Topologized Soft Fundamental Group of Soft Topological Group

Hiyam Hassan Kadhem,Noor Abdul Moneem Jawad

doi:10.24996/ijs.2020.61.2.21

Hiyam Hassan Kadhem, Noor Abdul Moneem Jawad

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https://doi.org/10.24996/ijs.2020.61.2.21

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Journal: Iraqi Journal of Science	Publication Date: Feb 28, 2020
License type: cc-by

Affiliation: University of Kufa

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In this paper, we show that each soft topological group is a strong small soft loop transfer space at the identity element. This indicates that the soft quasitopological fundamental group of a soft connected and locally soft path connected space, is a soft topological group.

Highlights

In this paper, we show that each soft topological group is a strong small soft loop transfer space at the identity element
The fundamental group awarded with the quotient topology is tempted by the natural surjective map where is the loop space of with compact-open topology, signified by and becomes a quasitopo-logical group [1,2]
Spanier [4] presented a different topology on the fundamental group which was called the whisker topology by Brodskiy et al [5] and signified by

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Introduction

The fundamental group awarded with the quotient topology is tempted by the natural surjective map where is the loop space of with compact-open topology, signified by and becomes a quasitopo-logical group [1,2]. Showed that the quasitopological fundamental group of a connected locally path connected, semi locally small generated space, is a topological group. Hamed Torabi [6] presented that the quasitopological fundamental group of a connected locally path connected topological group is a topological group. In this paper we will demonstrate that the soft quasitopological fundamental group of a soft connected locally soft path connected topological group is a soft topological group. Let be a soft locally path connected soft pointed space and be a soft path open cover of. Is a soft open subgroup of Definition 1.3: Let is the soft space of soft homotopy classes of based soft paths in. Definition 1.5: Let be a soft locally path connected space and let be a soft path open cover of by the soft neighborhoods excluding. Define as the subgroup of involving of the soft homotopy classes of soft loops that can be represented by a product of the following form where are arbitrary soft path starting at and each is a soft loop inside the soft open set for all

We call

Hence each generator of goes to which indicates that

Let be the soft isomorphism defined by

Note that

Now let be a soft path connected soft topological group and