Compact subsets of C λ,u (X)

The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in C λ,u (X) for a locally-λ space X when λ ⊇ 𝓕(X), for a hemi- λ λf -space X when λ ⊆ 𝓟 𝓢(X), and for a k-space X when λ ⊇ 𝓚(X). This paper also studies that every bounded subset of C λ,u (X) has compact closure for some classes of topological spaces X.