathbf{d}$ and $\mathbf{0}'$ as oracles. $\square$

The argument is uniform: there is a uniform procedure relative to $\mathbf{0}'$ that, given a compact presentation of any compact perfect Polish space $\mathcal{X}$, computes the jump of some Polish presentation of $\mathcal{X}$. We now observe that there cannot exist a uniform argument including non-perfect Polish spaces. Indeed, whether $X$ is not perfect is $\Sigma_2^0$ for a Polish presentation, so it is $\Sigma_1^0$ in its jump. If there was a uniform argument relative $\mathbf{0}'$, then being non-perfect would be $\Sigma_1^0(\mathbf{0}')$ for compact presentations, in particular it would be open in $\mathcal{K}(Q)$ endowed with the Hausdorff metric. However, the set of non-perfect compact sets is not open, as witnessed for instance by a sequence of segments shrinking to a singleton.

We actually show that Theorem 7.1 simply does not extend to non-perfect Polish spaces. Observe that a consequence of Theorem 7.1 is that if $X$ is a perfect Polish space with a $\mathbf{0}'$-computable compact presentation, then it has a low Polish presentation. We show that it fails for some non-perfect Polish space.

**Proposition 7.5.** There exists a (non-perfect) compact Polish space $\mathcal{X}$ with a $\mathbf{0}'$-computable compact presentation, but no low Polish presentation.

*Proof.* By Theorem 3.6 applied to $\mathbf{d} = \mathbf{0}'$, there is a space $\mathcal{X}$ having a $\mathbf{0}'$-computable compact presentation, but such that its Cantor-Bendixson derivative $\mathcal{X}'$ does not have a $\mathbf{0}''$-computable compact presentation. If $\mathcal{X}$ had a low Polish presentation then by Lemma 3.1, $\mathcal{X}'$ would have a $\mathbf{0}''$-computable compact presentation. $\square$

Therefore, for $\mathbf{d} = \mathbf{0}'$, this space $\mathcal{X}$ has a $\mathbf{d}$-computable compact presentation, but there is no $\mathbf{e}$ such that $X$ has an $\mathbf{e}$-computable Polish presentation and $\mathbf{e}' \le \mathbf{d} \oplus \mathbf{0}' = \mathbf{0}'$.
mapping $X$ to $X_0 \in \mathcal{U}_0$ (as $X$ is perfect, the computably overt set given by Lemma 6.6 can be replaced by the $\epsilon_0/2$-ball $\mathcal{B}_0$).

We now show that one can compute such an $f_0$. The space $\mathcal{P}_{\epsilon_0}$ of $\epsilon_0$-perturbations is a computable Polish space, the function $\Phi : \mathcal{P}_{\epsilon_0} \to \mathcal{V}(Q)$ mapping $f$ to $f(X)$ is $\mathbf{d}$-computable, so $\Phi^{-1}(\mathcal{U}_0)$ is a $\mathbf{d}$-effective open set, therefore one can $\mathbf{d}$-computably find some $f_0$ there.

We now pick a ball $\mathcal{B}_1$ of some radius $\epsilon_1$ (to be chosen, see below) around $X_0$, whose closure is contained in $\mathcal{B}_0 \cap \mathcal{U}_0$ and in which we are going to stay forever, so that in the limit, the copy $K$ of $X$ belongs to $\mathcal{U}_0$. We declare that $K \in \mathcal{U}_0$.

(2) Now assume that $\mathcal{B}_0$ does not intersect $\mathcal{U}_0$. We do nothing and proceed by picking a ball $\mathcal{B}_1$ of some radius $\epsilon_1$ around $X$ whose closure is contained in $\mathcal{B}_0$. In the sequel, we stay forever in this ball so in the limit, $K \in \mathcal{B}_0$ hence $K$ does not belong to $\mathcal{U}_0$. We declare that $K \notin \mathcal{U}_0$.

