Lusin and Suslin properties of function spaces

A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space X let Cp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_p(X)$$\end{document}, Ck(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_k(X)$$\end{document} and C↓F(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\downarrow }{\mathsf {F}}}(X)$$\end{document} be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, the compact-open topology, and the Fell hypograph topology, respectively. For a metrizable space X we prove the equivalence of the following statements: (1) X is σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-compact, (2) Cp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_p(X)$$\end{document} is Suslin, (3) Ck(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_k(X)$$\end{document} is Suslin, (4) C↓F(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\downarrow }{\mathsf {F}}}(X)$$\end{document} is Suslin, (5) Cp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_p(X)$$\end{document} is Lusin, (6) Ck(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_k(X)$$\end{document} is Lusin, (7) C↓F(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\downarrow }{\mathsf {F}}}(X)$$\end{document} is Lusin, (8) Cp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_p(X)$$\end{document} is Fσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\sigma $$\end{document}-Lusin, (9) Ck(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_k(X)$$\end{document} is Fσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\sigma $$\end{document}-Lusin, (10) C↓F(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\downarrow }{\mathsf {F}}}(X)$$\end{document} is Cδσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\delta \sigma }$$\end{document}-Lusin. Also we construct an example of a sequential ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}-space X with a unique non-isolated point such that the function spaces Cp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_p(X)$$\end{document}, Ck(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_k(X)$$\end{document} and C↓F(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\downarrow }{\mathsf {F}}}(X)$$\end{document} are non-Suslin.

In this paper we shall be interested in descriptive properties of the function spaces, i.e., properties that can be described in terms of Borel sets.
Let us recall that a set A in a topological space X is called -Borel if A belongs to the smallest σ -algebra of sets in X , containing all open subsets in X ; constructible if A belongs to the smallest algebra of sets in X , containing all open sets in X ; clopen if it is both open and closed; -an F σ -set if A is a countable union of closed sets in X ; -a G δ -set if A is a countable intersection of open sets in X ; -a C σ -set if A is a countable union of constructible sets in X ; -a C δ -set if A is a countable intersection of constructible sets in X ; -an C σ δ -set (resp. F σ δ -set) if A is a countable intersection of C σ -sets (resp. F σ -sets) in X ; -an C δσ -set (resp. G δσ -set) if A is a countable union of C δ -sets (resp. G δ -sets) in X .
De Morgan's laws imply that each constructible set in a topological space can be written as a finite union (U 1 ∩ F 1 ) ∪ · · · ∪ (U n ∩ F n ) of intersections U i ∩ F i of open and closed sets. For any set in a topological space we have the following implications.
For example, each metrizable space is perfect. In perfect spaces the above diagram simplifies to the following form. closed Let Γ be a class of Borel sets in topological spaces. A function f : X → Y between topological spaces is called Γ -measurable if for any open set U ⊆ Y the preimage f −1 [U ] is Borel of class Γ in X . In particular, a function f : X → Y is continuous if and only if it is G-measurable for the class G of open sets in topological spaces.
A topological space X is defined to be -Polish if it is homeomorphic to a separable complete metric space; -Suslin if it is the image of a Polish space under a continuous map; -Lusin if it is the image of a Polish space under a continuous bijective map; -Γ -Lusin for a Borel class Γ if it is the image of a Polish space P under a continuous bijective map f : P → X such that the inverse map f −1 : X → P is Γ -measurable.
In the role of the class Γ we shall consider the (additive) Borel classes G, F σ , C σ , C δσ , G δσ , B of open sets, F σ -sets, C σ -sets, C δσ -sets, G δσ -sets, Borel sets in topological spaces, respectively. For any topological space we have the implications: For regular spaces some implications in this chain turn into equivalences (see Theorem 3.4): By Lusin-Suslin Theorem [10, 15.1], a subspace X of a Polish space P is Lusin if and only if X is a Borel subset of P. By the famous result of Suslin [10, 14.2], each uncountable Polish space contains a Suslin subset, which is not Borel and hence is not Lusin. In the class of metrizable spaces, Lusin and Suslin spaces were defined by Bourbaki in [5].
It is well-known [7, 4.3.26] that each Polish (= G-Lusin) space X is a G δ -set in any Tychonoff space containing X as a subspace. In Theorem 3.1 we shall prove that each F σ -Lusin space X is a C σ δ -set in each Hausdorff space containing X as a subspace.
Let us observe that each Suslin space has a countable network, being a continuous image of a Polish space (which has a countable base).
We recall that a family N of subsets of a topological space X is called A topological space X is called cosmic if X is regular and has a countable network; -an ℵ 0 -space if X is regular and has a countable k-network.
For any topological space we have the following implications (see [8, §4]): Function spaces C ↓F (X ) and C k (X ) possessing countable networks were characterized in [11, Theorem 3.7 and 4.5] and [13] (see also [12, §4.1]). (1) the function space C ↓F (X ) has a countable network; (2) the function space C k (X ) is cosmic; The following characterization of cosmic spaces C p (X ) is well-known and can be found in [12, 4.1.3] or [1, I.1.3].

