Topologies between compact and uniform convergence on function spaces

Abstract

ABSTRACT. This paper studies two topologies on [6] to be the proper setting to study sequences of functions which converge uniformly on compact subsets. One of the distinguishing features of this topology is that whenever x is locally compact the compact-open topology on C(X) is the coarsest topology making the evaluation map e:X C(X)--. continuous (where e(x,])= f(x)).The compact-open topology and the topology of uniform convergence are equal if and only if x is compact. Because compactness is such a strong condition, there is a considerable gap between these two topologies. This gap was especially felt in [8] while studying the completeness of a normed linear space of continuous linear functionals on C(X) with the compact-open topology.Because of this, a new class of topologies was introduced in [7] on C*(X) to bridge the gap, where c*(x) is the set of bounded functions in C(x). This also generalized the a-compact-open topology

Quasi-uniform convergence topologies on function spaces- Revisited

“…[28], p. 9) in the case of Y a uniform space (see also [23]). These S A topologies are, in general, different from their corresponding uniform convergence topologies U A even in the case of Y a metric space.…”

On hereditary precompactness and completeness in quasi-uniform spaces

“…Let X be a Tychonoff space and λ ⊆ P(X). In addition to the t λ -topology on C(X, Y ), we can define (following the terminology of [22,17,29]) the notion of t λ * -topology on C(X, Y ) which has a subbase as the collection {N Here the modification f (A) ⊆ G in place of f (A) ⊆ G is due to McCoy and Ntantu ( [22]) who used it in order to generalize the compact-open topology to real-valued noncontinuous functions, to balance the disadvantage A is compact but f (A) is not compact. In this regard, we can also consider the topology of uniform convergence on elements of λ (the λ-topology) on C(X, Y ), denoted by C λ,u (X, Y ), which has a base at each f ∈ C(X, Y ) as the collection {< f, A, ε >: A ∈ λ, ε > 0}, where…”

“…other set-open topologies t λ (λ ⊆ P (X)) were investigated that lie between t k and t w (the largest set-open topology) (see, e.g., [9,10,15,17,5,27,29]). …”

The Generalized Pre-Open Compact Topology on Function Spaces

“…Recently in this direction, there have been some topologies introduced like σ-compact open topology, the topology of uniform convergence on σ-compact, the topology of uniform convergence on bounded subsets, bounded-open topology, pseudocompact-open topology, C-compact-open topology and compact-G δ -open topology. For more details we can see [6,7,9,10,11,12,13,14,15,18].…”