Dedekind built up the basic of mathematics inducing the concept called Schmitt. Not only Klein left many results in analysis but also he announced Erlangen Program , sorted the whole of geometry and served as a stepping stone for the new geometry. The research about geometry axiom became the basic of geometry axiomism. The modern mathematics progressing and advencing over and over again as it promoting his research. Topology Topology occupies a high position as an integraated mathematics in the 20th century.
Topology means position and form.
Started from Euler's polyhedron theorem, topology was developed through Poincare's algebraic topology and Brouwer's fixed point theorem. Topology is a feld of mathematics in which topological properties were treated. Many theorems including unicursal problem were known. As the area of mathematics extended to abstraction, so the object of mathematics expanded from concrete spaces to the abstract ones. These abstract spaces are called topological spaces.