Theorem 1.3 A Tychonoff space X is cosmic if and only if C p (X ) is cosmic.
The following fundamental result is due to Calbrix [6] (see also Theorem 9.7 in [9, p.208]).
For an Ascoli space X the Suslin property of the function space C k (X ) can be characterized in terms of R-universal ω ω -based uniformities.
Let us recall that a topological space X is called -Fréchet-Urysohn if for each set A ⊆ X and point a ∈Ā there exists a sequence {a n } n∈ω ⊆ A that converges to a; sequential if a subset A ⊆ X is closed in X if and only if A contains the limits of all sequences {a n } n∈ω ⊆ A that converge in X ; -a k-space if a subset A ⊆ X is closed in X if and only if for any compact subset K ⊆ X the intersection A ∩ K is closed in K ; -a k R -space if a function f : X → R is continuous if and only if for every compact subset The canonical map assigns to each point x the Dirac functional δ x : By [15], each Tychonoff k R -space is Ascoli. Therefore, for any Tychonoff space we obtain the following implications: None of these implications can be reversed, see Examples 1.6.18, 1.6.19 in [7,14], and [4, 6.7].
Next, we recall some information on ω ω -based uniformities. Here we consider ω ω as a partially ordered space endowed with the partial order ≤ defined by α ≤ β iff α(n) ≤ β(n) for all n ∈ ω.
A uniformity U on a set X is called ω ω -based if it has a base (U α ) α∈ω ω such that U β ⊆ U α for any α ≤ β in ω ω . For example, any metrizable uniformity is ω ω -based.
A uniformity U on a topological space X is called R-universal if it generates the topology of X and every continuous function f : X → R is uniformly continuous in the uniformity

U.
A topological space X is called universally ω ω -based if its universal uniformity is ω ω -based. The universal uniformity of X is generated by the family of all continuous pseudometrics on X . This theorem will be proved in Sect. 2. Now we discuss the descriptive properties of function spaces on Γ -quotient spaces.
We say that a topological space X is a quotient of a topological space M if there exists a surjective quotient map f : M → X . The quotient property of f means that a subset U ⊆ X is open if and only if its preimage f −1 [U ] in open in M.
A topological space X is called Γ -quotient for a Borel class Γ if X is a quotient of some space of class Γ in a compact metrizable space.
In particular, a G-quotient space is a quotient of a locally compact Polish space and an F σ -quotient space is a quotient of a σ -compact metrizable space.
We recall that a topological space is Lashnev if it is the image of a metrizable space under a continuous closed map. It is known (see [8, 11.3]  We recall that a Tychonoff space is universally ω ω -based if its universal uniformity of X has an ω ω -base. Therefore, for any Tychonoff space X we have the implications: where the last implication can be reversed for separable Lashnev spaces.
The following theorem is the main result of this paper. Theorem 1.7 For a Tychonoff space X consider the following statements: (1) X is G-quotient; (2) C k (X ) is Polish; Theorem 1.7 will be proved in Sect. 6 after some preliminary work done in Sects. 3, 4 and 5. This theorem implies the following characterization. Corollary 1.8 For a metrizable space X the following statements are equivalent: The implications of Theorems 1.7, 1.4 and 1.5 are represented in the following diagram holding for any Tychonoff space X .
is Suslin X is separable and has an R-universal ω ω -based uniformity The following proposition shows that the last implication in the diagram (established by Calbrix's Theorem 1.4) cannot be reversed even for countable Lashnev (and hence ℵ 0 -spaces) with a unique non-isolated point. This proposition also implies that Theorem 5.7.4 in [12] is incorrect (that theorem claims that for any sequential σ -compact ℵ 0 -space X the function space C k (X ) is Suslin).