In both cases, we have decided whether the set $K$ belongs to $\mathcal{U}_0$. We now iterate this process with $\mathcal{U}_1, \mathcal{U}_2$, etc., choosing $\epsilon_1, \epsilon_2, \dots$ smaller and smaller so that the composition of the $\epsilon_i$-perturbations converges to an injective function (the Inductive Convergence Criterion [44, Theorem 1.6.2]) tells us that we can always choose the next $\epsilon_i$ sufficiently small to ensure that the limit is an injective, and moreover $\epsilon_i$ can be chosen in a computable way), and taking the closure of $\mathcal{B}_{n+1}$ contained in $\mathcal{B}_n$. In the limit, we have built a copy $K$ of $X$ and computed its jump, using $\mathbf{d}$ and $\mathbf{0}'$ as oracles. $\square$

The argument is uniform: there is a uniform procedure relative to $\mathbf{0}'$ that, given a compact presentation of any compact perfect Polish space $\mathcal{X}$, computes the jump of some Polish presentation of $\mathcal{X}$. We now observe that there cannot exist a uniform argument including non-perfect Polish spaces. Indeed, whether $X$ is not perfect is $\Sigma_2^0$ for a Polish presentation, so it is $\Sigma_1^0$ in its jump. If there was a uniform argument relative $\mathbf{0}'$, then being non-perfect would be $\Sigma_1^0(\mathbf{0}')$ for compact presentations, in particular it would be open in $\mathcal{K}(Q)$ endowed with the Hausdorff metric. However, the set of non-perfect compact sets is not open, as witnessed for instance by a sequence of segments shrinking to a singleton.

We actually show that Theorem 7.1 simply does not extend to non-perfect Polish spaces. Observe that a consequence of Theorem 7.1 is that if $X$ is a perfect Polish space with a $\mathbf{0}'$-computable compact presentation, then it has a low Polish presentation. We show that it fails for some non-perfect Polish space.

**Proposition 7.5.** *There exists a (non-perfect) compact Polish space $\mathcal{X}$ with a $\mathbf{0}'$-computable compact presentation, but no low Polish presentation.*

*Proof.* By Theorem 3.6 applied to $\mathbf{d} = \mathbf{0}'$, there is a space $\mathcal{X}$ having a $\mathbf{0}'$-computable compact presentation, but such that its Cantor-Bendixson derivative $\mathcal{X}'$ does not have a $\mathbf{0}''$-computable compact presentation. If $\mathcal{X}$ had a low Polish presentation then by Lemma 3.1, $\mathcal{X}'$ would have a $\mathbf{0}''$-computable compact presentation. $\square$

Therefore, for $\mathbf{d} = \mathbf{0}'$, this space $\mathcal{X}$ has a $\mathbf{d}$-computable compact presentation, but there is no $\mathbf{e}$ such that $X$ has an $\mathbf{e}$-computable Polish presentation and $\mathbf{e}' \le \mathbf{d} \oplus \mathbf{0}' = \mathbf{0}'$.
References

[1] Djamel Eddine Amir and Mathieu Hoyrup. Strong computable type. *Computability*, 12(3):227–269, 2023. 6

[2] C. J. Ash and J. Knight. *Computable structures and the hyperarithmetical hierarchy*, volume 144 of *Studies in Logic and the Foundations of Mathematics*. North-Holland Publishing Co., Amsterdam, 2000. 2, 5, 13

[3] Jeremy Avigad and Vasco Brattka. Computability and analysis: the legacy of Alan Turing. In *Turing's legacy: developments from Turing's ideas in logic*, volume 42 of *Lect. Notes Log.*, pages 1–47. Assoc. Symbol. Logic, La Jolla, CA, 2014. 5

[4] Vasco Brattka and Peter Hertling. Handbook of Computability and Complexity in Analysis. 202x. 2, 3, 5

[5] Vasco Brattka, Peter Hertling, and Klaus Weihrauch. A tutorial on computable analysis. In *New computational paradigms*, pages 425–491. Springer, New York, 2008. 2, 5

[6] Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, and Arno Pauly. Connected choice and the Brouwer fixed point theorem. *J. Math. Log.*, 19(1):1950004, 46, 2019.

