Solve Special Case of Some Guran Problems

Throughout this paper, all topological groups are assumed to be topological differential manifolds and algebraically free, our aim in this paper is to prove the open problems number (7) and (8). Which are introduced by (Guran, I, 1998). In many cases of spaces and under a suitable conditions. therefore, we denote by I(X) and I(Y) to be a free topological groups over a topological spaces X and Y respectively where X and Y are assumed to be a nonempty sub manifolds Which are also a closed sub sets, and P is a classes of topological spaces, as a regular, normal, Tychonoff, lindelöf, separable connected, compact and Zerodimensional space, and we have tried to use a hereditary properties and others of these spaces, so we can prove the open problems in these cases and we have many results showed in this paper.


Introducing the Problem
The notion of free topological group F(X) over a topological space X was introduced by A. A. Markov.He proved that for any Tychonoff space X a free topological group F(X) exists and is unique, algebraically free and Tychonoff.
In this paper, the main idea which we use to prove the open problems number (7) and ( 8).Depend on a topological and hereditary properties of these spaces, and it also Depends on some properties of topological groups.Moreover, we replaced in this paper the free Topological inverse semi groups by the free Topological groups.Using these properties, we can determine some classes of topological spaces which satisfy the property in the texts of these open problems.

What is the Question?
The remained question is: can one prove or deny the statement: Where I(X) and I(Y) are a free Topological inverse semi groups over the topological spaces X and Y respectively.

What We Are Proving in This Paper?
In This paper, we prove that the open problems number ( 7) and ( 8) remain valid if the free Topological inverse semi groups replaced by the free Topological groups and all of these are assumed to be a topological differential manifolds.

Method
Firstly, we introduce some notions and terminology.

Definition
Let M be a manifold of dimensional n.A sub set See (Baker, 2002, p 186).

Theorem
Let M be a connected manifold and N M ⊆ be a non-empty sub manifold Which is also a closed sub set.If See (Baker, 2002, p 237).

Definition
A property S is said to be:

A topological property, if the following condition holds:
A space X has S, and X is homeomorphic to Y, then Y has S.

S is said to be a continuous invariant if the following condition holds:
A space X has S, and : 3. S is said to be a hereditary if whenever X has S, every subspace of X has S.

Definition
A topological group G is a group G with a (Hausdorff) topology such that the product mapping of G G × into G is jointly continuous and the inverse mapping of G onto itself associating 1 x − with an arbitrary x G ∈ is continuous.

Definition
Let X be a subspace of a topological group.G Assume that (1) The set X generates G algebraically, that is X G < >= (2) Each continuous mapping : Then  is called the Markov free topological group on X and is denoted by ( ) See (Lin, 2013, p 2).

Corollary
If  is a topological group which is T 1 , then G is completely regular and thus regular.

Results
Let  be a classes of topological spaces.Then we have the following results in many cases of spaces:

Theorem
If X is a zero-dimensional closed sub manifold of classes P, then the free Topological group I(X) is also zero-dimensional. Proof: If X is a zero-dimensional, then X has basis ( ) , is a basis of I(X), and we know from (Engelking, et al., 1986;Murdeshwar, 1990) that the sub set A X α ∩ is an open (closed) in the sub space X if and only if A α is an open (closed) set in the space I(X).In addition, X is closed and also X is open, since every element of X is contained in an open sub set of I(X) contained in X.Hence, the members of the basis { } I A α α∈ are Clopen sets in I(X), so I(X) is a zero-dimensional.

Theorem
Let I(X) is a connected and Tychonoff space of classes P. If I(X) is homeomorphic to I(Y), then , X Y P ∈ .

Proof:
Since I(X) is a connected space and X is closed as Well as open sub set in I(X) by Theorem (3.1), so ( ) X I X = by (Baker, 2002), then X P ∈ .
But I(X) is also a Tychonoff space, so I(X) is a 1 T -space, and because 1 T -space and connected space both have a topological property by & Sieklak, 1986), then so is I(Y) (being I(X) is homeomorphic to I(Y)); and we know by the properties of topological groups (Marris, 2003), that ( ) . Hence Y P ∈ .

Remark
Theorem (3.2) remains valid if P is a classes of ( 0,1, 2,3) . It follows directly of topological and hereditary properties of these spaces.

Theorem
If I(X) is homeomorphic to I(Y) and I(X) is a separable and Tychonoff space of classes P , then , X Y P ∈ .

Proof:
By theorem (3.1), X is an open sub set in a separable space I(X), and we know by (Murdeshwar, 1990), that the separability is open -hereditary then X is separable and Tychonoff space (because it is a hereditary property), thus is continuous surjection (being f is homeomorphism), where I(X) is separable, then so is I(Y), and because the space T -space and is completely regular then I(Y) is a Tychonoff space.Similarly, it is also easy to show that Y is separable and a Tychonoff space (it follows from hereditary properties).Thus Y belongs to the classes P.

Theorem
Let I(X) be a separable and compact space of classes P. If I(X) is homeomorphic to I(Y), then , X Y P ∈ . Proof: Since X is a closed sub set in I(X), then it is a compact, but  is also open in I(X) by theorem (3.1), so X is a separable.Hence X P ∈ .Otherwise, ( ) ( ) is continuous surjection, where I(X) is a separable and compact space, then so is I(Y) and Y because Y is open as well as closed sub set in I(Y).Thus Y P ∈ .

Theorem
If I(X) is homeomorphic to I(Y) and I(X) is a lindelöf and compact space of classes P, then , X Y P ∈ . Proof: X is a closed sub set in the lindelöf space I(X).Therefore, X is itself lindelöf by ( Engelking, et al.,1986;Murdeshwar, 1990).Since I(X) is also compact, X is compact.Thus X P ∈ .Furthermore, ( ) ( ) Therefore, I(Y) is a lindelöf and compact space.Hence, Y is lindelöf and is compact, too.It comes easily from closed-hereditary.Thus Y P ∈ .

Theorem
Let I(X) be a compact and normal space of classes P.
If X is a closed sub manifold in I(X), then X is compact and normal by closed-hereditary.Hence X P ∈ .Suppose ( ) ( ) is homeomorphism between the spaces () and I(Y).Then  is continuous surjection.Hence, I(Y) is compact since I(X) is.But also I(Y) is a completely regular space (being I(Y) is a topological group).Hence, I(Y) is a regular space, and because the regular and compact space is normal by (Murdeshwar, 1990).Thus I(Y) is normal.So the closed sub manifold  belongs to the classes P, it follows easily from closed-hereditary by (Murdeshwar, 1990).

Remark
Theorem (3.6) is true for regular and compact spaces of classes P.

Problem ( 8 ):
Describe classes P of Topological spaces having the following property If I(X) is homeomorphic to I(Y) and X P ∈ then Y P ∈ .

3.∈
M is locally Euclidean of dimension n, i.e. for every m M where U is an open neighborhood of m in M, V an open subset of ℝ  , and :U V