Proposition 1.10 Let X = M/A be the quotient space of a metrizable space M by a closed nowhere dense subset A ⊂ M. If the function space C k (X ) is Suslin, then the space M is σ -compact and hence X is an F σ -quotient space.
Proof The quotient space X = M/A of the metrizable space M is a sequential Tychonoff space and hence Ascoli. If the function space C k (X ) is Suslin, then X cosmic by Theorem 1.3, and by Theorem 1.5, the topology of X is generated by some ω ω -based uniformity. By Corollary 8.2.3 [2], the set A is σ -compact. By Theorem 1.4, the cosmic space X is σ -compact and so is its open subspace X \{A}, which is homeomorphic to M\A.

Example 1.11
Let ω <ω = n∈ω ω n be the family of all functions x : n → ω defined on finite ordinals n ∈ ω. Let ω ≤ω = ω <ω ∪ω ω . For any function x ∈ ω ≤ω defined on an ordinal n ≤ ω and any ordinal k ≤ ω denote by x k the restriction of x to the ordinal k ∩ n = min{n, k}. The set ω ≤ω carries a natural partial order ≤ defined by x ≤ y iff there exists an ordinal n ≤ ω such that x = y n. The space ω ≤ω is endowed with the topology τ generated by the countable base consisting of the sets ↑x = {y ∈ ω ≤ω : x ≤ y} where x ∈ ω <ω . It is easy to see that (ω ≤ω , τ ) is a Polish space, ω <ω is a dense discrete subspace in ω ≤ω and ω ω is a closed nowhere dense subset in ω ≤ω . Consider the quotient space X = ω ≤ω /ω ω and observe that X is a countable sequential ℵ 0 -space with a unique non-isolated point. Since the space ω ≤ω is not σ -compact, the function spaces C p (X ), C k (X ) and C ↓F (X ) are not Suslin according to Proposition 1.10. In Sect. 7 we shall present an alternative self-contained proof of this fact.
Observe that the space X = M/A in Proposition 1.10 is Lashnev, i.e., the image of a metrizable space under a closed continuous map.

Problem 1.12 Assume that a Tychonoff space X is Lashnev and its function space C k (X ) is
Suslin. Is X an F σ -quotient space?

The Suslin property of the function spaces on Ascoli spaces
In this section we shall prove Theorem 1.5. To prove the "if" part of this theorem, assume that a Tychonoff space X is separable and has an R-universal ω ω -based uniformity. By Theorem 7.5.1(18) of [2], the function space C k (X ) is Suslin.
To prove the "only if" part of Theorem 1.5, assume that the space X is Ascoli and the function space C k (X ) is Suslin. By Theorem 1.2, the space X is separable. By the definition of an Ascoli space, the canonical map δ : For every α ∈ ω ω consider the entourage of the diagonal in X × X . The continuity of the map δ : is an open neighborhood of the diagonal in X × X . It is easy to see that (U α ) α∈ω ω is an ω ω -base of some uniformity U on X . To see that this uniformity in R-universal, we need to show that every function f ∈ C(X ) is uniformly continuous with respect to the uniformity U. Given any ε > 0, find α ∈ ω ω such that 2 −α(0) < ε and f = ξ(α N). Then for any pair which means that f is uniformly continuous.

