The Fundamental Algebraic Group of Topological Spaces

In the study of topology, we are often interested in understanding and classifying the internal structure of topological spaces. Algebraic topology is the application of abstract algebra to topology in order to further identify the structure of topological spaces by developing a correspondence between topological spaces and certain groups called homotopy groups. In this paper, we will examine the construction and nature of the first homotopy group, which is more commonly known as the fundamental group of a topological space. This paper is intended for undergraduate students of group theory with little to no background in point-set topology and will provide an introduction to algebraic topology and homotopy groups. We will first briefly cover the basics of point-set topology, then use these concepts to facilitate a rigorous study of the construction of the fundamental group. Afterwards, we will examine various proofs describing the nature of the fundamental group and demonstrate basic methods of calculation of the fundamental group of a topological space.