A study on some new types of functions in topological spaces

Abstract

The Research reported in this thesis deals with some new class of newlinefunctions using sets in topological spaces Their characterizations various newlineproperties and their interrelations have been discussed Somewhat newlinecontinuous function and slightly continuous function have been newlineintroduced which are weaker than continuous functions It is observed that newlineevery continuous function is somewhat continuous function Example is newlinegiven to show that the converse is not true It is also discussed that newlinecomposition of two somewhat continuous function need not be somewhat newlinecontinuous illustration is given to strengthen the statement newlineThe concepts of resolvable and irresolvable are defined and newlinetheir properties are discussed in topological spaces It is seen that every newlineslightly continuous function is slightly continuous but the converse is not newlinetrue It is also discussed that composition of slightly continuous function is newlinenot slightly continuous The concept of neighbourhood system have been newlineintroduced Some of the properties of derived set and boundary of a set newlinehave been discussed newlineCompletely irresolute function and completely weakly newlineirresolute function have been defined which are weaker than irresolute newlinefunction Their interrelations and impact on various separation axioms have newlinebeen established It is observed that every irresolute function is weakly newlineirresolute function and it is proved through an illustration that the converse newlineis not true It is proved that strongly continuous function is completely newlineirresolute and completely irresolute function is irresolute newlineCounterexamples are given to prove that the reverse implications are not newlinepossible Equivalent conditions for the existence of the functions some of newlinetheir properties their compositions with various functions are seen If is newlinealmost open surjection and g is any function such that is completely newlineirresolute then is completely weakly irresolute Weak forms of open function such as somewhat open weakly newlineopen contra open and quasi open function have been defined It is newlineobserved that composition of closed fu