Given M and N Hausdorff topological spaces, we study topologies on the space C-0 (M; N) of continuous maps f : M? N. We review two classical topologies, the "strong" and the "weak" topology. We propose a definition of "mild topology" that is coarser than the "strong" and finer than the "weak" topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M? N, and affine actions when N = R-n. To define the "mild topology" we use "separation functions;" these "separation functions" are somewhat similar to the usual "distance function d(x, y)" in metric spaces (M, d), but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define "set separation functions" that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C-0(M; N). Other examples are the Sorgenfrey line and the topology of topological manifolds.

Separation functions and mild topologies

Mennucci, Andrea Carlo Giuseppe
2023

Abstract

Given M and N Hausdorff topological spaces, we study topologies on the space C-0 (M; N) of continuous maps f : M? N. We review two classical topologies, the "strong" and the "weak" topology. We propose a definition of "mild topology" that is coarser than the "strong" and finer than the "weak" topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M? N, and affine actions when N = R-n. To define the "mild topology" we use "separation functions;" these "separation functions" are somewhat similar to the usual "distance function d(x, y)" in metric spaces (M, d), but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define "set separation functions" that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C-0(M; N). Other examples are the Sorgenfrey line and the topology of topological manifolds.
2023
Settore MAT/05 - Analisi Matematica
continuous functions; proper maps; strong topology; weak topology; metrization; quasi metrics; asymmetric metrics; topological manifolds
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/129262
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