Equicontinuity criteria for metric-valued sets of continuous functions

Abstract

Abstract. Combining ideas of Troallic [20] and Cascales, Namioka, and Vera [3], we prove several characterizations of almost equicontinuity and hereditarily almost equicontinuity for subsets of metric-valued continuous functions when they are defined on aČech-complete space. We also obtain some applications of these results to topological groups and dynamical systems.

Equi-Baire 1 families of functions

“…Note that in [7] the authors defined the notion of almost equicontinuity for a family of continuous functions, which is similar to the notion of equi-cliquishness. Definition 6.3.…”

“…Definition 6.3. [7] Let X and (M, d) be a Hausdorff, completely regular space and a metric space, respectively, and let C(X, M ) denote the set of all continuous functions from X to M . A subset G ⊂ C(X, M ) is said to be almost equicontinuous if G is equicontinuous on a dense subset of X.…”

Interpolation sets in spaces of continuous metric-valued functions

“…By Theorem B, this implies that L| Y is equicontinuous on Y . Applying [16,Theorem B], it follows that G is hereditarily equicontinuous on X, which implies that G is equicontinuous because G consists of group homomorphisms.…”

“…Since any compact topology is (Hausdorff) minimal, this would imply that the Bohr topology would coincide with the original topology of G on the compact subset A G and, as a consequence, on A. Thus A would have property P in G.So, assume wlog that A is not precompact in G. As in the proof of Theorem D, if we take the abelianČech-complete group G and inject G in C( G, T) by means of the evaluation map E : G → G ⊆ C( G, T), it follows that A is not equicontinuous on G. By[16, Cor. 2.4], it follows that there exists a countable subset F ⊆ A and a separable compact subset X ⊆ G such that F is not equicontinuous on X.…”