The properties of a topological space that were developed so far have been depended on the choice of topology, the collection of open sets. Taking a different tack, we introduce a different structure, algebraic in nature, associated to a space together with a choice of base point . The structure will allow us to bring to bear the power of algebraic arguments   . The fundamental group was introduced by Poincairé in his investigations of the actions of a group on a manifold. In a long paper entitled Analysis Situs, Poincare introduced the concept of the fundamental group of a topological space. The research begun with a heuristic introduction, using functions (not necessarily single valued) on a manifold defined by equations between coordinates . It assumed that these functions satisfy certain differential equations, where are known single-valued differentiable functions of and which satisfy certain integrability conditions . Then it considered that the transformations of produces result if one traces their values along a closed loop. Dugundji, put, for the first time, a topology on fundamental groups of certain spaces and deduced a classification theorem for connected covers of a space.
Furthermore, topologists of the early 20th century dreamed of a generalisation to higher dimensions of the non-abelian fundamental group, for applications to problems in geometry and analysis for which group theory had been successful.
For some decades now, the theorem is a group, established to be a fundamental group with respect to “ ” in the interval [0, 1] . It has been used to prove some mathematical concepts such as connectedness, metric space, isomorphism, Cech homotopy etc. The theorem has led few researchers to work within this interval [0, 1]  . Cannon and Conner (2005) used the fact that is a group to work in one dimensions. Their research examined the fundamental groups of complicated one-dimensional spaces. In attempt to prove some other mathematical concepts, it was established that if X is a space of dimension at most 1, then, the fundamental group is isomorphic to a subgroup of the first Cech homotopy group based on finite open covers. Consequently, for a one-dimensional continuum X, the fundamental group is isomorphic to a subgroup of the first Cech homotopy group . A potentially new approach to homotopy theory derived from the expositions in Brown’s (1968) and Higgins’ (1971), which in effect suggested that most of 1-dimensional homotopy theory can be better expressed in terms of groupoids rather than groups .
This led to a search for the uses of groupoids in higher homotopy theory, and in particular for higher homotopy groupoids. The basic intuitive concept was generalising from the usual partial compositions of homotopy classes of paths to partial compositions of homotopy classes (of some form) of complexes. But a search for such constructs proved abortive for some years from 1966 .
Recently, the concept of fundamental group was used on the ageing process of human. The homotopy relates the topological shape of the infant to the topological shape of the adult. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. This complex connected cylindrical shape of the body , described by the homotopic functions provided the ageing process in vertical interval . But this work is limited to the interval .
Therefore this research intends to establish the proof that is a group in other domains other than [0, 1]. In this paper, we defined homotopy and presented some group properties. We then described the fundamental group and its related properties such as group homomorphisms. We also looked at useful definitions and theorems that play an important role in computing fundamental groups, and our main result was actually built upon these theorems. In the end, we established the proof that the equivalent class is a fundamental group in the interval .
Throughout this paper we assumed the knowledge of basic algebra and general topology.
Let X be a topological space. A path in X from x0 to x1 is a continuous map such that and . We say that x0 is the initial point and x1 the final point .
2.2. Homotopic Path Concatenation
Suppose X is a space and is a choice of base point in X. The space of based loops in X denoted , is the subspace of ,
Composition of loops determines a binary operation . We restrict the notion of homotopy when applied to the space of based loops in X in order to stay in that space during the deformation.
Given two based loops and , a loop homotopy between them is a homotopy of paths with , and . That is, for each , the path is a loop at .
Continuous mappings and induce well-defined functions and by and for .
Proof. We need to show that if , then and . Fixing a homotopy with and , then the desired homotopies are and .
To a space X we associate a space particularly rich in structure, the mapping space of paths in X, . Recall that is the set of continuous mappings with the compact-open topology. The space has the following properties:
1) X embeds into by associating to each point to the constant path, for all .
2) Given a path , we can reverse the path by composing with . Let .
3) Given a pair of paths for which , we can compose paths by
Thus, for certain pairs of paths and , we obtain a new path . Composition of paths is always defined when we restrict to a certain subspace of .
Let X be a topological space, and a point in X. The fundamental group of X is the set of path homotopy classes of loops based at , together with the operation “ ”. We denote it by .
