Feral dual spaces and (strongly) distinguished spaces C(X)

Following Dieudonné and Schwartz a locally convex space is distinguished if its strong dual is barrelled. The distinguished property for 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} of continuous real-valued functions over a Tychonoff space X is a peculiar (although applicable) property. It is known that 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 distinguished if and only if 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 large in RX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^X$$\end{document} if and only if X is a Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}-space (in sense of Reed) if and only if the strong dual of 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} carries the finest locally convex topology. Our main results about spaces whose strong dual has only finite-dimensional bounded sets (see Theorems 2, 7 and Proposition 4) are used to study distinguished spaces 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} with the compact-open topology. We also put together several known facts (Theorem 6) about distinguished 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} with self-contained full proofs.

If X is a Tychonoff space then C p (X ) and C k (X ) denote the linear space C(X ) of all real-valued continuous functions defined on X equipped with the pointwise convergence and the compact-open topology, respectively.
The simplest examples of distinguished C p (X ) spaces which are not semi-reflexive are those with X any countable nondiscrete Tychonoff space (see [12]). Several recently obtained results about distinguished spaces C p (X ) have been proved in articles [12,15,16,24,25,32]. We know that C p (X ) is distinguished if and only if the strong dual L β (X ) of C p (X ) (i.e. the topological dual C p (X ) of C p (X ) with the strong topology β(C p (X ) , C p (X ))) carries the finest locally convex topology, see [12,15]. In [24] we proved that C p (X ) is distinguished if and only if X is a -space (in the sense of Reed [35]). This theorem apparently provides a nice connection with problems from the set theory and related with -sets, λ-sets, and Q-sets X , and corresponding distinguished spaces C p (X ).
A similar characterization of distinguished spaces C p (X ) in term of the space X has been proved in [16].
In Sect. 2 we put together (Theorem 6) several equivalent conditions for C p (X ) to be distinguished with self-contained proofs.
Being motivated by results from papers mentioned above we propose Definition 1 A locally convex space E is strongly distinguished if its strong dual E β caries the finest locally convex topology (i.e. any absolutely convex and absorbing subset of E β is a neighbourhood of zero).
Following [22] a locally convex space is feral if every bounded set in E is finitedimensional. Hence a locally convex space E carries the finest locally convex topology if and only if E is feral and E is bornological (i.e. every locally bounded linear map from E into any locally convex space is continuous). Recall that a linear map between locally convex spaces is locally bounded if it maps bounded sets into bounded sets; clearly, any continuous linear map is locally bounded.
In [12] we showed that C p (X ) is strongly distinguished if and only if it is distinguished, see also [15]; in [13, page 392] we proved that for each Tychonoff space X the strong dual L β (X ) of C p (X ) is feral. Consequently, C p (X ) is strongly distinguished if and only if its strong dual L β (X ) is bornological.
In [26] we proved the following