Some results on Lusin and Suslin spaces
Proof Write X as the image of a Polish space P under a continuous bijective map f : P → X with F σ -measurable inverse f −1 : X → P. Fix a complete metric d that generates the topology of the Polish space X .
For every n ∈ ω fix a countable open cover U n of X by sets of d-diameter < 2 −n . By the choice of the map f , for every U ∈ U n the image f [U ] is an F σ -set in X . Consequently, f [U ] = F n,U for some countable family F n,U of closed sets in X . Let F n = U ∈U n F n,U and F = n∈ω F n . Therefore, F is a countable family of closed sets in X . It follows that each set F ∈ F is equal to the intersection X ∩F of X and the closureF of F in the space Y . Observe that for any sets Consequently, Then for every n ∈ ω there exists a set F n ∈ F n such that y ∈F n . For the set F n find an We claim that for every n ∈ ω the intersection F 0 ∩ · · · ∩ F n is not empty. Assuming that this intersection is empty, we would conclude that y ∈F 0 ∩ · · · ∩F n is contained in Y \ A, which contradicts the choice of y. Therefore, the family of closed sets (F n ) n∈ω is centered and so is the family By the Hausdorff property of Y , the point f (x) has an open neighborhood V ⊆ Y whose closure does not contain the point y. By the continuity of f at x, there exists n ∈ ω such that We do not know if Theorem 3.1 generalizes to higher Borel classes.
The following characterization of Suslin spaces was proved in [3, 2.5].

Theorem 3.3 A cosmic space X is Suslin if and only if it is the image of a Suslin space Z under a surjective Borel map f : Z → X.
A similar characterization holds for Lusin spaces. (1) X is Lusin;

Borel properties of the identity maps between various function spaces
It is clear that for any Tychonoff space X the identity maps C k (X ) → C p (X ) and C k (X ) → C ↓F (X ) are continuous.

Lemma 4.1 For any
Proof By Theorem 1.2, the function space C k (X ) has a countable network and hence is hereditarily Lindelöf. Then it suffices to find a subbase of the topology of C k (X ) consisting of the sets which are Borel in the topology of the space C p (X ). We claim that the standard subbase of C k (X ) has this property. Given a compact set K ⊆ X and a real number r , we need to check that the open sets are Borel in C p (X ). The compact subset K of the ℵ 0 -space has a countable network and hence is separable. Consequently, we can fix a countable dense set {x m } m∈ω in K . Now observe that the sets are Borel of type F σ in C p (X ).

Lemma 4.2 For any cosmic space X , the identity map C
Proof By Theorem 1.3, the function space C p (X ) has a countable network and hence is hereditarily Lindelöf. Then it suffices to find a subbase of the topology of C p (X ) consisting of sets which are of type C σ in the Fell hypograph topology. We claim that the standard subbase of C p (X ) has this property. Given a point x ∈ X and a real number r , we need to check that the open sets The set x; r is open in C ↓F (X ) and hence C σ by the definition of the Fell hypograph topology. Since x; r + 1 n , the set K ; r is Borel of type F σ in C ↓F (X ).

Lemma 4.3 For any
Proof By Theorem 1.2, the function space C k (X ) has a countable network and hence is hereditarily Lindelöf. Then it suffices to find a subbase of the topology of C k (X ) consisting of sets which are of type F σ in the Fell hypograph topology. We claim that the standard subbase of C k (X ) has this property. Given a nonempty compact set K ⊆ X and a real number r , we need to check that the open sets are Borel of class C σ in C ↓F (X ). The set K ; r is open in C ↓F (X ) by the definition of the Fell hypograph topology. The compact subset K of the ℵ 0 -space has a countable network {K n } n∈ω consisting of closed (and hence compact) sets in K . Since the set K ; r is Borel of type F σ in C ↓F (X ).

Lemma 5.1 For any F σ -quotient Tychonoff space X , the function spaces C p (X ) and C k (X ) are F σ -Lusin and C ↓F (X ) is C δσ -Lusin.
Proof By the definition of an F σ -quotient space, there exists a quotient surjective map q : M → X defined on a σ -compact metrizable space M. First we establish two properties of the quotient map q.

