A Study on the Concept of Connected Set

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. Let be a topological space. A connected set in is a set which cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set . Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. The space is a connected topological space if it is a connected subset of itself. The real numbers are a connected set, as are any open or closed interval of real numbers. The (real or complex) plane is connected, as is any open or closed disc or any annulus in the plane. The topologist's sine curve is a connected subset of the plane. An example of a subset of the plane that is not connected is given by Geometrically, the set is the union of two open disks of radius one whose boundaries are tangent at the number 1. An open set S is called disconnected if there are two open, non-empty sets U and V such that:

1. U V = 0 2. U V = S

A set S (not necessarily open) is called disconnected if there are two open seats U and V such that 1. (U S) # 0 and (V S) # 0

2. (U S) (V S) = 0 3. (U S) (V S) = S

If S is not disconnected it is called connected. Note that the definition of disconnected set is easier for an open set S. In principle, however, the idea is the same: If a set S can be separated into two open, disjoint sets in such a way that neither set is empty and both sets combined give the original set S, then S is called disconnected. To show that a set is disconnected is generally easier than showing connectedness: if you can find a point that is not in the set S, then that point can often be used to 'disconnect' your set into two new open sets with the above properties. Hence, as with open and closed sets, one of these two groups of sets are easy:

• open sets in R are the union of disjoint open intervals

• connected sets in R are intervals

The other group is the complicated one:

• closed sets are more difficult than open sets (e.g. Cantor set)

• disconnected sets are more difficult than connected ones (e.g. Cantor set)

In fact, a set can be disconnected at every point. A set S is called totally disconnected if y V, and (U S) (V S) = S. Intuitively, totally disconnected means that a set can be be broken up into two pieces at each of its points, and the breakpoint is always 'in between' the original set.

Proof. We can combine our particularly good understanding of connected subsets in the real line with Theorem. On the face of it, all this says is that the continuous image of an interval is also an interval-cut, but perhaps not very useful. However, if we combine this with the defining property of an interval, we obtain an extremely useful theorem indeed: the Intermediate Value Theorem. We start by proving the theorem, and the remainder of the chapter will be taken up with two applications of this theorem: a quick construction of the logarithmic functions, and a method for calculating roots of equations. The method of bisection is a very simple and effective means of estimating the roots of equations. Unlike other methods for finding roots, which may be faster, this is the only method which is guaranteed to work for all continuous functions. The theoretical underpinning of the method of bisection is the Intermediate Value Theorem. This tells us that if we know that a function is positive at one point, and negative at another, then a root of the function (a point at which it is zero) must lie between these two points. As an example of how to use this algorithm, we shall approximate the square root of 2 to within 0.1.

• Shami, P.A. and Waqar, A. (2009). Human Resource Development. Academy of Educational Planning and Management, Islamabad.

• Sharma. J.P. (2012) Growth and Development of In-service Education for Secondary School Teachers in the State of Bihar. In Third Survey of Research in Education. M.B. Buch (Ed.), New Delhi: NCERT.

 Shastri. C. (2007) Human Resource Development and Educational Planning in Kumaun. In Third Survey of Research in Education. M.B. Buch (Ed.), New Delhi: NCERT.

Implementation and Related Potential Outcomes. Unpublished Dissertation Abstracts International May.

• Singh, S.A. (2010) A Critical Study of the Programmes of the Pre-Service and Inservice Education of Teachers of Higher Education in India. In Third Survey of Research in Education. M.B. Buch (Ed.), New Delhi: NCERT. Pp.846-A, 847-A.

• Smith, C.A. (2012) Teacher’s Perceptions of Staff Development Activities. A Descriptive Evaluative Model of Teacher Education. Unpublished Dissertation Abstracts International, 62(12), June, p.4128-A.

• Smith, S. (2012) International Technology in Tennessee -12 Public School Classrooms: A Training Needs Assessment. Unpublished Dissertation Abstracts International, 62(7), January, p.2394-A.

• Snow, D.B. (2011) Teacher Induction: A Research-Based Framework. Unpublished Dissertation Abstracts International, 61(12), June, p.4637-A.

• Snyman, S.E. (2010) A Human Resource Management Plan for Peer Support Accompanying Programme: An Educational Management Perspective.

• Unpublished Dissertation Abstracts International 60(12), June p.4273-A.

• Sriyan, D. S. (2007). Human Resources Development for Competitiveness: A Priority for Employers. ILO Workshop on Employers’ Organizations in Asia Pacific in the Twenty First Century, Turin, Italy.

• Stevenson, J. M. (2009). The modern city as a multicultural metropolis: Consumer, civility and collegial triangles. Education, 119(4), 710+. Retrieved April 11, 2005, from Questia database, http://www.questia.com.

• Stockton, J.J. (2005) A Comparison of Levels of Satisfaction Regarding Human Resource Development Among Employees of North Arkansas Community /Technical College. Dissertation Abstracts International, 110(1), November.

• Suder Shan. K.N. and Singh.U.K. (2014) Global Education: An Analysis.

• The Economist, 28 September 2006 p.43.

• Thomas, L.E (2008) The Rationale, Design, and Implementation of an In-service Training Programme and Direct Support System for Continuing Training of Teachers in the Human Development Programme. Unpublished Dissertation Abstracts International 59(4), October. P.1074-A.

• Thompson, H.L (2012) A Descriptive Evaluative Model of Teacher Education. Unpublished Dissertation Abstracts International, 62(12), June, p.4128-A.

• Thurow L. (2012). Head to Head: The Coming Economic Battle Among Japan. Europe and America Marrow, New York.

 Thurston, P.W. (2010) Clarifying the Structure of Justice Using Fairness Perceptions of Perfromance Appraisal Practices. Unpublished Dissertation Abstracts International, 63(1), July, p.277-A.