*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.

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

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