Theorem 1 ( [26, Theorem 1])
For a Tychonoff space X the following are equivalent: (1) The space C k (X ) is strongly distinguished.
(2) X is a -space and every compact subset of X is finite.
(3) The space C k (X ) is large in R X .
Since a linear space with the finest locally convex topology is feral, Theorem 1 suggests the following natural Problem 1 Characterize locally convex spaces whose strong dual is feral.
This problem is solved in Theorem 7 providing some applications, see Corollary 7 (extending [11,Theorem 3.3]). The main application of Theorem 7 extends Theorem 1, essentially for (1) ⇒ (2), and is formulated below: Theorem 2 For a Tychonoff space X the following are equivalent: (1) The strong dual of C k (X ) is feral.
(2) Every compact subset of X is finite.
Recall that a locally convex space E is quasi-barrelled if every absolutely convex closed set in E absorbing bounded sets in E is a neighbourhood of zero. It is not clear if the quasibarrelledness of C k (X ) in Theorem 2 can be omitted. Note however that for each Tychonoff space X the strong bidual M β (X ) of C p (X ) is feral if and only if X is finite, see Corollary 4.
Recall that C k (X ) is quasi-barrelled if and only if X is a W -space i.e. every b-bounding subset B of X is relatively compact, see [34,Theorem 10.1.21]; the b-boundedness of B means that for every bounded subset Observe that the item (2) in Theorem 2 does not guarantee that the strong dual of C k (X ) is bornological, see Example 1.
Krupski and Marciszewski proved that for any infinite compact spaces X and Y the spaces Corollary 3.2], see also [28] for motivations and results around this line of research).
Applying Theorem 2 we extend this result by proving the following.
Corollary 1 Let X and Y be Tychonoff spaces. Let C w (Y ) be the space C k (Y ) with its weak topology σ (C k (Y ), C k (Y ) ). If there exists a continuous linear map from C p (X ) onto C w (Y ), then every compact subset of Y is finite. In particular, the spaces C p (X ) and C w (Y ) are not isomorphic, if Y contains an infinite compact subset.
In fact we prove a more general statement, see Corollary 5.
Clearly, a decreasing sequence (X n ) of subsets of X is point-finite if and only if ∞ n=1 X n = ∅.
A topological space X is called (a) a -space if every decreasing point-finite sequence (X n ) of subsets of X admits a decreasing point-finite open expansion (U n ), see [24]; (b) a strong -space if every decreasing point-finite sequence (X n ) of subsets of X admits a decreasing compact-finite open expansion (U n ).
Theorems 1, 2 and 6 imply the following Corollary 2 For a Tychonoff space X the following are equivalent: (3) The strong dual C k (X ) β of C k (X ) is feral and the strong dual L β (X ) of C p (X ) is bornological. (4) X is a -space and every compact subset of X is finite. (5) X is a strong -space. Note however that there exist (non-complete) Montel spaces (hence distinguished) whose strong dual is not bornological, see [34,Example 4.7.8]. Clearly, for every discrete space X and for every countable Tychonoff space X the space C p (X ) is distinguished, since every discrete space and every countable Tychonoff space are -spaces.
On the other hand, we show that Example 1 There exists an uncountable pseudocompact Haydon space X with all compact sets finite such that C k (X ) (= C p (X )) is not distinguished.
Recall that any Haydon space X is a subspace of βN with X ⊃ N.
Another example is related with the paper [20].

Example 2
The space ω * = (βN \ N) contains a dense countably compact subspace X such that every compact subset of X is finite and C k (X ) (= C p (X )) is not distinguished.
The subspace ( ∞ ) p = {(x n ) ∈ R N : sup n |x n | < ∞} of R N is distinguished. For all Haydon spaces X there exists a continuous open linear map from C p (X ) onto ( ∞ ) p (see [23,Theorem 1.3]), but for some Haydon spaces X the space C p (X ) is not distinguished (by Example 1). We note also the following more general fact involving the case C k (X ). Proposition 1 Let X be a subspace of βN with X ⊃ N. Then In particular, the spaces C p (X ) and C k (X ) have infinite-dimensional quotients that are distinguished although these spaces can be not distinguished.
We assume that all locally convex spaces are Hausdorff and over the field R of real numbers.

A few facts about distinguished spaces
A locally convex space E is called quasi-normable [34,Definition 8.3.34] if for every neighbourhood of zero U in E there exists a neighbourhood of zero V such that for every λ > 0 there exists a bounded set B in E with V ⊆ B + λU .
The class of quasi-normable spaces contains, for example, (D F)-spaces, metrizable spaces C k (X ), spaces C n ( ) for open subsets of R N , as well as all Fréchet-Montel spaces (see [19,33]).
Grothendieck showed that a metrizable locally convex space E is distinguished if and only if its strong dual E β = (E , β(E , E)) is bornological, [34,Theorem 8.3.44]. Heinrich [18] observed that each metrizable quasi-normable locally convex space satisfies the density condition what implies that every metrizable quasi-normable locally convex space is distinguished. In particular, the strong dual of a distinguished metrizable locally convex space can be described as a regular (L B)-space, see [34, Observation 8.5.14 (e)].
Since each space C k (X ) is quasi-normable (see [19, 10.8.2 Theorem]) and any metrizable quasi-normable space is distinguished, we have Note however that non-metrizable and distinguished spaces C k (X ) which are not strongly distinguished do exist, see Example 7.
Proposition 2 implies the following

Theorem 3 C k (X ) is distinguished for any locally compact paracompact space X .
Proof It is known that X is the direct sum ⊕ t∈T X t of locally compact σ -compact spaces, see [9].
Note that the strong dual C k (X ) β of C k (X ) is a complete strict (L B)-space, i.e., complete strict inductive limit of a sequence of Banach spaces what is a consequence of the fact that C k (X ) β is bornological (by applying the Grothendieck theorem, see [27, 23.7, 29.3]).
Below we provide two concrete examples of metrizable dense subspaces of R N which are strongly distinguished.