[7] Vasco Brattka and Gero Presser. Computability on subsets of metric spaces. *Theoretical Computer Science*, 305(1-3):43–76, 2003. 8

[8] Douglas Cenzer. Π<sub>1</sub><sup>0</sup> classes in computability theory. In *Handbook of computability theory*, volume 140 of *Stud. Logic Found. Math.*, pages 37–85. North-Holland, Amsterdam, 1999. 13

[9] Joe Clanin, Timothy H. McNicholl, and Don M. Stull. Analytic computable structure theory and L<sup>p</sup> spaces. *Fund. Math.*, 244(3):255–285, 2019. 3

[10] Rodney G. Downey and Alexander G. Melnikov. Computably compact metric spaces. *Bulletin of Symbolic Logic*, pages 1–59, 2023. 3, 4, 6, 8, 15, 16, 17, 22

[11] Samuel Eilenberg and Norman Steenrod. *Foundations of Algebraic Topology*. Princeton University Press, Princeton, 1952. 19

[12] R. Engelking. *Dimension theory*. Mathematical Studies. North-Holland Publishing Company, 1978. 24

[13] Lawrence Feiner. Hierarchies of Boolean algebras. *J. Symbolic Logic*, 35:365–374, 1970. 11

[14] Ekaterina B. Fokina, Valentina Harizanov, and Alexander Melnikov. Computable model theory. In *Turing's legacy: developments from Turing's ideas in logic*, volume 42 of *Lect. Notes Log.*, pages 124–194. Assoc. Symbol. Logic, La Jolla, CA, 2014. 2, 4

[15] Noam Greenberg, Alexander Melnikov, Andre Nies, and Daniel Turetsky. Effectively closed subgroups of the infinite symmetric group. *Proceedings of the American Mathematical Society*, 146:5421–5435, 2018. 3
References

[1] Djamel Eddine Amir and Mathieu Hoyrup. Strong computable type. *Computability*, 12(3):227–269, 2023. 6

[2] C. J. Ash and J. Knight. *Computable structures and the hyperarithmetical hierarchy*, volume 144 of *Studies in Logic and the Foundations of Mathematics*. North-Holland Publishing Co., Amsterdam, 2000. 2, 5, 13

[3] Jeremy Avigad and Vasco Brattka. Computability and analysis: the legacy of Alan Turing. In *Turing's legacy: developments from Turing's ideas in logic*, volume 42 of *Lect. Notes Log.*, pages 1–47. Assoc. Symbol. Logic, La Jolla, CA, 2014. 5

[4] Vasco Brattka and Peter Hertling. Handbook of Computability and Complexity in Analysis. 202x. 2, 3, 5

[5] Vasco Brattka, Peter Hertling, and Klaus Weihrauch. A tutorial on computable analysis. In *New computational paradigms*, pages 425–491. Springer, New York, 2008. 2, 5

[6] Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, and Arno Pauly. Connected choice and the Brouwer fixed point theorem. *J. Math. Log.*, 19(1):1950004, 46, 2019.

[7] Vasco Brattka and Gero Presser. Computability on subsets of metric spaces. *Theoretical Computer Science*, 305(1-3):43–76, 2003. 8

[8] Douglas Cenzer. Π<sub>1</sub><sup>0</sup> classes in computability theory. In *Handbook of computability theory*, volume 140 of *Stud. Logic Found. Math.*, pages 37–85. North-Holland, Amsterdam, 1999. 13

[9] Joe Clanin, Timothy H. McNicholl, and Don M. Stull. Analytic computable structure theory and L<sup>p</sup> spaces. *Fund. Math.*, 244(3):255–285, 2019. 3

[10] Rodney G. Downey and Alexander G. Melnikov. Computably compact metric spaces. *Bulletin of Symbolic Logic*, pages 1–59, 2023. 3, 4, 6, 8, 15, 16, 17, 22

[11] Samuel Eilenberg and Norman Steenrod. *Foundations of Algebraic Topology*. Princeton University Press, Princeton, 1952. 19

[12] R. Engelking. *Dimension theory*. Mathematical Studies. North-Holland Publishing Company, 1978. 24

[13] Lawrence Feiner. Hierarchies of Boolean algebras. *J. Symbolic Logic*, 35:365–374, 1970. 11

