# David Delali Zigli, William Obeng-Denteh and Isaac Owusu-Mensah, On the Candid Appraisal of the Proof of $ \prod_{1}(S,x_{o}) $ as a Fundamental Group with respect to "$ \circ $", Volume 2017, Number 6, Pages 20-26, 2017

**Abstract:** It is known that a homeomorphism between two topological spaces preserves all the topological properties such as connectedness and compactness. To a map of spaces we can associate a homomorphism of groups such that compositions of maps yield compositions of homomorphisms of groups. Then any thing we can say about a topological situation gives information about the algebraic one. In this paper we studied and proved with self explained diagrams that $\prod_1(S,x_0)$ is a fundamental group with respect to "$ \circ $" from the homotopical point of view. We made it more comprehensible, self explanatory and user friendly with clear diagrams. This is of interest because it can be used as a great tool in mathematics to prove many facts.

**Keywords: **Homotopy, Fundamental group, Homomorphism, Homeomorphism, Juxtaposition.

**How to cite this article:**

David Delali Zigli, William Obeng-Denteh and Isaac Owusu-Mensah, On the Candid Appraisal of the Proof of $ \prod_{1}(S,x_{o}) $ as a Fundamental Group with respect to "$ \circ $", International Journal of Advances in Mathematics, Volume 2017, Number 6, Pages 20-26, 2017.