TYPES OF TOPOLOGICAL SPACES AND THEIR INTERRELATIONSHIPS Research Paper

TYPES OF TOPOLOGICAL SPACES AND THEIR INTERRELATIONSHIPS - Research Paper Example Indeed, the nature is chaotic and therefore the ‘good mathematics’ is not always sufficient. Topology is a branch of mathematics that deals with the study of spaces and shapes. Certainly, the human mind is suited for a two dimensional space. Consequently, it more difficult to study spaces of higher orders hence, the need to apply abstract tools. The beauty about mathematics is mathematicians avoid natural problems, instead they create and solve problems to represents the natural world. Therefore, much of the work done on topology is an artificial creation that resembles real world problem. Topology has significant applications in other branch of mathematics such as geometry and algebra. Major mathematical problems that can be solved using topology include continuity, connectedness, and  boundary. The interesting aspect of topology is not the development of mathematical solutions, but how different mathematician approach a topology problem. This has led to the developme nt of different topologies namely T1 – T4. This paper explores the different types of topology and their relationships. Definition 1.1. Let be a set and a collection of subsets of such that the following properties hold. I. The empty set and the space II. If , then III. If for , then The collection is referred to as a topology on and the pair is referred to as a topological space. ... However, this definition does not endow a topological space with ‘nice’ properties similar to those found in metric spaces. For example in a metric space, every convergent sequence always converges to a unique limit. However, this is not necessarily true in topological spaces. To recover these properties, we need to supply enough open sets to the space. Thus, separation axioms classify topological spaces according to their sufficiency in open sets. Definition 2.2. A topological space is called a T0- space if for every two distinct points there exist an open set such that i. p lies in U and q does not lie in U. ii. q lies in U and p does not lie in U. Definition 2.3. T1 (Frechet) A topological space is called a T1- space if for every pair of points there exists such that Definition 2.4. T2 (Hausdorff) A topological space is called a T2-space or Hausdorff if for every pair of points there exists open sets such that, and. Definition 2.5. T3 (Regular) A T1 is called a T3 or a regular space if for every point and a closed set with there exists open sets such that and and. Definition 2.6. T3  1/2 (Completely Regular or Tychonoff) A T1 – space is called completely regular or Tychnoff if for every point and a closed set with there exist a continuous function such that and. Definition 2.7. T4 (Normal) A T1 – space is a T4 space if for every pair of disjoint closed sets A and B , there exists open sets such that , and . Remark 2.1 All T1 spaces are T0 but the converse is not true The discreet topology is T0 but not T1 All completely regular spaces are also T3 Every metric space is T4 Theorem 2.1. In any Hausdorff space, sequences have at most one limit It follows that every finite set is a T2 space is