This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms

The “incipit” of the homeomorphism group theory resides in the early seminal work of Birkoff [

As rim-compactness is a weak and peripherical compactness property, one might think any further relaxation as impossible. But, we show that rim-compactness for

The issues so far discussed lead us to show:

On the other hand, if

The uniform topologies so far considered are totally bounded, and the concept of totally bounded uniformity can be dually recast as EF-proximity and then as strong inclusion, [

Again in local compactness, in the paper [

Firstly, we give some useful background and summarise a number of already stated basic facts. Definitions and terminology quoted below are drawn by [

Let

A net

Topologies on

Any admissible topology on

Let

Topologies on

Let

Let

Let

Of course, every admissible group topology makes the evaluation function as a group action.

There is always on

Of course, every admissible group topology on

Let

if

Secondly, we differentiate the topologies on

Let

Let

Let

It is known that, having been given a topological characterisation, the fine topology

Let

In [

In the metric setting three of the examined methods collapse in just one because of the two following circumstances. The former one is why the open-cover topology and the limitation topology agree: any open cover in a metric space

If

Implicitly due to Birkhoff, a natural way to get admissible group topologies works efficiently. Whenever

Let

Starting with a totally bounded uniformity we construct a

Let

Let

The family

The uniformity

Any self-homeomorphism of

The uniformity

For every uniformity

Let

In the case

Let

The previous result can be summarised as follows.

A uniform topology on

In the direction of extending Arens' result beyond the class of locally compact spaces, it comes as very natural idea to

We are now able to give a very simple description as set-open topologies for

When

Let

Unfortunately, we have no hope for minimality of

For that, we focus our attention on the class of rim-compact

Trying to capture minimality in local connectedness we get a previous basic result. Let

If

A result about local compactness involving as particular case the real line and more generally connected non compact Lie groups is the following.

Let

Whenever

A relationship with local compactness resides in the following.

If

In a more general context in which unfortunately the group topologies do not have a simple description and a convergence strategy, even though rather technical, has to be managed we have the following.

If

By essentially using the previous basic result, then we construct in two steps a

Suppose

Whenever

Let

The rational case apparently is singular. First, since any two nonempty open subspaces in

Let

Remember that

Of course, the main issue in the rational case is the following one.

Any admissible group topology

We now investigate whether the

Let

Let

For each

Let

The Whitney topology on

Now, denote

It is easily verified that

The uniformity

The topology generated by the base

The full homeomorphism group

In the rational case, the proof strategy is based on the property

If

If

If

Recall that a Tychonoff space

If

Supposing

Recall that a Weil uniformity is

Let

The left, the right, and the two-sided uniformities associated with

Let

Let

Let

We conclude with the following.

If

If

Now, we carry on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure. Let

Let

Whenever

We say that a class

If

Every class of metrics

If

As rim-compactness is a weak and peripherical compactness property, one might think of any further relaxation as impossible. But, we show that rim-compactness for

Since, if

Let us turn now our attention to

Let us focus on the second half

The identification leads to a natural embedding of

We now recall the notion of product metric on a product space. Let

If we suppose

Every admissible group topology on

In looking for topologies of uniform convergence on members of a given family, containing all compact sets, which are admissible group topologies, we focus beyond local compactness. In order to do so, we follow as suggestive example that of bounded sets of an infinite dimensional normed vector space carrying as proximity the metric proximity associated with the norm. We emphasise first that local compactness of

Uniformities, proximities, and

Let

The following properties are equivalent:

It is to be reminded that a

It is remarkable that, for each positive integer

The concept of EF-proximity can be recasted as

Furthermore, later on we essentially use the following

Let

Since the uniform topologies so far considered are relative to totally bounded uniformities, it is worthwhile to reformulate them as proximal set-open topologies. To unify the concepts of compact-open topology, bounded-open topology, and topology of proximity convergence [

Let

The proximal set-open topologies have remarkable properties [

Let

Blending proximity with boundedness gives local proximity. Local proximities play the same role in the construction of

Let

A non empty collection

(a)

The elements of

We expressly remark that we look at a local proximity as localisation of an EF-proximity modulo of a free regular filter [

It is remarkable that the boundedness in a local proximity space

For a Tychonoff space

Let

Let

Furthermore, given that a proximity-isomorphism or

If

We summarise the previous two results as follows.

If

Whenever

Whenever

Whenever

Now, assume that a

It is to be recalled that a local proximity space

Whenever

Nevertheless, what stated above is sufficient to draw the following final issue.

If

This final result can be recasted as follows.

Whenever