Comprehensive Study on Investigation on Topology of the Line and Plane

Abstract

In this paper, we have discussed the definitions and the basic properties of open sets, closed sets, accumulation points or limit points and sequence. Sets may be neither open nor closed. The reader should not confuse the concept “limit point of a set” with the different, though related, concept “limit of a sequence”. Some of the solved and supplementary problems will show the relationship between these two concepts. Observe that  (a_n ∶n ∈ N) denotes a sequence and is a function. On the other hand, {a_n ∶n ∈ N} denotes the range of the sequence and is a set. We have given several characterizations of these sets. We discuss the definitions of bounded sequence, convergent sequence, Cauchy sequence and their relations. Then we can study every bounded sequence of real numbers contains a convergent subsequence and every Cauchy sequence of real numbers converges to a real number. Let A be a bounded, infinite set of real numbers. Then A has at least one accumulation point. We express the definition of topology, usual topology and topological space. The creation of topology the science of spaces and figures that remains unchanged under continuous deformations represents a phenomenon of this kind, but of a distinctly modern variety. Then we begin our study of some properties topological spaces by making the idea of being connected that is being in one piece. We observe that for the usual topology on the line R and in the plane R². Finally, we express the characterizations of the discrete topological space and indiscrete topological space.