[14] Ekaterina B. Fokina, Valentina Harizanov, and Alexander Melnikov. Computable model theory. In *Turing's legacy: developments from Turing's ideas in logic*, volume 42 of *Lect. Notes Log.*, pages 124–194. Assoc. Symbol. Logic, La Jolla, CA, 2014. 2, 4

[15] Noam Greenberg, Alexander Melnikov, Andre Nies, and Daniel Turetsky. Effectively closed subgroups of the infinite symmetric group. *Proceedings of the American Mathematical Society*, 146:5421–5435, 2018. 3
[16] Noam Greenberg and Antonio Montalbán. Ranked structures and arithmetic trans-finite recursion. *Transactions of the American Mathematical Society*, 360(3):1265–1307, 2008. **3, 10, 13**

[17] Vassilios Gregoriades, Tamás Kispéter, and Arno Pauly. A comparison of concepts from computable analysis and effective descriptive set theory. *Math. Structures Comput. Sci.*, 27(8):1414–1436, 2017. **5**

[18] Matthew Harrison-Trainor, Alexander Melnikov, and Keng Meng Ng. Computability of Polish spaces up to homeomorphism. *The Journal of Symbolic Logic*, 85(4):1664–1686, 2020. **3, 11, 12**

[19] Allen Hatcher. *Algebraic Topology*. Algebraic Topology. Cambridge University Press, 2002. **20**

[20] Denis R. Hirschfeldt, Bakhadyr Khoussainov, Richard A. Shore, and Arkadii M. Slinko. Degree spectra and computable dimensions in algebraic structures. *Ann. Pure Appl. Logic*, 115(1-3):71–113, 2002. **2**

[21] Mathieu Hoyrup, Takayuki Kihara, and Victor Selivanov. Degree spectra of homeomorphism types of polish spaces. arXiv preprint arXiv:2004.06872, 2020.

[22] Mathieu Hoyrup, Takayuki Kihara, and Victor Selivanov. Degrees of non-computability of homeomorphism types of polish spaces. In Marcella Anselmo, Gi-anluca Della Vedova, Florin Manea, and Arno Pauly, editors, *Beyond the Horizon of Computability*, pages 189–192, Cham, 2020. Springer International Publishing.

[23] Yu. L. Ershov, S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek, editors. *Handbook of recursive mathematics. Vol. 2*, volume 139 of *Studies in Logic and the Foundations of Mathematics*. North-Holland, Amsterdam, 1998. Recursive algebra, analysis and combinatorics. **2**

[24] Witold Hurewicz and Henry Wallman. *Dimension Theory (PMS-4)*. Princeton University Press, 1941. **5, 13, 14, 15, 17**

[25] Zvonko Iljazovic. Isometries and computability structures. *J. Univers. Comput. Sci.*, 16(18):2569–2596, 2010. **8**

[26] Carl G. Jockusch, Jr. and Robert I. Soare. Boolean algebras, Stone spaces, and the iterated Turing jump. *J. Symbolic Logic*, 59(4):1121–1138, 1994. **4**

[27] Hiroyasu Kamo. Effective Dini's theorem on effectively compact metric spaces. *Electronic Notes in Theoretical Computer Science*, 120:73–82, 2005. Proceedings of the 6th Workshop on Computability and Complexity in Analysis (CCA 2004). **8**

[28] Alexander S. Kechris. *Classical Descriptive Set Theory*. Springer, January 1995. **17**

[29] Julia F. Knight and Michael Stob. Computable Boolean algebras. *J. Symbolic Logic*, 65(4):1605–1623, 2000. **3**

[30] Peter la Roche. Effective galois theory. *The Journal of Symbolic Logic*, 46(2):385–392, 1981. **15**
[16] Noam Greenberg and Antonio Montalbán. Ranked structures and arithmetic trans-finite recursion. *Transactions of the American Mathematical Society*, 360(3):1265–1307, 2008. **3, 10, 13**

[17] Vassilios Gregoriades, Tamás Kispéter, and Arno Pauly. A comparison of concepts from computable analysis and effective descriptive set theory. *Math. Structures Comput. Sci.*, 27(8):1414–1436, 2017. **5**

[18] Matthew Harrison-Trainor, Alexander Melnikov, and Keng Meng Ng. Computability of Polish spaces up to homeomorphism. *The Journal of Symbolic Logic*, 85(4):1664–1686, 2020. **3, 11, 12**

