The study of non-Newtonian calculi was started in 1972 by Grossman and Katz. These calculi provide an alternative to the classical calculus and they include the geometric, anageometric and bigeometric calculi, etc. Recently, C, akmak and Ba, sar ( 2002) have studied the concept of non-Newtonian metric. Also they have given the triangle and Minkowski's inequalities in the sense of non-Newtonian calculus. In this paper, we introduce a fixed point theory by defining some topological structures of the relevant non-Newtonian metric space.