Claim 5.2 For any sequence
Proof If the set Ω = {n ∈ ω : x n = x} is infinite, then take any point z ∈ q −1 (x) and put z k = z for all k ∈ Ω. It is clear that the sequence (z k ) k∈Ω converges to z and So, we assume that the set Ω is finite. Then the set A = {x n : n ∈ ω \ Ω} is not closed in X and by the quotient property of q, the preimage q −1 (A) is not closed in M. Since M is metrizable, there exists a sequence {z k } k∈ω ⊆ q −1 [A], convergent to a point z / ∈ q −1 [A]. The continuity of q implies that q(z) ∈Ā \ A = {x}.

Claim 5.3 For every compact set K ⊆ X and a cover U of the set q −1 [K ] by open subsets of M there exists a finite subfamily
Proof Since the σ -compact space M is Lindelöf, the open cover U of the closed set q −1 [K ] contains a countable subcover V. We can choose an enumeration {U n } n∈ω of the countable family V such that q −1 [K ] ⊆ ∞ n=k U n and hence K ⊆ ∞ n=k q[U n ] for every k ∈ ω. To finish the proof of the claim, it suffices to find n ∈ ω such that K ⊆ k≤n q[U k ].
Assuming that no such number n exists, for every n ∈ ω we can choose a point x n ∈ K \ k≤n q[U k ]. By the compactness of K , the sequence (x n ) n∈ω accumulates at some point In the Polish space ω ω × R D consider the G δ -subset Observe that for every (α, f ) ∈ P and every n ∈ ω the restriction f D ∩ M n is a uniformly continuous function, which admits a uniformly continuous extensionf n : M n → R to M n (by the density of D ∩ M n in M n ). Taking into account that D ∩ M n ⊆ D ∩ M n+1 , we conclude thatf n =f n+1 M n , which allows us to define a functionf : M → R such thatf M n = f n for all n ∈ ω. We claim that the functionf is continuous. Indeed, for any x ∈ M and any ε > 0, we can find n ∈ ω such that x ∈ M n and 1 2 n < 1 3 ε. We claim that |f (x) −f (y)| < ε for any y ∈ M with d(x, y) < 1 2 α(n) . Find a number m ≥ n such that y ∈ M m . By the continuity of the mapf M m and the density of Therefore the functionf is continuous. Next, we show thatf (x) =f (y) for any x, y ∈ M with q(x) = q(y). Assuming thatf (x) =f (y), we can find n ∈ ω such that x, y ∈ M n and | f (x) − f (y)| > 3 2 n . Then (x, y) ∈ Q n . By the density of D in M, there exist points x ∈ D(x; 1 2 α(n) ) and y ∈ D(y; 1 2 α(n) ) such that | f (x ) − f (y )| = |f (x ) −f (y )| > 3 2 n . But this contradicts the inclusion (α, f ) ∈ P. This contradiction shows thatf =f •q for some functionf : X → R. Since the map q is quotient, the functionf : X → R is continuous. So, we can consider the function ξ : P → C(X ) assigning to each pair (α, f ) the (unique) continuous functionf : X → R such that f =f • q D.

Claim 5.4 The function ξ : P → C(X ) is surjective.
Proof Let ϕ : X → R be any continuous function. For every n ∈ ω, the continuity of the function ψ = ϕ • q at points of the compact set M n yields a number α(n) ∈ ω such that |ψ(x)−ψ(y)| ≤ 2 −n for any x ∈ M n and y ∈ M with d(x, y) < 2 −α(n) . We can assume that α(n) is the smallest possible number with this property. Then there exists x ∈ M n and y ∈ M such that d(x, y) < 2 1−α(n) and |ψ(x) − ψ(y)| > 2 −n . By the density of the sets D ∩ M n in M n and D in M, there are points x ∈ M n ∩ D and y ∈ D such that d(x , y ) < 2 1−α(n) and |ψ(x ) − ψ(y )| > 2 −n . It is easy to see that the pair (α, ψ D) belongs to the first two sets in the definition of the set P.
Observe that for the function f = ψ D, we getf = ψ and ϕ =f = ξ(α, f ), which means that the function ξ is surjective.
Proof By [8, 11.3], the F σ -quotient space X is an ℵ 0 -space. By Theorem 1.2, the function space C k (X ) is cosmic and hence hereditarily Lindelöf. So, it suffices to show that for any nonempty compact set K ⊆ X and any real number r the sets ξ −1 K ; r and ξ −1 K ; r are open in P.
For every z ∈ q −1 [K ] we can find a number k ≥ n such that z ∈ M k . By the density of the set D ∩ M k in M k , there exists a point x ∈ D ∩ M k ∩ q −1 [U ] ∩ B(z; 2 −α(k) ). Then z ∈ B(x; 2 −α(k) ). Therefore, By Claim 5.3, there exists m ≥ n and a finite subset

