P-Topological Groups in Nonstandard Analysis

The aim of this paper is to introduce and study a new class of topological groups called P-topological group. By using some nonstandard techniques, we investigated some properties of P-monads in P-topological group.

In 1982, Mashhour A.S.et al [6], defined a new version of nearly open sets which is significant notion to the field of general topology called preopen sets .
There are several important concepts in topology in which can be defined in terms of preopen sets.
In 2000 Dontchev J.et al [3 ] introduced the concept of pre --open sets , in this work we use the notion of pre --open sets to define and study a new type of topological groups called P-topological group, also we study some properties of P-monads in P-topological group. For this investigation, we need the following basic background in general topology and nonstandard analysis.

2-Basic Backgrounds in General Topology:-
Throughout this work, (X,) or (simply X) denotes a standard topological space on which no separation axioms are assumed unless explicitly stated, we recall the following definitions, notational conventions and characterizations.
The closure (resp., interior) of a subset A of a space X is denoted by ClA(resp. IntA).

Definition 2.1: A subset A of a space X is said to be
• preopen set [6] if and only if A IntClA.
• preclosed set [2] if and only if X A is preopen set. Equivalently, ClIntA A.

Definition 2.2[4]:
A topological space (X,) is called p-regular(resp., p*-regular) if and only if for each xX and each closed (resp., preclosed )set F such that xF, there exist two disjoint preopen sets A and B such that xA and FB.

Theorem 2.6 [1]:
i) If X is extremely disconnected, then PO(X)= PδO(X). ii) If X is p -regular space , then   PO(X). iii) If X is p* -regular space, then PO(X)=PO(X).

Proposition 2.7 [1]
:Let X1 and X2 be two topological spaces and be the topological product, let for i=1,2 then .

Definition 2.8:
A mapping is said to be i) Pδ-irresolute [4], if for each ii) Pδ**-continuous [4], if for each iii) completely preirresolute [2 ], if for each iv) faintly precontinuous [2],if for each v) strongly faintly precontinuous [2], if for each Note that to define a P -topological group, we introduce the following new type of continuity in topological spaces called p -irresolute function, some characterizations and relations are obtained for this definition. iii) The inverse image of every pre--closed set in Y is pre--closed set in X. iv) , for each subset A of X. v) , for each subset B of Y. vi) , for each subset B of Y. Proof: Straightforward. Theorem 2.11:If X is extremely disconnected space, then every pδ-irresolute mapping is p -irresolute. Proof: Let be a pδ-irresolute mapping, and let Then, by Theorem 2.5(iii) we have , since f is pδ-irresolute function, then Since X is extremely disconnected space, by Theorem 2.6(i), we get . Hence f is p -irresolute. Theorem 2.12: If X is p-regular space, then every pδ**-continuous mapping is p -irresolute mapping. Proof: The proof is similar to Theorem 2.11 Theorem 2.13: If X is p*-regular space, then every completely preirresolute mapping is p -irresolute mapping. Proof: It follows directly from Theorem 2.6(iii) and their definitions. Theorem 2.14: For any mapping , the following statements are true i) Every p -irresolute is faintly precontinuous function. ii) Every strongly faintly precontinuous is p -irresolute. Proof: The proof is easy, and therefore is omitted.

Basic Backgrounds in Nonstandard analysis:
In this section, we use E. Nelson's Nonstandard Analysis construction, based on a theory called internal set theory IST [7] . The axioms of IST is the axiom of Zermelo-Frankel with the axiom of choice (briefly ZFC) together with three axioms which are the transfer principle (T), the idealization principle (I) , and the standardization principle (S), are stated by the following Transfer principle Let be an internal formula with free variables Only then Example 3.1: Consider the following statement: Applying transfer principle, we have Thus, we may assert that R has a unique multiplicative identity. Furthermore, recalling that we can identify R st as a subset of R, we can say that this is 1.
The primary use of the transfer principle is that if one wishes to prove a theorem about the standard universe, it suffices to prove an analogous theorem with standard parameters in the enlarged universe.

Idealization Principle (I)
Let B(x,y) be an internal formula with free variables x ,y and with possibly other free variables then .

Standardization Principle (S)
Let F(Z) be a formula, internal or external with free variables z and with possibly other free variables. Then, . Every set or element defined in a classical mathematics is called standard. Any set or formula which does not involve new predicates "standard, infinitesimal, limited, unlimited is called internal, otherwise it is called external Definition 3.2 [5]: Let (X,) be a standard topological space. Then, the P-monad at a standard point aX is defined as follows P-monad = {pClA ; }, and is denoted by µp(a).

Theorem 3.3[5]:
Let (X,) be a standard topological space, and let aX be any element. Then, there exists a standard preopen H such that pClH  µp(a).   The converse part is obvious.

Theorem 4.6:
Let be a standard p-topological group, then the following mappings are p-homeomorphism. i) , defined by ii) .defined by iii) .defined by iv) . defined by are p-homeomorphism, for a fixed Proof: As a sample we proof (i) , then , since and is a group, Therefore, a=e, which is contradiction

Some properties of P-monads in p-topological groups:
In this section, we give some properties of p-monads in p-topological groups, by using nonstandard techniques. since PO(X) PO(X), and since V pClV, we have }= . If we replace a by . Which implies that .

Theorem 5.3:
Let a and b be any two standard points in standard p-topological group then .