Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Separation functions and mild topologies

created by mennucci on 04 Dec 2021
modified on 29 Mar 2023


Published Paper

Inserted: 4 dec 2021
Last Updated: 29 mar 2023

Journal: Analysis and Geometry in Metric Spaces
Year: 2023
Doi: 10.1515/agms-2022-0149


Given $M$ and $N$ Hausdorff topological spaces, we study topologies on the space \(C^0(M; N )\) of continuous maps \(f:M\to N\). We review two classical such topologies, the 'strong' and '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 \(f:M\to N\), and affine actions when \(N={\mathbb R}^n\).

To define the 'mild topology' we use 'separation functions'.

'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; with some additional hypotheses we can define 'set separation functions' that prove that the topology is T6; with some more hypotheses it is possible to prove that the topology is metrizable.

We provide some examples of usages of separation functions: one is the aforementioned case of the mild topology on \(C^0(M; N )\). Another one can be used on topological manifolds.

Keywords: continuous/proper functions, strong/weak topology, quasi metrics, manifolds