Example 2.2. In a second-countable $T_0$ space X, the equality relation is $\boldsymbol {\Pi }^0_2$ in $X^2$ :

$$ \begin{align*} x = y \iff \forall \text{ basic open } U\, (x \in U \iff y \in U). \end{align*} $$

Example 3.4. If we Morleyize the fragment $\mathcal {F}$ consisting of all atomic and negations of atomic formulas, we obtain a theory $\mathcal {T}'$ modulo which the $\Sigma _\alpha , \Pi _\alpha\ \mathcal {L}'$ -formulas, in our “non-Hausdorff” sense, are equivalently the $\Sigma _\alpha , \Pi _\alpha\ \mathcal {L}$ -formulas in the traditional sense.

Example 3.5. Suppose $\mathcal {L}$ is a countable propositional language, consisting only of nullary relation symbols. It is best to regard such a language as being $0$ -sorted, so that an “ $\mathcal {L}$ -structure” has no underlying set, and is simply a truth assignment $\mathcal {M} : \mathcal {L} \to 2$ . The $\Sigma _1$ formulas define the open sets of a topology, namely, the Sierpiński power topology on $\mathbb {S}^{\mathcal {L}}$ (also known as the Scott topology). The $\Sigma _\alpha , \Pi _\alpha $ formulas define precisely the $\boldsymbol {\Sigma }^0_\alpha , \boldsymbol {\Pi }^0_\alpha $ subsets of $\mathbb {S}^{\mathcal {L}}$ . Thus, quasi-Polish spaces are up to homeomorphism precisely the spaces of models of countable $\Pi _2$ propositional theories.

The Morleyization of the preceding remark corresponds in this case to changing topology on $\mathbb {S}^{\mathbb {N}}$ to turn it into $2^{\mathbb {N}}$ , as in (2.8). More generally, Morleyization of an arbitrary countable fragment $\mathcal {F}$ corresponds to changing topology to make countably many Borel sets open as in (2.7).

(In Definition 4.12, we relate Morleyization of non-propositional languages to change of topology.)

Example 4.10. For a countable propositional language $\mathcal {L}$ , regarded as a 0-sorted first-order language as in Example 3.5, an étale $\mathcal {L}$ -structure $p : \mathcal {M} \to X$ consists simply of an open set $P^{\mathcal {M}} \subseteq X$ for each 0-ary relation symbol $P \in \mathcal {L}$ , hence literally corresponds to a continuous map $X \to \mathbb {S}^{\mathcal {L}}$ (whose Pth coordinate is the indicator function of $P^{\mathcal {M}}$ ).

Example 6.1 (space of structures on $\mathbb {N}$ ).

Let $\mathcal {L}$ be a countable language,

$$ \begin{align*} X := \prod_{n\text{-ary fn } f \in \mathcal{L}} \mathbb{N}^{\mathbb{N}^n} \times \prod_{n\text{-ary rel } R \in \mathcal{L}} 2^{\mathbb{N}^n} \end{align*} $$

be the zero-dimensional Polish space of $\mathcal {L}$ -structures on $\mathbb {N}$ , with the product topology, and

$$ \begin{align*} M := X \times \mathbb{N} \end{align*} $$

equipped with the first projection $p : M \to X$ ; this is clearly étale. On each fiber $M_x = \{x\} \times \mathbb {N} \cong \mathbb {N}$ of M, we may put the structure $\mathcal {M}_x := (M_x,x)$ ; that is,

$$ \begin{align*} P^{\mathcal{M}}(x, a_0, \dotsc, a_{n-1}) &:= x(P)(a_0, \dotsc, a_{n-1}) \end{align*} $$

for each symbol (function or relation) $P \in \mathcal {L}$ , using here the “based” representation (4.4) of $M^n_X$ . This defines an étale $\mathcal {L}$ -structure $\mathcal {M}$ over X. The isomorphism groupoid is

where $S_\infty $ is the infinite symmetric group of all bijections $\mathbb {N} \cong \mathbb {N}$ , acting on X via pushforward of structure (the logic action). Up to topological groupoid isomorphism, we may forget about the second coordinate y above, yielding (the action groupoid of the logic action).

