At the end of this course the students will be able to setup and seek the solutions of several equations in several. Unlike other courses of its level, linear algebra embodies a circle of theoretical ideas which necessitate careful definitions, and statements and proofs of theorems, as well as a body of computational techniques that can serve both the theory itself and its application. Further they will be able to compute eigenvectors. The last part of the course will enable the students to work with and linear operators and their adjoints.
Motivation and introduction, Sets and their operations, Countable and uncountable sets, Cardinal and transfinite numbers, Topological spaces, Open and closed sets, interior, Closure and boundary of a set, Neighborhoods and neighborhood systems, Isolated points, Some topological theorems, Topology in terms of closed sets, Limit points, Derived and perfect sets, Dense sets and separable spaces, Topological bases, Criteria for topological bases, Local bases, First and second countable spaces, Relationship between sparability and second countablity, Relative or induced topologies, Necessary and sufficient condition for a subset of a subspace to be open in the original space, Induced bases. Metric spaces, Topology induced by a metric, Equivalent topologies, Formulation with closed sets, Cauchy sequence, Complete metric spaces, Characterization of completeness, Cantor’s intersection theorem, Completion of metric space, Metrizable spaces. Continuous functions, Various characterizations of continuous functions, Geometric meaning, homeomorphisms, Open and closed continuous functions, Topological properties and homeomorphisms, Separation axioms, T1 and T2 spaces and their characterization, Regular and normal spaces and their characterizations, Urysohn’s lemma, Urysohn’n metrizablity theorem (without proof). Compact spaces their characterization and some theorems, Construction of compact spaces, Compactness in metric spaces, Compactness and completeness, Local compactness, Connected spaces, Characterization and some properties of connected spaces.