Claim 5.7
The function ξ −1 : Proof Since the Polish space P is hereditarily Lindelöf, it suffices to show that for any (n, m) ∈ ω, point x ∈ D and real number r , the images of the subbasic open sets under the map ξ are F σ -sets in C p (X ). We shall prove that the images of these sets are open or closed (and hence F σ ) in C p (X ).
Observe that the set are open in C p (X ).

Claim 5.8
For every x, y ∈ X and ε > 0 the set Proof Let Q be the set of rational numbers and Q = {( p, q) ∈ Q × Q : p + ε < q}. Observe that is of type C σ in C ↓F (X ).

Claim 5.9
The function ξ −1 : Proof Similarly as in Claim 5.7, it suffices to check that for any (n, m) ∈ ω, point x ∈ D and real number r , the images of the subbasic open sets P n≤m , P n≥m , P x>r , P x<r under the map ξ are C δσ -sets in C ↓F (X ). We shall prove that the images of these sets are of type C σ or C δ in C ↓F (X ). By Claim 5.8, the set . Claim 5. 10 The function space C p (X ) is F σ -Lusin.
Proof By Claims 5.4, 5.5 and 5.6, the map ξ : P → C k (X ) is bijective and continuous. Since the identity map C k (X ) → C p (X ) is continuous, the map ξ : P → C p (X ) is continuous as the composition of two continuous maps. By Claim 5.7, the inverse map ξ −1 : C p (X ) → P is F σ -measurable, which implies that the space C p (X ) is F σ -Lusin.

Claim 5.11
The function space C k (X ) is F σ -Lusin.
Proof By Claims 5.4, 5.5 and 5.6, the map ξ : P → C k (X ) is bijective and continuous. By Claim 5.7 the map ξ −1 : C p (X ) → P is F σ -measurable. The continuity of the identity map C k (X ) → C p (X ) implies that the map ξ −1 : C k (X ) → P is F σ -measurable (as the composition of a continuous and F σ -measurable maps). Now we see that the Polish space P and the map ξ : P → C k (X ) witness that the space C k (X ) is F σ -Lusin.

Claim 5.12
The function space C ↓F (X ) is C δσ -Lusin.
Proof By Claims 5.4, 5.5 and 5.6, the map ξ : P → C k (X ) is bijective and continuous. Since the identity map C k (X ) → C ↓F (X ) is continuous, the map ξ : P → C ↓F (X ) is continuous (as the composition of two continuous maps). By Claim 5.9, the inverse map ξ −1 : C ↓F (X ) → P is C δσ -measurable, which implies that the space C p (X ) is C δσ -Lusin.