This $\mathcal {M}$ does not have $\Sigma _1$ saturations, however, but only $\Sigma _2$ saturations. The complement of the diagonal in $M^2_X$ is open and -invariant, but not $\Sigma _1$ -definable, since $\ne $ is not $\Sigma _1$ ; similarly for negations of any relation symbols in $\mathcal {L}$ . In fact, this is the only issue: by definition of the product topology on X, a basic open set in $M^n_X \cong X \times \mathbb {N}^n$ is of the form

for some fixed $\vec {a} \in \mathbb {N}^{n+m}$ and finite conjunction $\phi (y_n, \dotsc , y_{n+m-1})$ of atomic $\mathcal {L}$ -formulas and their negations; the -saturation of this set is easily seen to be defined by the $\Sigma _2$ formula

(This is the base case of the proof of the classical Lopez-Escobar theorem; see [Reference Kechris33, Proposition 16.9].)

We may thus Morleyize all negated atomic formulas, as in Example 3.4, to obtain an expanded language $\mathcal {L}' \supseteq \mathcal {L}$ and a $\Pi _2\ \mathcal {L}'$ -theory $\mathcal {T}'$ . Let $\mathcal {M}'$ be the unique expansion of $\mathcal {M}$ to an étale model of $\mathcal {T}'$ , by interpreting the newly added relation symbols as the complements of the atomic formulas interpreted in $\mathcal {M}$ (which are clopen). Then $\mathcal {M}'$ has $\Sigma _1$ saturations, namely, given by the Morleyized formulas $\psi '$ corresponding to the above $\psi $ (as in Definition 3.3). A $\Pi _2$ axiomatization of the fibers of $\mathcal {M}'$ is given by $\mathcal {T}'$ together with the “infinite models” axioms for each $n \in \mathbb {N}$

(6.2) $$ \begin{align} \exists x_0, \dotsc, x_{n-1} \bigwedge_{i \ne j} (x_i \ne x_j). \end{align} $$

If we started with a $\Pi _2\ \mathcal {L}$ -theory $\mathcal {T}$ which already has only infinite models, then we may restrict this $\mathcal {M}'$ to the models of $\mathcal {T}$ (as in Remark 5.15) to obtain a parametrization of the models of the Morleyized $\mathcal {T}$ ; if furthermore $\mathcal {T}$ already implied that every negated atomic formula is equivalent to a $\Sigma _1$ formula, then there is no need to Morleyize.

Example 6.3 (space of structures on $\le \mathbb {N}$ with $\Delta _1$ sizes).

For each $N \le \mathbb {N}$ (i.e., $N \in \mathbb {N} \cup \{\mathbb {N}\}$ ), define an étale structure $\mathcal {M}_N \to X_N$ over the space $X_N$ of $\mathcal {L}$ -structures on N, exactly as above. We may then simply take the disjoint unions $X := \bigsqcup _{N \le \mathbb {N}} X_N$ and $\mathcal {M} := \bigsqcup _{N \le \mathbb {N}} \mathcal {M}_N$ . The groupoid consists of the disjoint union of the logic actions of each symmetric group $S_N$ on $X_N$ .

However, each $X_N \subseteq X = M^0_X$ is clopen invariant, while “the model has size N” is $\Sigma _3$ -definable for finite N (by a conjunction of a sentence (6.2) and a negation of such a sentence) and $\Pi _3$ -definable for infinite N (by the infinite conjunction of all (6.2)); thus this $\mathcal {M}$ only has $\Sigma _4$ saturations. Even if we first Morleyize $\ne $ , i.e., use the traditional $\Sigma _1$ that includes $\ne $ , we still only get $\Sigma _3$ saturations. (It is easy to see that Morleyizing the above sentences (6.2), their subformulas, their negations, and their conjunction, thereby rendering each $X_N \subseteq X\ \Sigma _1$ -definable, is sufficient to yield $\Sigma _1$ saturations, since each $\mathcal {M}_N$ individually has $\Sigma _1$ saturations by the same argument as in the preceding example.)

Example 6.4 (space of structures on $\le \mathbb {N}$ with $\Delta _1$ finite sizes).

Let $\overline {\mathbb {N}} := \{0, 1, 2, \dotsc , \mathbb {N}\}$ , with the topology of the one-point compactification of $\mathbb {N}$ , i.e., $0, 1, 2, \dotsc $ are isolated points converging to $\mathbb {N}$ . We may realize the disjoint union X in the preceding example as

regarded now as a closed subspace of the compact zero-dimensional space $\overline {\mathbb {N}} \times \prod _{R \in \mathcal {L}} 2^{\mathbb {N}^n}$ , and

$$ \begin{gather*} M := \{(N, x, a) \in X \times \mathbb{N} \mid a \in N\} \subseteq \overline{\mathbb{N}} \times \prod_{R \in \mathcal{L}} 2^{\mathbb{N}^n} \times \mathbb{N}, \\ R^{\mathcal{M}}(N, x, a_0, \dotsc, a_{n-1}) := x(R)(a_0, \dotsc, a_{n-1}) \quad \text{for }R \in \mathcal{L}. \end{gather*} $$