Example 3
The spaces ( ∞ ) p and (c 0 ) p = {(x n ) ∈ R N : x n → 0} with the topologies induced from R N are dense large subspaces of R N , so they are strongly distinguished. On the other hand, does not exist a Tychonoff space X such that ( ∞ ) p is isomorphic to C p (X ). In fact, as easily seen, Distinguished spaces C p (X ) are totally different from distinguished spaces C k (X ), see Theorem 6. Recall first the following general Theorem 4 [15,36] The following assertions are equivalent for any locally convex space E.
(1) E has the finest locally convex topology.
(2) Every absolutely convex absorbing subset of E is a neighbourhood of zero.
(3) E is the strong dual of the product R dim E . (4) E is barrelled and admits a continuous basis [36].
Clearly, any linear functional on a linear space E is continuous in the finest locally convex topology ξ on E; if E is infinite-dimensional, then ξ is non metrizable (since every metrizable infinite dimensional locally convex space admits a discontinuous linear functional).
On the other hand, in [15, Theorem 9] we proved

Theorem 5 The homeomorphic copy of X in the dual C p (X ) with the weak * -topology is a continuous basis in L β (X ).
Recall (see [22]) that a locally convex space E is called primitive if for any increasing sequence (E n ) of linear subspaces of E with ∞ n=1 E n = E, a linear functional f on E is continuous if and only if its restrictions to E n , n ∈ N, are continuous.
Proofs of some implications are original.

Theorem 6
For a Tychonoff space X the following are equivalent: (3) C p (X ) and R X have the same strong duals.
(3) ⇒ (9) follows by reflexivity of R X . (9) ⇒ (10) is obvious. ∞)) is an open neighbourhood of x in X . Put U n = x∈X n V x for n ∈ N. Clearly, (U n ) is a decreasing sequence of open subsets of X and X n ⊂ U n for n ∈ N. Let n ∈ N. For x ∈ X n and z ∈ V x we have g x (z) > n, so sup g∈B |g(z)| > n for z ∈ U n . Let y ∈ X .
For some m ∈ N we have sup g∈B |g(y)| ≤ m, since B is bounded in C p (X ). It follows that y / ∈ U m , so y / ∈ ∞ n=1 U n . Thus ∞ n=1 U n = ∅.
Then (X n ) is a decreasing sequence of subsets of X with empty intersection. Thus there exists a decreasing sequence (U n ) of open subsets of X with an empty intersection such that X n ⊂ U n for n ∈ N. For any We have ψ(x) < ϕ(x) for every x ∈ X . Indeed, let x ∈ X and n ∈ N with n − 1 ≤ |ψ(x)| < n. Then x ∈ X n ⊂ U n and x / ∈ U ϕ(x)+1 , so n < ϕ(x) + 1. Thus ψ(x) < n ≤ ϕ(x). Clearly, the set B = {g ∈ C p (X ) : |g| ≤ ϕ} is bounded in C p (X ). We shall prove that A ⊂ cl R X (B). Let f ∈ A. Let W be a neighbourhood of f in R X . Then there exists a finite subset K of X such that Let {V x : x ∈ K } be a family of pairwise disjoint open subsets of X with For every x ∈ K there exists a continuous function  Let (X n ) be a decreasing point-finite sequence of subsets of X 0 = X . Put P n = [X n−1 \X n ], n ∈ N. Then (P n ) is a partition of X . Let (U n ) be a point-finite open expansion of (P n ) and V n = ∞ m=n+1 U m , n ∈ N. Then (V n ) is a decreasing point-finite open expansion of (X n ), since X n = ∞ m=n+1 P m ⊂ ∞ m=n+1 U m = V n , n ∈ N.
On the other hand, notice the following fact implying that the item (11) in Theorem 6 cannot be replaced by M β (X ) is Baire. [12,Theorem 3.9] proved that the strong dual L β (X ) of C p (X ) is always distinguished (i.e. M β (X ) is always barrelled) and then Ferrando and Saxon asked [16,Problem 11] if the Baire property of M β (X ) implies that C p (X ) is distinguished. The answer is negative (as noticed in [16,Addendum]) since M(ω 1 ) is a Baire space [16, Corollary 21] but C p (ω 1 ) is not distinguished [24].