Proof of Theorem 1.7
In this section, for a Tychonoff space X we shall prove the implications of Theorem 1.7.
(2) ⇒ (3) Assume that the function space C k (X ) is Polish. The continuity of the identity map C k (X ) → C ↓F (X ) implies that the space C ↓F (X ) is Lusin. Since the identity map (2) ⇒ (1) If the function space C k (X ) is Polish, then by Corollary 5.2.5 in [12], X is a cosmic hemicompact k-space. The hemicompactness of X yields an increasing sequence (K n ) n∈ω of compact sets in X such that each compact subset of X is contained in some set K n . Consider the locally compact subspace M = n∈ω (K n × {n}) of the product X × ω where the ordinal ω is endowed with the discrete topology. Let q : M → X be the natural projection. We claim that the map q is quotient. Indeed, take any subset A ⊆ X such that the preimage q −1 [A] is closed in X . Then A ∩ K n is closed in K n for every n ∈ ω. Since each compact set K ⊆ X is contained in some K n , the intersection (1) ⇒ (2) Assume that X is G-quotient and find a quotient surjective map q : M → X defined on an open subspace M of a compact metrizable space. Write the locally compact space M as the countable union M = n∈ω U n of an increasing sequence (U n ) n∈ω of open sets such that each set U n has compact closure U n , contained in U n+1 . By Claim 5.3, for every compact set K ⊆ X there exists a number n ∈ ω such that K ⊆ q[U n ] ⊆ q[U n ]. Now we see that the sequence of compact sets (q[U n ]) n∈ω witnesses that the space X is hemicompact. By Theorem 11.3 [8], X is a cosmic k-space, and by Corollary 5.2.5 of [12], for the cosmic hemicompact k-space X , the function space C k (X ) is Polish.
(8) ⇒ (6) Assume that the space C ↓F (X ) is Lusin. Then C ↓F (X ) has a countable network and X is an ℵ 0 -space by Theorem 1.2. By Theorem 1.3, the function space C p (X ) is cosmic. By Lemma 4.2, the identity map C ↓F (X ) → C p (X ) is Borel and by Theorem 3.4, the cosmic space C p (X ) is Lusin.
(6) ⇒ (7) Assume that C p (X ) is Lusin and X is an ℵ 0 -space. By Theorem 1.2, the function space C k (X ) is cosmic. By Lemma 4.1, the identity map C p (X ) → C k (X ) is Borel and by Theorem 3.4, the cosmic space C k (X ) is Lusin.
(11) ⇒ (9) Assume that the space C ↓F (X ) is Suslin. Then C ↓F (X ) has a countable network and X is an ℵ 0 -space by Theorem 1.2. By Theorem 1.3, the function space C p (X ) is cosmic. By Lemma 4.2, the identity map C ↓F (X ) → C p (X ) is Borel and by Theorem3.3, the space C p (X ) is Suslin.
(9) ⇒ (10) Assume that C p (X ) is Suslin and X is an ℵ 0 -space. By Theorem 1.2, the function space C k (X ) is cosmic. By Lemma 4.1, the identity map C p (X ) → C k (X ) is Borel and by Theorem 3.3, the space C k (X ) is Suslin.
(10) ⇒ (12) If the space C k (X ) is Suslin, then so is the space C p (X ) (being a continuous image of C k (X ). By Calbrix's Theorem 1.4, the space X is σ -compact.

Discussing Example 1.11
In this section we prove that for the quotient space X = ω ≤ω /ω ω from Example 1.11, the function spaces C p (X ), C k (X ) and C ↓F (X ) are non-Suslin.
For a topological space T denote by T the set of non-isolated points of T and observe that is a closed linear subspace of C p (T ).
We recall that the discrete subspace ω <ω of the Polish space ω ≤ω = ω <ω ∪ ω ω carries the partial order ≤ defined by x ≤ y iff x = y n for some n ∈ ω. Endowed with this partial order, the set ω <ω is a tree (which means that for any x ∈ ω <ω the set ↓x = {y ∈ ω <ω : y ≤ x} is finite and linearly ordered). A subtree T of ω <ω is well-founded if it contains no infinite linearly ordered subsets.
For any function f ∈ M 0 (ω ≤ω ) the preimage f −1 (1) is a well-founded subtree of the tree ω <ω . So, the space M 0 (ω ≤ω ) can be identified with the space W F of well-founded trees on ω. By [10, 32.B], the space W F is coanalytic but not analytic and so is the space M 0 (ω ≤ω ). Let us recall that a subset A of a Polish space P is analytic (resp. coanalytic) if the space A (resp. P \ A) is Suslin.

Claim 7.2
The function spaces C p (X ), C k (X ) and C ↓F (X ) are non-Suslin.
Proof The space C p (X ) is not Suslin since it contains the closed subspace C p (X ) which is not Suslin by Claim 7.1. By Theorem 1.7, the space C k (X ) and C ↓F (X ) are not Suslin, too.
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