1. Introduction

A topological group is one of the most interesting topics. Equipped with two compatible structures (group and topology), it admits a beautiful set of properties that are not readily available in either of the two structures separately. One of which is the universal hausdorff group which is the quotient of the closure of the identity element.

A topological group is a set G on which two structures are given, a group structure and a topology, such that the group operations are compatible. Specifically, the mapping from the direct product into G must be continuous. A subgroup H of a topological group G is a topological group in the with respect to the relative topology. The quotient space of cosets is given the quotient topology with respect to the canonical mapping from G onto G/H. If H is a normal subgroup of G, then G/H (the quotient group of G by H) is a topological group. Taking the closure of the identity we get a normal subgroup of the topological group and hence we get a topological group of the resulting quotient group, this group is the main focus of this study.

Continuous action of a topological group is simply a group action which is also continuous. Continuity of the action map can give extra properties to both of the topological and algebraic structures, which is a fact well used in this paper.

This research consists of four section; section one consist the basic definition of topological group, hausdorff topological space,…etc. in section two we study the universal hausdorff group also given some theorems and lemma. In section three we study the Continuous Action of Topological Group, in section fourwe study Continuous Action of Topological Groups and induced an action of the universal hausdorff group on that space.

2. Basic Concepts

Definition 1.1 [6]: A topological group is a set G th two structures:

i. G is a group,

ii. G is a topological space,

Such that the two structures are compatible i.e., the multiplication map nd the inversion map are both continuous.

Definition 1.2 [6]: amorphism of topological groups is a continuous group homomorphism.

Notation: if G is a topological group then:

Remark 1.3: The maps defined by are homeomorphisms (not necessarily a homomorphism) called left and right translation respectively.

Definition 1.4 [3]: a subgroup H of a group G is a normal subgroup of G if for every .

Definition 1.5 [3]: If G is a group, H is a normal subgroup, then the collection of cosets of H denoted by G/H is a group called the quotient group.

Remark 1.6: If H is a normal subgroup of G hen it is also a topological group, and the quotient map is both continuous and open.

Remark 1.7: Let in has kernel K and image with quotient map and inclusion map then in such that The map is a bijection and if is open or closed it is a - iisomorphism.

Definition 1.8 [2]: a topological space X is a space if every singleton subset of of X is closed.

Definition 1.9 [4]: a topological space X is Hausdorff if each pair of distinct points belong respectively to disjoint open sets.

Definition 1.10 [2]: a subset A of a topological space X is compact if every cover of A is reducible to a finite cover.

3. The Universal Hausdorff Group

Theorem 2.1 [6]: let G e a topological group, E is the closure of the identity element e, then E is a normal subgroup of G.

Proof:

Now if then and are closed sets containing e and hence contain E.

Also if again if then so that E is a group.

Normality follows for if

Lemma 2.2: let then E has the property that if A is any closed set, then

Proof: let

Suppose that but is a closed set containing e then which is a contradiction hence .

Theorem 2.3: let be the quotient group induced by E, then has the property that if is in , where H is Hausdorff¹, in such that where is the quotient map.

Proof: let then A is a closed normal subgroup of G so that by lemma 2.2

Now A has the universal property that such that

.

Define by we shall show that is well defined.

Let such that we must show that Now so that is well defined.

Also is a group homomorphism for if we have

Also we have so that and hence we have if is open we have and considering the fact that are continuous and open maps respectively we have is open so that is continuous and hence a -morphism.

Set then for all so that , also being the composition of -morphism, is a -morphism.

Uniqueness follows for if such that then for all

Definition 2.4: as stated above is called the universal Hausdorff group.

4. Continuous Action of Topological Groups

Definition. 3.1 [4]: Let G be a group and X a set. An action of G on M is a map satisfying:

i. and

ii.

Definition 3.2 [1]: A topological group action is a group action such that the map is continuous.

Definition 3.3 [5] A continuous action is said to be proper if the map defined by is a proper map, that is inverse of compact sets is compact.

Definition 3.4 [5]: An action is said to be faithful if for any in G there exist an x in X such that .

Theorem. 3.5: let be a topological group G action on the space X then the stabilizer of any point in X is a closed subset of G, i.e. is closed for all .

Proof: since the action map is continuous then the restriction map is continuous,

Now the pre-image of the point x is the set which is a closed subset of . But the map defined by is also continuous, so that the pre-image of the set is just and hence is closed.

Corollary: The kernel of a continuous action on a space X is a closed normal subgroup of G.

Proof: by theorem. 3.2 we have is closed in G so that is closed in G. Also being the kernel of the homomorphism (the space of all homeomorphisms ), then A is a normal subgroup of G.

Remark 3.6: A, as defined above, has the universal property that the quotient space also acts continuously on the space X. this action is faithful and is given by .

5. Action of the Universal Hausdorff Group

Further next we shall construct an action of the universal hausdorff group.

Remark 4.1: if G is a topological group acting on the space X then as in the above remark we get an action of the quotient of the kernel of the action, also we obtain, as in theorem 2.3 a continuous homomorphism

Theorem 4.2: let G be a topological group acting on the space X, E is the universal hausdorff group, then there is an action of the quotient group on the space X such that the diagram commute.

Proof: by the corollary of theorem 3.3 the kernel A of the action is a closed normal subgroup of G so that by lemma 2.2 we have .

Define by , we shall show that is well defined.

Let such that we must show that. Now so that f is well defined.

Also f is a group homomorphism for if we have

Now define by .

We shall proof that is an action map, considering the diagram below let

Also and by 1 and 2 we have

Furthermore then by is a group action.

Continuity follows from the fact that is the composition of the two continuous mappings and . Then is a continuous action of on X.

Now , this proofs that the diagram commute.

Theorem 4.3: if is an action of a topological group G on a then the induced action of the universal hausdorff group is also proper.

Proof: since the action of G is proper then the map defined by is proper.

Now and so that the below diagram commute, now let V be a compact subset of since f is proper then is compact in finally since are continuous mappings then is compact making a proper map which in turn imply that is a proper action.

Remark 4.4: being also topological groups, everything we mentioned is also valid to Lie groups; however, lie groups and more generally Manifolds are often considered (by some authors) as Hausdorff spaces, a convention that imply that the closure of the identity is merely the trivial group, which means that the universal hausdorff group is equivalent to the topological group itself.

6. Conclusions

All group actions considered are continuous. For any topological group acting on a space the action of the universal hausdorff group have been constructed and proven to be continuous.

Note

1. It is enough for the proof for H to be merely . However, for any topological group the property of being is equivalent to the property of being hausdorff.