[19] Allen Hatcher. *Algebraic Topology*. Algebraic Topology. Cambridge University Press, 2002. **20**

[20] Denis R. Hirschfeldt, Bakhadyr Khoussainov, Richard A. Shore, and Arkadii M. Slinko. Degree spectra and computable dimensions in algebraic structures. *Ann. Pure Appl. Logic*, 115(1-3):71–113, 2002. **2**

[21] Mathieu Hoyrup, Takayuki Kihara, and Victor Selivanov. Degree spectra of homeomorphism types of polish spaces. arXiv preprint arXiv:2004.06872, 2020.

[22] Mathieu Hoyrup, Takayuki Kihara, and Victor Selivanov. Degrees of non-computability of homeomorphism types of polish spaces. In Marcella Anselmo, Gi-anluca Della Vedova, Florin Manea, and Arno Pauly, editors, *Beyond the Horizon of Computability*, pages 189–192, Cham, 2020. Springer International Publishing.

[23] Yu. L. Ershov, S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek, editors. *Handbook of recursive mathematics. Vol. 2*, volume 139 of *Studies in Logic and the Foundations of Mathematics*. North-Holland, Amsterdam, 1998. Recursive algebra, analysis and combinatorics. **2**

[24] Witold Hurewicz and Henry Wallman. *Dimension Theory (PMS-4)*. Princeton University Press, 1941. **5, 13, 14, 15, 17**

[25] Zvonko Iljazovic. Isometries and computability structures. *J. Univers. Comput. Sci.*, 16(18):2569–2596, 2010. **8**

[26] Carl G. Jockusch, Jr. and Robert I. Soare. Boolean algebras, Stone spaces, and the iterated Turing jump. *J. Symbolic Logic*, 59(4):1121–1138, 1994. **4**

[27] Hiroyasu Kamo. Effective Dini's theorem on effectively compact metric spaces. *Electronic Notes in Theoretical Computer Science*, 120:73–82, 2005. Proceedings of the 6th Workshop on Computability and Complexity in Analysis (CCA 2004). **8**

[28] Alexander S. Kechris. *Classical Descriptive Set Theory*. Springer, January 1995. **17**

[29] Julia F. Knight and Michael Stob. Computable Boolean algebras. *J. Symbolic Logic*, 65(4):1605–1623, 2000. **3**

[30] Peter la Roche. Effective galois theory. *The Journal of Symbolic Logic*, 46(2):385–392, 1981. **15**
[31] Martino Lupini, Alexander Melnikov, and Andre Nies. Computable topological abelian groups. *Journal of Algebra*, 615:278–327, 2023. 3, 4, 15, 16, 17, 22, 23

[32] Timothy H. McNicholl and Donald M. Stull. The isometry degree of a computable copy of $\ell^{p1}$. *Computability*, 8(2):179–189, 2019. 3

[33] Alexander G. Melnikov. Computably isometric spaces. *J. Symbolic Logic*, 78(4):1055–1085, 2013. 3

[34] A. G. Melnikov. New degree spectra of Polish spaces. *Siberian Mathematical Journal*, 62(5):882–894, 2021. 3, 4, 17, 19, 23

[35] Alexander Melnikov. Computable topological groups and Pontryagin duality. *Transactions of the American Mathematical Society*, 370(12):8709–8737, 2018. 3

[36] Y. N. Moschovakis. Recursive metric spaces. *Fund. Math.*, 55:215–238, 1964. 3

[37] Yiannis N. Moschovakis. *Descriptive set theory*, volume 155 of *Mathematical Surveys and Monographs*. American Mathematical Society, Providence, RI, second edition, 2009. 3

[38] S. P. Odintsov and V. L. Selivanov. The arithmetical hierarchy and ideals of enumerated Boolean algebras. *Sibirsk. Mat. Zh.*, 30(6):140–149, 1989. 11, 12

[39] Arno Pauly, Dongseong Seon, and Martin Ziegler. Computing haar measures. In Maribel Fernández and Anca Muscholl, editors, *28th EACSL Annual Conference on Computer Science Logic, CSL 2020, January 13-16, 2020, Barcelona, Spain*, volume 152 of *LIPIcs*, pages 34:1–34:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. 8