A basic open set in $\overline {\mathbb {N}}$ is some finite Boolean combination of the sets $\{n, n+1, \dotsc , \mathbb {N}\}$ which are axiomatized by the sentences (6.2). Thus, a basic open set in $M^n_X$ is

(6.5)

for some $\vec {a} \in \mathbb {N}^{n+m}$ , finite conjunction $\phi (y_n, \dotsc , y_{n+m-1})$ of atomic $\mathcal {L}$ -formulas and their negations, and finite conjunction $\psi $ of the sentences (6.2) and their negations. Note that in addition to the condition “ $a_0, \dotsc , a_{n-1} \in N$ ”, for each non-negated atomic formula in $\phi $ , each of its arguments among $a_n, \dotsc , a_{n+m-1}$ is also implicitly required to be in N. (Once we remember these implicit conditions, we may assume $\phi $ contains no equalities, which have constant truth value given $\vec {a}$ .) All of these conditions of the form “ $a_i \in N$ ” together mean $N \ge \max _i (a_i+1)$ , which can itself be expressed by a single sentence $\theta $ of the form (6.2). Thus the saturation of the above set is defined by

(6.6)

which is $\Sigma _3$ ; it suffices to Morleyize negations of atomic formulas and of (6.2). As before, if we are only interested in parametrizing models of a $\Pi _2$ theory modulo which negations of atomic formulas and “the model has size $\le n$ ” are already $\Sigma _1$ -definable, then there is no need to Morleyize anything.

Example 6.7 (space of structures on $\le \mathbb {N}$ with $\Sigma _1$ lower bounds on size).

Modify the preceding example by putting the Scott topology on $\overline {\mathbb {N}}$ , with open sets $\{n, n+1, \dotsc , \mathbb {N}\}$ . Then the $\psi $ in (6.6) above will only be a conjunction of the sentences (6.2), no longer their negations, and so the final formula defining the saturation will be $\Sigma _2$ ; it suffices to Morleyize negations of atomic formulas.

If we also replace $2$ with $\mathbb {S}$ in the definition of X, then the $\phi $ in (6.6) will become basic (and equality-free); and so we only need to Morleyize $\ne $ . In other words, we obtain a parametrization of any $\Pi _2$ theory modulo which $\ne $ is $\Sigma _1$ -definable. (This parametrization was used in [Reference Chen14]; its restriction to the countably infinite models was used in [Reference Bazhenov, Fokina, Rossegger, Soskova and Vatev5].)

Example 6.9 (space of partially enumerated structures).

Let $\mathcal {L}$ be a countable relational language, and let $\mathcal {L}' := \{{\sim }\} \sqcup \mathcal {L}$ where $\sim $ is a binary relation symbol. Construct as in the second part of Example 6.7 a parametrization $\mathcal {M}' \to X'$ of all $\mathcal {L}'$ -structures which has $\Sigma _1$ saturations modulo $\ne $ . Let $X \subseteq X'$ be the structures in which $\sim $ is an equivalence relation and all relations in $\mathcal {L}$ are $\sim $ -invariant. Then $\sim ^{\mathcal {M}'}$ , which is open in $(M')^2_{X'}$ , restricted to the fibers over X is an open fiberwise equivalence relation on the étale space $M'|X \to X$ , i.e., only equating pairs of elements in the same fiber. It follows that the quotient $M := (M'|X)/({\sim ^{\mathcal {M}'}}|X) \to X$ is also an étale space, and that the quotient map $M'|X \twoheadrightarrow M$ is open. Let $\mathcal {M}$ be the étale structure on M descended from $\mathcal {M}'|X$ .

Explicitly, the space X consists of tuples $(N, {\sim }, x)$ , consisting of $N \in \overline {\mathbb {N}}$ (with the Scott topology), an equivalence relation $\sim $ on N, and a $\sim $ -invariant $\mathcal {L}$ -structure x on N (both with the Sierpiński topology), so that x descends to the quotient $N/{\sim }$ ; this descended structure is then the fiber $\mathcal {M}_{(N, {\sim }, x)}$ of $\mathcal {M} \to X$ . It is clear that $\mathcal {M}$ parametrizes all countable $\mathcal {L}$ -structures up to isomorphism, since for any countable $\mathcal {L}$ -structure, we may enumerate it with an initial segment $N \le \mathbb {N}$ and then lift the structure to N.