Remark 1 Ferrando and Kakol
For spaces C k (X ) note the following simple

Proposition 3 Let X be a Tychonoff space. The space C k (X ) is feral if and only if X is finite.
Proof Clearly, C k (X ) is feral if X is finite. Assume that X is infinite. If X is pseudocompact, then C k (X ) admits a stronger normed topology. If X is not pseudocompact, then C k (X ) contains an isomorphic copy of R N ([21, Theorem 2.12]). Thus C k (X ) is not feral.

Feral dual spaces
Two locally convex spaces E and F are called bornologically isomorphic if there exists a linear bijective map T : E → F such that T and T −1 are locally bounded; E is a free locally convex space if E carries the finest locally convex topology.
In order to prove Theorem 2 we need the following simple fact which is probably known.

Lemma 1 Let E and F be locally convex spaces. Assume that there exists a continuous linear surjection T : E → F which is bounded covering, i.e. for every bounded set B in F there exists a bounded set A B in E with T (A B ) = B. Then the strong dual E β of E contains an isomorphic copy of the strong dual F β of F.
Proof The adjoint map T * : for any ξ ∈ F β . It follows that T * is an isomorphism onto its range, so E β contains an isomorphic copy of F β .
Dealing with locally convex spaces we have the following characterization for quasibarrelled spaces carrying the weak topology.

Theorem 7 For a locally convex space E the following are equivalent:
(1) E β is feral.
(1) ⇒ (3) First we show that E μ = (E, μ(E, E )) is quasi-barrelled. Let B be a bounded subset of E β . Then B is finite-dimensional, so the closure W of the absolutely convex hull of B in (E , σ (E , E)) is compact. Thus the set Let E ν be the space E with the finest locally convex topology. The identity map I : E ν → E β is a continuous linear surjection. Since every bouned subset of E β is finite-dimensional, I is bounded covering. By Lemma 1 we derive that the strong bidual (E β ) β is isomorphic to a subspace of (E ν ) β . On the other hand, E ν is the direct sum t∈X R, where X is a Hamel basis of E ; so its strong dual (E ν ) β is isomorphic to the product R X , which clearly carries the weak topology. Hence (E β ) β = (E , β(E , E ) carries the weak topology and β(E , E )| E = σ (E , E), since every subspace of a locally convex space with the weak topology has the weak topology.
Thus σ (E, E ) = μ(E, E ), so E is quasi-barrelled and carries the weak topology.
(3) ⇒ (1) Let X be a Hamel basis of E with the topology induced from (E , σ (E , E)). The linear map ψ : is an isomorphism between E and ψ(E), and ψ(E) is dense in C p (X ). Thus we can identify E with a dense subspace of the product R X and E with (R X ) .
Let B be a bounded subset of E β . Since E is quasi-barrelled, B is equicontinuous, so there exists a neighbourhood U of zero in E such that B ⊂ U • ⊂ E = (R X ) . Clearly, the closure V of U in R X is a neighbourhood of zero in R X . Let f ∈ B and v ∈ V . Then there exists a net (u t ) t∈T in U convergent to v in R X . Hence the net ( f (u t )) t∈T is convergent to [19,Corollary 11.7.3]) and carries the weak topology, Theorem 7 applies the following known result (see [13]).

Corollary 4 Let X be a Tychonoff space. The strong bidual M β (X ) of C p (X ) is feral if and only if X is finite.
Proof (⇒) The strong dual L β (X ) of C p (X ) is quasi-barrelled (by Theorem 7) and complete (by [12,Proposition 3.10]). Consequently, L β (X ) is barrelled, so M β (X ) is the product R X (by Theorem 6). Since M β (X ) is feral, X is finite. (⇐) is obvious.
The following proposition provides a stronger version of Corollary 1.

Proposition 4 Let E be a locally convex space. If there exists a Tychonoff space X and a locally bounded linear map T from C p (X ) onto E, then the strong dual E β of E is feral.
Proof Let υ X be the realcompactification of the space X .

Corollary 5 Let X and Y be Tychonoff spaces.
If there exists a locally bounded linear map T from C p (X ) onto C k (Y ), then every compact subset of Y is finite.
Proof By Proposition 4 the strong dual of C k (Y ) is feral. Hence, by Theorem 2, each compact subset of Y is finite. Proposition 4 suggests the following.