Given two loop classes and we define:
2) The inverse of is given by , that is , where .
If X is a convex subset of , and if , . This defines a homotopy between and .
The relation is an equivalence relation on the set, , of continuous mappings from X to Y.
Proof. Let be a given mapping. The homotopy is a continuous mapping and so .
If and is a homotopy between and , then the mapping given by is continuous and a homotopy between and that is .
Finally, for and , suppose that is a homotopy between and , and is a homotopy between and . Define the homotopy by
Since , the piecewise definition of H gives a continuous function. By definition, and and so .
The equivalence classes of maps from X to Y as in the Theorem 2.2.7 are called the homotopy classes ad denoted by [f] as the homotopy class of the map f. While we use the notation when is a loop in X based at .
2.3. Homotopy Classes
The equivalent classes determined by homotopy modulo on the collection of all closed paths f on S based at are called homotopy classes of . The collection of these homotopy classes is denoted by .
3. Main Results
Path Concatenation in [m, n]
1) If we define the juxtaposition of f and g as follows:
Thus and “ ” is a binary operation on .
2) If , then let
Let be topological spaces and let A a subset of X. Then is ann equivalence relation on the set of maps from X to Y which agree with with a given map on A.
Proof. For notational convenience, drop the subscript A from the notation.
1) Reflexive property : Define . This is the composition of f with the projection of on X. Since it is a composition of two continuous maps, it is continuous.
2) Symmetric property : Suppose is a homotomy (relative to A) of f tog. Let . The and similarly for . Also, if for , the same is true for . is a composition of two continuous maps. What are they?
3) Transitive property : This is somewhat harder. Let be a homotopy (relative to A) from f to g, and let be such a homotopy of g to h. Define
Note that the definitions agree for . We need to show H is continuous.
(Eckmann-Hilton). Let G be a group space and be the identity point. Then is abelian.
Proof. To show is abelian, we will show that for any two loops we have . Indeed for this, we construct a homotopy between the two loops above.
(Van Kampen’s theorem). Suppose where each contains a green basepoint so that each is path connected and each is path connected. We have homomorphisms induced by the inclusions and homomorphisms induced by the inclusions .
1) The homomorphism is surjective.
2) If further each is path connected, then the kernel of is the minimal normal subgroup generated by all the elements of the form for so induces an isomorphism .
is a fundamental group with respect to “ ” in the general interval .
1) “ ” is associative. We need to show that for .
We define a homotopy between and as follows:
Then the following is true:
Thus and for all .
Also and for all .
2) We show that the constant mapping is such that [c] is the identity element of with respect to “ ”. Thus we must show that for any . Let be defined as follows:
and if [i.e. ]. Thus and for all . More so, and for all .
3) Finally we want to show that each homotopy class has an inverse such that . Thus we want to show that if there exist a such that . Let . Since ,
by definition, we have
We then define homotopy h between and c as follows:
Since f and g are continuous, h is continuous and we have
Thus and . Also for all . Hence .
This paper, in full accordance with the principles of homotopy, has been able to establish the proof that is a fundamental group in the general interval , .
In general, depends upon . However, in the case of an arc-wise-connected space S, we can show that is independent of .
Sincere thanks to Mrs. Belinda Zigli and Prof. William Obeg-Denteh for their immersed supports. The authors wish to acknowledge the support of their respective Universities and the anonymous referees for their helpful comments towards the improvement of the paper.
 Brew, L., Obeng-Denteh, W. and Zigli, D.D. (2019) Application of Homotopy to the Ageing Process of Human Body within the Framework of Algebraic Topology. Journal of Mathematics Research, 11, 21-25.
 Zigli, D.D., Brew, L. and Otoo, H. (2020) Homotopical Proof of Π1 (S, xo)as a Fundamental Group with respect to “。” in the Interval [0,n]. International Journal of Algebra, 14, 191-201.
 Zigli, D.D., Obeng-Denteh, W. and Owusu-Mensah, I. (2017) On the Candid Appraisal of the Proof of Π1 (S, xo) as a Fundamental Group with Respect to “。”. International Journal of Advances in Mathematics, 2017, 20-26.