We now check that $\mathcal {M}$ has $\Sigma _1$ saturations. A basic open set in $M^n_X$ is the image under the quotient map of one in $(M')^n_{X'}$ as in (6.5) restricted to X, where (in the notation of that example) $\vec {a}$ is fixed, $\phi $ is a basic equality-free formula and $\psi $ only asserts that “the model has size $\ge \dotsb $ ” as in Example 6.7, since we used the Scott topology for $X'$ . We claim that the saturation of such an image in $M^n_X$ is defined by (6.6) with all occurrences of $\sim $ in $\phi $ replaced with $=$ and all $\ne $ replaced with $\top $ . Suppose $([b_0],\dotsc ,[b_{n-1}]) \in (N'/{\sim '})^n = M^n_{(N',{\sim '},x')}$ satisfies this modified version of (6.6), as witnessed by $([b_n],\dotsc ,[b_{n+m-1}]) \in (N'/{\sim '})^m = M^m_{(N',{\sim '},x')}$ ; we must find $N, {\sim }, x$ such that $(N, {\sim }, x, a_0, \dotsc , a_{n-1})$ is in (6.5) and there is an isomorphism of the quotient structures $(N,x)/{\sim } \cong (N',x')/{\sim '}$ mapping $[a_i] \mapsto [b_i]$ for $i < n$ . Let $N \subseteq \mathbb {N}$ be an initial segment such that

1. (i) if $N' = \emptyset $ , then $N = \emptyset $ ;

2. (ii) each $a_0, \dotsc , a_{n+m-1} \in N$ (this is consistent with (i) since $[b_0], \dotsc , [b_{n+m-1}] \in N'/{\sim '}$ );

3. (iii) N is at least as big as required by $\psi $ (consistent with (i) since $N'/{\sim '} \models \psi $ );

4. (iv) is at least as big as .

By virtue of the $\bigwedge _{a_i = a_j} (y_i = y_j)$ in (6.6), $a_i \mapsto [b_i]$ is a well-defined function $\{a_0,\dotsc ,a_{n+m-1}\} \to \{[b_0],\dotsc ,[b_{n+m-1}]\} \subseteq N'/{\sim '}$ ; extend it to a surjection $h : N \twoheadrightarrow N'/{\sim '}$ , which is easily seen to be possible using the above conditions. Now define , and lift the rest of the quotient structure on $(N',x')/{\sim '}$ to the structure x on N. Then h descends to an isomorphism $N/{\sim } \cong N'/{\sim '}$ mapping $[a_i] \mapsto [b_i]$ , whence $(N, {\sim }, x, a_0, \dotsc , a_{n-1})$ is in (6.5).

As usual, by further restricting this construction to the models of a $\Pi _2$ theory $\mathcal {T}$ , we obtain that every $\Pi _2$ theory has a quasi-Polish parametrization with $\Sigma _1$ saturations. As in Remark 6.8, we may further pull back to a zero-dimensional Polish $Z \twoheadrightarrow X$ ; however, unlike there, we can no longer assume that the underlying étale space M is also zero-dimensional.

Example 6.12 (space of marked structures).

Let $\mathcal {L}$ be an arbitrary countable language, possibly with function symbols. Let T be the set of $\mathcal {L}$ -terms over countably many variables $a_0, a_1, \dotsc $ . Let $X'$ be the space of enumerated $\mathcal {L}$ -structures as in Remark 6.11, where function symbols in $\mathcal {L}$ are encoded via their graphs, and where the role of the index set $\mathbb {N}$ is replaced by T; let $\mathcal {M}' \to X'$ be the corresponding étale structure. Thus a point in $X'$ consists essentially of an equivalence relation ${\sim }$ on T and an $\mathcal {L}$ -structure on $T/{\sim }$ . Let $X \subseteq X'$ be the $\boldsymbol {\Pi }^0_2$ subspace where $\sim $ is a congruence with respect to the usual syntactic action of function symbols on terms, and where the quotient structure on $T/{\sim }$ interprets each function symbol via this syntactic action. In other words,

Let $\mathcal {M} := \mathcal {M}'|X$ , so that $\mathcal {M}_{(\sim ,x)} = (T/{\sim }, \widetilde {x})$ is said generated structure.