Problem 2 Does there exist a locally convex space E whose strong dual is feral such that E is not continuous [not locally bounded] linear image of C p (X ) for any Tychonoff space X ?
Recall that the strongly distinguished space ( ∞ ) p is not isomorphic to any space C p (X ) Theorem 7 the space C k (X ) carries the weak topology, so C k (X )/M carries the weak topology, too. We have a contradiction, since C k (X )/M is isomorphic to the infinite-dimensional Banach space C k (K ). (⇐) follows by Corollary 3, since C k (X ) = C p (X ).
(II). Assume that E = C k (X ) is quasi-barrelled. Then E is a subspace of (E β ) β ([19, Proposition 11.2.2]). Hence, if (E β ) β is feral, then E is feral, too; so X is finite, by Proposition 3. The converse is obvious.
Another consequence of Theorem 7 yields the following

Corollary 6 An infinite-dimensional metrizable locally convex space E is strongly distinguished if and only if it is isomorphic to a dense subspace of R N . Hence a Fréchet space F is strongly distinguished if and only if F is finite-dimensional or F is isomorphic to R N .
Proof (⇒) By Theorem 7 and its proof, E carries the weak topology and it is isomorphic to a dense subspace of R X for some infinite set X . E is metrizable, so X is countable.
(⇐) By [34, Observation 8.3.23 (b)] E is large in R N , so the strong duals of E and R N coincide and are isomorphic to ϕ, the ℵ 0 -dimensional linear space with the finest locally convex topology.
Recall that a locally convex space admits a fundamental bounded resolution if there exists a family {B α : α ∈ N N } of bounded sets such that B α ⊂ B β for all α ≤ β and each bounded set in E is contained in some B α . Clearly every metrizable locally convex space admits such resolution, see [21,Lemma 15.2]. A locally convex space E has a N N -base if there exists a base {U α : α ∈ N N } of neighborhoods of zero such that U β ⊂ U α for all elements α ≤ β in N N . Clearly, every metrizable locally convex space has an N N -base, [21].
In [11,Theorem 3.3] we proved that C p (X ) admits a fundamental bounded resolution if and only if X is countable (if and only if C p (X ) is metrizable).
Using Theorem 7 we extend the above result and supplement Corollary 6.

Corollary 7 An infinite-dimensional quasi-barrelled locally convex space E is isomorphic to a dense subspace of R N if and only if E has a fundamental bounded resolution and E caries the weak topology.
Proof (⇒) is clear. (⇐) By the proof of Theorem 7, E is isomorphic to a dense subspace of R X for some infinite set X . Assume that X is uncountable.

Proof of Proposition 1 and Examples
Proof of Proposition 1 (1) Consider three cases: (1.1). X is not pseudocompact. Then the space C k (X ) contains a complemented copy of the space R N , see [21,Theorem 2.12] and [34,Corollary 2.6.5]. Hence there exists a continuous open linear map from C k (X ) onto R N .
(1.2). X is pseudocompact and every compact subset of X is finite. Since N ⊂ X ⊂ βN, then N is C * -embedded into X . Applying [3, Theorem 1] we get that C p (X ) has a quotient C p (X )/W isomorphic to the subspace ( ∞ ) p of R N (endowed with the product topology), where W = n { f ∈ C p (X ) : x∈F n f (x) = 0} and (F n ) is a sequence of non-empty, finite and pairwise disjoint subsets of N with lim n |F n | = ∞. Hence C k (X )(= C p (X )) admits a continuous open linear map onto ( ∞ ) p .
(1.3). X is pseudocompact and contains an infinite compact subset K . By [23, Proposition 2.9] the restriction map T : If X is pseudocompact, then C p (X ) has a quotient isomorphic to ( ∞ ) p (see (1.2)), so C p (X ) admits a continuous open linear map onto ( ∞ ) p .
Below we provide concrete situations where C k (X ) is distinguished but its strong dual C k (X ) β is not feral.

