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

Human Resource Management Research Paper Example | Topics and Well Written Essays - 2500 words

Human Resource Management - Research Paper Example The case study is all about the Jet Blue Airways which was founded by David Neeleman in the year 2000. The company has been trying to provide superior service to its staffs and has been constantly struggling for finding the right candidate who could blend with the culture of the organization. Rigorous training was provided to the staffs depending upon the duties and responsibilities. Moreover, the employees or the crew members went out of their way by helping the passengers; and the pilots used to interact with the customers and also engage themselves in activities such as cleaning up the plane for the next flight. The airport manager was found issuing the boarding passes. While recruiting the candidate, the company looked for people who had a positive attitude as the candidate was often asked to do certain things that were not a part of the job description. The airlines basically looked for the five values in the people they hired and these traits were given the first preferences. T he company’s recruitment and selection process have been quite complex. The selection process of the Jet Blue Airways has been focused on the ability of the applicant to handle the situation on several hypothetical situations. Even during the selection process the company tried its best to find the candidates’ abilities to imbibe the values of the Jet Blue. It was also found that despite having revolved around the values of the organization, the employees reported that they were unhappy with the leaders. From the above paragraph, it is evident that the company had a very complex recruitment and selection process. Moreover, it has been focusing more on the values during the selection and the recruitment process. Although, the culture of the organization has helped the company to perform well in the near past, however, it is not always possible for a candidate to have all the five values that the companies look for while recruiting and selecting the candidate. The compan y may face several problems if they have such tight recruitment and selection process. The company also faced problems of leadership and teamwork despite the leaders getting the best training. The company always realized the need to focus on the training of the leaders as they would guide their followers to adhere with the culture of the organization.