Note that $X \subseteq X'$ is not -invariant; thus we cannot immediately conclude that $\mathcal {M}$ has $\Sigma _1$ saturations by Remark 5.15. Nonetheless, this is the case, as we now check. Chasing through Remark 6.11, Example 6.9, and ultimately (6.5), a basic open set in $M^n_X$ is

(6.13)

for some fixed terms $t_0, \dotsc , t_{n+m-1} \in T$ and basic $(\mathcal {L} \cup \{\sim \})$ -formula $\phi $ not mentioning $=$ ; by absorbing function applications into $t_n, \dotsc , t_{n+m-1}$ , we may assume $\phi $ only mentions relation symbols in $\mathcal {L}$ and $\sim $ . Let $k \in \mathbb {N}$ be sufficiently large so that $t_0, \dotsc , t_{n+m-1}$ only mention the variables $a_0, \dotsc , a_{k-1}$ . Then the -saturation of (6.13) is defined by

(6.14)

where $\phi '$ is $\phi $ with all occurrences of $\sim $ replaced with $=$ . Indeed, if $\mathcal {N}$ is a countable nonempty (which the fibers of $\mathcal {M}$ all are) $\mathcal {L}$ -structure which satisfies this formula under an assignment $\vec {x} \mapsto \vec {b} \in N^n$ , as witnessed by some $h : \{a_0, \dotsc , a_{k-1}\} \to N$ , we may extend h to the remaining variables $a_k, a_{k+1}, \dotsc $ to yield an enumeration of N, and then to all terms to obtain a surjective homomorphism $h : T \twoheadrightarrow N$ with respect to the function symbols in $\mathcal {L}$ , and finally pull back the relations as well as $=$ in $\mathcal {N}$ along h to obtain $({\sim },x) \in X$ such that $h : T \twoheadrightarrow N$ descends to an isomorphism $\mathcal {M}_{({\sim },x)} \cong \mathcal {N}$ and $({\sim },x,[t_0],\dotsc ,[t_{n-1}])$ is in (6.13) where $t_0, \dotsc , t_{n-1}$ are any h-preimages of  $\vec {b}$ .

Example 7.2. Taking $\mathcal {T}$ to be a propositional theory, as in Example 3.5 (and the arities $n_i$ above to all be $0$ ), we recover the Baire category theorem for quasi-Polish spaces as a special case.

Example 8.5. If $X = 1$ and $\mathcal {M}$ is a countable structure with $\Sigma _1$ -definable automorphism orbits, as in Remark 5.3, then Theorem 8.4 says that every continuous action of $\operatorname {\mathrm {Aut}}(\mathcal {M})$ on a countable discrete space A is named by some $\Sigma _1$ imaginary of $\mathcal {M}$ . To see this directly: for every $a \in A$ , the stabilizer $\operatorname {\mathrm {Aut}}(\mathcal {M})_a \subseteq \operatorname {\mathrm {Aut}}(\mathcal {M})$ is a clopen subgroup, hence contains some basic clopen subgroup $\operatorname {\mathrm {Aut}}(\mathcal {M},\vec {b})$ , the automorphism group of $\mathcal {M}$ expanded with finitely many fixed constants $\vec {b} \in M^n$ ; then the orbit $\operatorname {\mathrm {Aut}}(\mathcal {M}) \cdot a$ is a definable quotient of the $\Sigma _1$ -definable orbit $\operatorname {\mathrm {Aut}}(\mathcal {M}) \cdot \vec {b}$ . The above proof can be seen as the natural generalization of this to étale structures.

Example 9.4. The construction from each integral domain $\mathcal {R}$ of its field of fractions is specified by an interpretation $\mathcal {F} : (\text {theory of fields}) \to (\text {theory of integral domains})$ .

• • The underlying imaginary is $F = \phi /\epsilon $ where $\phi , \epsilon $ are the formulas in the language of rings:

  $$ \begin{align*} \phi(x,y) &:= (y \ne 0), & \epsilon(x_1,y_1,x_2,y_2) := (x_1y_2 = x_2y_1). \end{align*} $$

  Thus given an integral domain $\mathcal {R}$ , the field of fractions $\mathcal {F}^{\mathcal {R}}$ has underlying set

  

• • The operation $+$ of fields is interpreted as the definable function $+^{\mathcal {F}} = \psi /\epsilon ^3$ where

  $$ \begin{align*} \psi(x_1,y_1,x_2,y_2,x_3,y_3) &:= \epsilon(x_1y_2+x_2y_1, y_1y_2, x_3, y_3). \end{align*} $$

  In other words, for an integral domain $\mathcal {R}$ , this defines the graph of $+$ on $\mathcal {F}^{\mathcal {R}}$ :

  $$ \begin{align*} [(a_1,b_1)] + [(a_2,b_2)] = [(a_3,b_3)] \iff (a_1b_2+a_2b_1, b_1b_2) \sim (a_3,b_3). \end{align*} $$