KUK - LMS

Syllabus

MODULE NO / UNIT	COURSE SYLLABUS CONTENTS OF MODULE	NOTES
Course Code: MMATH20-105 Course Name: Topology
1	Definition and examples of topological spaces, neighbourhoods, neighbourhood system of a point and its properties, interior point and interior of a set, interior as an operator and its properties, definition of a closed set as complement of an open set, limit point (accumulation point) of a set, derived set of a set, adherent point (closure point) of a set, closure of a set, closure as an operator and its properties, dense sets and separable spaces. Base for a topology and its characterization, base for neighbourhood system, sub-base for a topology. relative (induced) topology and subspace of a topological space. Alternate methods of defining a topology using properties of neighbourhood system, interior operator, closed sets, Kuratowski closure operator. comparison of topologies on a set, about intersection and union of topologies, the collection of all topologies on a set as a complete lattice.
2	First countable, second countable, their relationships and hereditary property. about countability of a collection of disjoint open sets in a separable and a second countable space, Lindelof theorem. Definition, examples and characterizations of continuous functions, composition of continuous functions, open and closed functions, homeomorphism. Tychonoff product topology, projection maps, their continuity and openness, Characterization of product topology as the smallest topology such that the projections are continuous, continuity of a function from a space into a product of spaces. T0 , T1, T2 spaces, productive property of T1 and T2 spaces.
3	Regular and T3 separation axioms, their characterization and basic properties i.e. hereditary and productive properties. quotient topology w.r.t. a map, continuity of function with domain a space having quotient topology, about Hausdorffness of quotient space. Completely regular and Tychonoff (T 3 1/2), spaces, their hereditary and productive properties. Embedding lemma, Embedding theorem, normal and T4 spaces, Urysohn’s Lemma, complete regularity of a regular normal space, Tietze’s extension theorem (statement only).
4	Definition and examples of filters on a set, finer filter, ultra filter (u.f.) and its characterizations, Ultra Filter Principle (UFP). image of a filter under a function. convergence of filters: limit point (cluster point) and limit of a filter and relationship between them, Continuity in terms of convergence of filters. Hausdorffness and filter convergence. Compactness: Definition and examples of compact spaces, compactness in terms of finite intersection property (f.i.p.), continuity and compact sets, compactness and separation properties. regularity and normality of a compact Hausdorff space. compactness and filter convergence, Tychonoff product theorem.