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.
We show that the fact that X has a compact resolution swallowing the compact sets characterizes those C c .X/ spaces which have the so-called G-base. So, if X has a compact resolution which swallows all compact sets, then C c .X / belongs to the class G of Cascales and Orihuela (a large class of locally convex spaces which includes the (LM) and (DF)-spaces) for which all precompact sets are metrizable and, conversely, if C c .X/ belongs to the class G and X satisfies an additional mild condition, then X has a compact resolution which swallows all compact sets. This fully applicable result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space C c .X / in terms of topological properties of X and leads to a nice theorem of Cascales and Orihuela stating that for X containing a dense subspace with a compact resolution, every compact set in C c .X / is metrizable.In what follows, unless otherwise stated, X will be a Hausdorff completely regular space and C p .X/ and C c .X / will denote the space C.X/ of all real-valued continuous functions defined on X provided with the pointwise convergence topology and with the compact-open topology, respectively.Let us recall that a family A D ¹A˛W˛2 N N º of subsets of a set X is called a resolution of X if S ¹A˛W˛2 N N º D X and A˛Â Aˇfor˛Ä( coordinatewise), see [9, Chapter 3]. A resolution A of a topological space X is called compact if it consists of compact sets.A locally convex space (lcs) E is said to have a G-base (or a G-basis) (see [9, Chapter 1]) if there exists a basis ¹U˛W˛2 N N º of (absolutely convex) neighborhoods of the origin in E, such that UˇÂ U˛whenever˛Äˇ.An lcs E is said to belong to the class G if its topological dual E 0 has a resolution ¹A˛W˛2 N N º such that for every˛2 N N each sequence in A˛is
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}.
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