Example 4
Let X be an uncountable hemicompact space. Then the space C k (X ) is distinguished but its strong dual C k (X ) β is not feral.
Proof X is hemicompact, i.e. X is covered by a sequence of compact sets (K n ) such that each compact set in X is contained in some K n , so C k (X ) is metrizable. Applying Proposition 2 we infer that C k (X ) is distinguished. Since X is uncountable, for some n ∈ N the set K n is infinite. By Theorem 2, C k (X ) β is not feral.
A topological space X is called a Q-space if each subset of X is G δ . Recall that a normal space X is a Q-space if and only if X is strongly splittable, i.e. for every f ∈ R X there exists a sequence ( f n ) n in C p (X ) such that f n → f in R X , see [39,Problems 445,447]. Using Theorem 2 one gets the following Proposition 5 For a normal space X the assertions are equivalent.
(1) X is a Q-space and every compact subset of X is finite.
is a neighbourhood of zero in C p (X ). Thus the topological spaces C k (X ) and C p (X ) are equal, so any compact subset of X is finite.
Example 1 uses the following scheme of constructing uncountable pseudocompact spaces without infinite compact subsets due to Haydon, [17].
A is an infinite subset of N}) be topologized as a subspace of βN. To simplify the notation we will call such spaces the Haydon spaces. It is known that each Haydon space X is an uncountable pseudocompact space and each compact subset of X is finite, see [17].
A point x of a topological space X is said to be a a weak P-point if for any countable subset F of (X \ {x}) we have x / ∈ F. Clearly, any countable set of weak P-points of X is discrete. By [30], see also [20], the set of all weak P-points of the space ω * is dense in ω * .
Thus for any infinite subset A of N there exists an element u A ∈ A ∩ ω * , that is a weak P-point of X . Then any countable subset of the set Z := {u A : A is an infinite subset of N} is discrete. Thus any countable subset of the Haydon space Y = (N ∪ Z ) is scattered.
We will use the following two results. In [24] Kakol and Leiderman proved the following

Proof of Example 1
The compact space ω * has no isolated points. It is well-known (and easy to prove), that every compact space without isolated points contains a non-empty countable subset without isolated points, too. Thus there exists a non-empty countable subset P of ω * without isolated points; clearly, P is not scattered. Let P = {p n : n ∈ N} and A n = {k ∈ N : k > n} for n ∈ N. Since p n ∈ (A n \ N), n ∈ N, there exists a Haydon space X , that contains P. By Theorem 9, the space C k (X ) (= C p (X )) is not distinguished.

Proof of Example 2
In [20] Juhasz and van Mill proved that the space ω * contains a dense subspace X such that X is countably compact and non-scattered but all countable subsets of X are scattered. Observe that every compact subset of X is finite. Indeed, if a compact subset A of X ⊂ β N is infinite, then A contains a copy of βN, but β N contains a countable subset which is not scattered; a contradiction. By Theorem 8, the space C k (X ) (= C p (X )) is not distinguished. Example 6 Every uncountable-dimensional subspace of C k (R N ) contains a metrizable compact infinite-dimensional set and the strong dual of C k (R N ) admits an infinite-dimensional compact set. In particular, C k (R N ) and its strong dual are not feral.
Proof The first claim follows from the fact that C k (R N ) admits an N N -base (by [10]) and then we apply [4,Theorem 1.2]. The other claim follows from Theorem 2.
By the theorem of Heinrich (see Sect. 2), we know that every quasi-normable metrizable locally convex space is distinguished.

Problem 3 Is every quasi-normable locally convex space with a N N -base a distinguished space?
Note that every (L B)-space is quasi-normable ( [34]) and each (L B)-space has a N N -base ( [21]). Moreover, each space C p (X ) is quasi-normable ( [15]) and C p (X ) has an N N -base if and only if C p (X ) is metrizable ( [21]). Recall also that C k (R N ) is quasi-normable and has an N N -base by applying the main theorem of [10].

Example 7
There exists a non-metrizable distinguished space C k (X ) which is not strongly distinguished.
Proof By [21, Example 2.4], there exists a Tychonoff space X such that C k (X ) is a (d f )space but not (D F)-space. Then C k (X ) is not metrizable, since any metrizable (d f )-space is a (D F)-space. By [21, Theorem 2.14], the strong dual of C k (X ) is a Fréchet space, so C k (X ) is distinguished but not strongly distinguished.
Note also that C k (R N ) is not a (d f )-space; it is even not covered by a sequence of bounded sets. Indeed, this follows directly from [23, Lemma 2.3].

Problem 4
Is the space C k (X ) distinguished when X is metrizable? In particular, are the spaces C k (R N ) and C k (Q) distinguished?

Problem 5 Characterize distinguished spaces C k (X ) in terms of X .
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