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Denote by C k [M] the C k -stable closure of the class M of all metrizable spaces, i.e., C k [M] is the smallest class of topological spaces that contains M and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces C k (X, Y ) with Lindelöf domain in this class. We show that the class C k [M] coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces C k (X, Y ) with a separable metrizable space X and a metrizable space Y . We say that a topological space Z is Ascoli if every compact subset of C k (Z) is evenly continuous; by the Ascoli Theorem, each k-space is Ascoli. We prove that the class C k [M] properly contains the class of all Ascoli ℵ 0 -spaces and is properly contained in the class of P-spaces, recently introduced by Gabriyelyan and Kakol. Consequently, an Ascoli space Z embeds into the function space C k (X, Y ) for suitable separable metrizable spaces X and Y if and only if Z is an ℵ 0 -space.1991 Mathematics Subject Classification. 46E10 and 54C35 and 54E18.

Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent:We prove that L(X) is a (DF )-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF )-space but is not a c0-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df )-space but is not a quasi-(DF )-space. If X is a µ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford-Pettis property for every Tychonoff space X. If X is a sequential µ-space (for example, metrizable), then L(X) has the sequential Dunford-Pettis property iff X is discrete.2000 Mathematics Subject Classification. Primary 46A03, 46A08; Secondary 54C35.

Abstract. We define and study the free topological vector space V(X) over a Tychonoff space X. We prove that V(X) is a kω-space if and only if X is a kω-space. If X is infinite, then V(X) contains a closed vector subspace which is topologically isomorphic to V(N). It is proved that if X is a k-space, then V(X) is locally convex if and only if X is discrete and countable. If X is a metrizable space it is shown that: (1) V(X) has countable tightness if and only if X is separable, and (2) V(X) is a k-space if and only if X is locally compact and separable. It is proved that V(X) is a barrelled topological vector space if and only if X is discrete. This result is applied to free locally convex spaces L(X) over a Tychonoff space X by showing that: (1) L(X) is quasibarrelled if and only if L(X) is barrelled if and only if X is discrete, and (2) L(X) is a Baire space if and only if X is finite.

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The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space C c (X) of all continuous real-valued functions with the compact-open topology on some classes of topological spaces X including Čech-complete Lindelöf spaces. Being motivated also by several results providing various concepts of networks we introduce the class of P-spaces strictly included in the class of ℵ-spaces. This class of generalized metric spaces is closed under taking subspaces, topological sums and countable products and any space from this class has countable tightness. Every P-space X has the strong Pytkeev property. The main result of the present paper states that if X is an ℵ 0 -space and Y is a P-space, then the function space C c (X, Y ) has the strong Pytkeev property. This implies that for a separable metrizable space X and a metrizable topological group G the space C c (X, G) is metrizable if and only if it is Fréchet-Urysohn. We show that a locally precompact group G is a P-space if and only if G is metrizable.

Following [3] we say that a Tychonoff space X is an Ascoli space if every compact subset K of C k (X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k R -space, hence any k-space, is Ascoli.Let X be a metrizable space. We prove that the space C k (X) is Ascoli iff C k (X) is a k R -space iff X is locally compact. Moreover, C k (X) endowed with the weak topology is Ascoli iff X is countable and discrete.Using some basic concepts from probability theory and measure-theoretic properties of ℓ 1 , we show that the following assertions are equivalent for a Banach space E: (i) E does not contain isomorphic copy of ℓ 1 , (ii) every real-valued sequentially continuous map on the unit ball B w with the weak topology is continuous,We prove also that a Fréchet lcs F does not contain isomorphic copy of ℓ 1 iff each closed and convex bounded subset of F is Ascoli in the weak topology. However we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff either F is finite dimensional or F is isomorphic to the product K N , where K ∈ {R, C}.