On isocompactness of function spaces (original) (raw)
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Note on function spaces with the topology of pointwise convergence
Archiv der Mathematik, 2003
The note contains two examples of function spaces C p (X) endowed with the pointwise topology. The first example is C p (M), M being a planar continuum, such that C p (M) m is uniformly homeomorphic to C p (M) n if and only if m = n. This strengthens earlier results concerning linear homeomorphisms. The second example is a non-Lindelöf function space C p (X), where X is a monolithic perfectly normal compact space all linearly orderable closed subspaces of which are metrizable. This example is obtained under the additional set-theoretical axiom ♦. This solves a problem of Arhangel skiȋ.
Compact covers and function spaces
Journal of Mathematical Analysis and Applications, 2014
For a Tychonoff space X, we denote by C p (X) and C c (X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of C p (X), we extend Okunev's results by showing that if there exists a surjection from C p (X) onto C p (Y) (resp. from L p (X) onto L p (Y)) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from C c (X) onto C c (Y), then Y is a separable completely metrizable space. We study also the non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by p-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and p-equivalent to a first countable Y , then Y is metrizable (V. Valov). Some additional results are presented.
Topology and its Applications, 1997
For a completely regular space X and a normed space E let Ck (X, E) (respectively C,(X, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (respectively pointwise convergence) topology. We prove that some topological properties P satisfy the following conditions: (1) if Ck(X, E) and Ck(Y,F) (respectively C&(X, E) and C,(Y,F)) are linearly homeomorphic, then X E P if and only if Y E P; (2) if there is a continuous linear surjection from Ck (X, E) onto Cp(Y, F), then Y E P provided X E P; (3) if there is a continuous linear injection from Ck (X, E) into C,(Y, F), then X has a dense subset with the property P provided Y has a dense subset with the same property. 0 1997 Elsevier Science B.V.
The pseudocompact-open topology on C (X)
2006
Abstract. This paper studies the pseudocompact-open topology on the set of all real-valued continuous functions on a Tychonoff space and compares this topology with the compactopen topology and the topology of uniform convergence. In the second half, the induced map, as well as the metrizability of this topology, is studied.
On countable bounded tightness for spaces Cp(X)
Journal of Mathematical Analysis and Applications, 2003
It is well known that the space C p ([0, 1]) has countable tightness but it is not Fréchet-Urysohn. Let X be a Cech-complete topological space. We prove that the space C p (X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet-Urysohn if and only if C p (X) has countable bounded tightness, i.e., for every subset A of C p (X) and every x in the closure of A in C p (X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided.
Closure properties of function spaces
Applied General Topology, 2003
In this paper we investigate some closure properties of the space Ck(X) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology.
Basic properties of XXX for which spaces Cp(X)C_p(X)Cp(X) are distinguished
2021
In our paper [18] we showed that a Tychonoff space X is a ∆-space (in the sense of [20], [30]) if and only if the locally convex space Cp(X) is distinguished. Continuing this research, we investigate whether the class ∆ of ∆-spaces is invariant under the basic topological operations. We prove that if X ∈ ∆ and φ : X → Y is a continuous surjection such that φ(F ) is an Fσ-set in Y for every closed set F ⊂ X , then also Y ∈ ∆. As a consequence, if X is a countable union of closed subspaces Xi such that each Xi ∈ ∆, then also X ∈ ∆. In particular, σ-product of any family of scattered Eberlein compact spaces is a ∆-space and the product of a ∆-space with a countable space is a ∆-space. Our results give answers to several open problems posed in [18]. Let T : Cp(X) −→ Cp(Y ) be a continuous linear surjection. We observe that T admits an extension to a linear continuous operator T̂ from R onto R and deduce that Y is a ∆-space whenever X is. Similarly, assuming that X and Y are metrizable s...
A characterization of XXX for which spaces Cp(X)C_p(X)Cp(X) are distinguished and its applications
Proceedings of the American Mathematical Society, Series B, 2021
We prove that the locally convex space C p (X) of continuous realvalued functions on a Tychonoff space X equipped with the topology of pointwise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45-60]. As an application of this characterization theorem we obtain the following results: 1) If X is aČech-complete (in particular, compact) space such that C p (X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell-Mrówka type X, the space C p (X) is distinguished. 3) If X is the compact space of ordinals [0, ω 1 ], then C p (X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that C p (X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.
Countability properties of the pseudocompact-open topology on C (X): A comparative study
2007
The main goal of this paper is to study the countability properties, such as the countable chain condition, Lindeöf property and second countability of the pseudocompact-open topology on C (X), the set of all continuous real-valued functions on a Tychonoff space X. But in order to make this study fruitful, these countability properties of the pseudocompact-open topology are compared with those of the point-open and compact-open topologies on C (X).