# Separation functions and mild topologies

created by mennucci on 04 Dec 2021

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Submitted Paper

Inserted: 4 dec 2021
Last Updated: 4 dec 2021

Year: 2021

Abstract:

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.