In Banach theorem, a metric on X is used in the essential statement that f is a contraction. The unit ball in a Euclidean space is also a metric space and the metric topology determines the continuity of continuous functions, however the core of Brouwer theorem. is a topological property of the unit ball, namely the unit ball is compact and contractible. It is unbearable to differentiate two fixed point theories in a precise way and it is not easy to define certain topics belong to which branch. In general, the fixed point theory is regarded as a branch of topology. But due to profound effect on topics related to non-linear analysis or dynamic systems, many parts of the Fixed Point theory can be considered as a outlet of analysis. In the early 19th century, the problem of stability of solar system resurfaced, which was subsequently referred to the mathematicians. This was then duly noted by Henry Poincare ́ who propounded (without Proof) that no smooth vector field exists on a sphere having no sources or sinks. Motivated by Poincare, another mathematician L.E. Jan Brouwer observed the former’s Theorem based on a cup of coffee and came up with his own Theorem which was supposedly better.

**Author(s) Details:**

** Arti Saxena, **

School of Engineering and Technology, Manav Rachna International Institute of Research and Studies Faridabad, Haryana, India.

** Poonam Rani,**

Department of Humanities and Applied Sciences, Echelon Institute of Technology, Faridabad, Haryana, India.