[40] Marian B. Pour-El and J. Ian Richards. *Computability in analysis and physics*. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1989. 3

[41] V. L. Selivanov. Algorithmic complexity of algebraic systems. *Mat. Zametki*, 44(6):823–832, 863, 1988. 11

[42] Victor Selivanov. On degree spectra of topological spaces. *Lobachevskii Journal of Mathematics*, 41 (2):252-259, 2020. 3

[43] Rick L. Smith. Effective aspects of profinite groups. *The Journal of Symbolic Logic*, 46(4):851–863, 1981. 15

[44] Jan van Mill. *The infinite-dimensional topology of function spaces*, volume 64 of *North-Holland Mathematical Library*. North-Holland Publishing Co., Amsterdam, 2001. 5, 28, 30, 32

[45] Mariko Yasugi, Takakazu Mori, and Yoshiki Tsujii. Effective properties of sets and functions in metric spaces with computability structure. *Theoretical Computer Science*, 219(1-2):467–486, 1999. 8

[46] Klaus Weihrauch. *Computable analysis*. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 2000. An introduction. 2, 3, 5
[31] Martino Lupini, Alexander Melnikov, and Andre Nies. Computable topological abelian groups. *Journal of Algebra*, 615:278–327, 2023. **3, 4, 15, 16, 17, 22, 23**

[32] Timothy H. McNicholl and Donald M. Stull. The isometry degree of a computable copy of $\ell^{p1}$. *Computability*, 8(2):179–189, 2019. **3**

[33] Alexander G. Melnikov. Computably isometric spaces. *J. Symbolic Logic*, 78(4):1055–1085, 2013. **3**

[34] A. G. Melnikov. New degree spectra of Polish spaces. *Siberian Mathematical Journal*, 62(5):882–894, 2021. **3, 4, 17, 19, 23**

[35] Alexander Melnikov. Computable topological groups and Pontryagin duality. *Transactions of the American Mathematical Society*, 370(12):8709–8737, 2018. **3**

[36] Y. N. Moschovakis. Recursive metric spaces. *Fund. Math.*, 55:215–238, 1964. **3**

[37] Yiannis N. Moschovakis. *Descriptive set theory*, volume 155 of *Mathematical Surveys and Monographs*. American Mathematical Society, Providence, RI, second edition, 2009. **3**

[38] S. P. Odintsov and V. L. Selivanov. The arithmetical hierarchy and ideals of enumerated Boolean algebras. *Sibirsk. Mat. Zh.*, 30(6):140–149, 1989. **11, 12**

[39] Arno Pauly, Dongseong Seon, and Martin Ziegler. Computing haar measures. In Maribel Fernández and Anca Muscholl, editors, *28th EACSL Annual Conference on Computer Science Logic, CSL 2020, January 13-16, 2020, Barcelona, Spain*, volume 152 of *LIPIcs*, pages 34:1–34:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. **8**

[40] Marian B. Pour-El and J. Ian Richards. *Computability in analysis and physics*. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1989. **3**

[41] V. L. Selivanov. Algorithmic complexity of algebraic systems. *Mat. Zametki*, 44(6):823–832, 863, 1988. **11**

[42] Victor Selivanov. On degree spectra of topological spaces. *Lobachevskii Journal of Mathematics*, 41 (2):252-259, 2020. **3**

[43] Rick L. Smith. Effective aspects of profinite groups. *The Journal of Symbolic Logic*, 46(4):851–863, 1981. **15**

[44] Jan van Mill. *The infinite-dimensional topology of function spaces*, volume 64 of *North-Holland Mathematical Library*. North-Holland Publishing Co., Amsterdam, 2001. **5, 28, 30, 32**

[45] Mariko Yasugi, Takakazu Mori, and Yoshiki Tsujii. Effective properties of sets and functions in metric spaces with computability structure. *Theoretical Computer Science*, 219(1-2):467–486, 1999. **8**

[46] Klaus Weihrauch. *Computable analysis*. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 2000. An introduction. **2, 3, 5**