Topological Properties of Spaces of Continuous Functions

Abstract. In this paper we extend the theory of strong uniform continuity and strong uniform convergence, developed in the setting of metric spaces in [13,14], to the uniform space setting, where again the notion of shields plays a key role. Further, we display appropriate bornological/variational modifications of classical properties of Alexandroff [1] and of Bartle [7] for nets of continuous functions, that combined with pointwise convergence, yield continuity of the limit for functions between metric spaces.2000 AMS Classification: Primary 40A30, 54C35; Secondary 54E15, 54C08

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“…Let B be a bornology on X. The classical topology of uniform convergence T B on Y X [22] has as a base for its entourages…”

Section: Preliminariesmentioning

confidence: 99%

The topology of theρ-hausdorff distance

Attouch

Lucchetti

Wets

1991

Annali di Matematica pura ed applicata

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“…The study of the above concepts for the function spaces with various topologies has a rich history and is also nowadays an active area of research, see [1,16,19,30] and references therein. For Tychonoff topological spaces X and Y , we denote by C k (X, Y ) and C p (X, Y ) the space C(X, Y ) of all continuous functions from X into Y endowed with the compact-open topology or the pointwise topology, respectively.…”

Section: Introductionmentioning

confidence: 99%

The Ascoli property for function spaces

Gabriyelyan

Grebík

Ka̧kol

et al. 2016

Topology and its Applications

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Abstract. The paper deals with Ascoli spaces Cp(X) and C k (X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of C k (X) is evenly continuous, essentially includes the class of k R -spaces. First we prove that if Cp(X) is Ascoli, then it is κ-Fréchet-Urysohn. If X is cosmic, then Cp(X) is Ascoli iff it is κ-Fréchet-Urysohn. This leads to the following extension of a result of Morishita: If for aČech-complete space X the space Cp(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then Cp(X) is Ascoli iff X is scattered. (b) If X is aČech-complete Lindelöf space, then Cp(X) is Ascoli iff X is scattered iff Cp(X) is Fréchet-Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent:is an Ascoli space. The Asoli spaces C k (X, I) are also studied.

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“…The cellularity c(Z), on the other hand, is defined as the supremum of the cardinalities of all systems of pairwise disjoint open sets. Firstly, it follows from [HV1, Theorem 3.8] and from [MN,Theorem 4.5.2] of M c C o y and N t a n t u that the weight satisfies: …”

Section: Some Notes On Densely Continuous Forms Peter Vadovičmentioning

confidence: 99%