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Part 2 consists of three units:
Unit 4 – Real Analysis
In this unit we define and give examples of general metric spaces which show that they occur abundantly in mathematics. We then look at the structure of a general metric space a long the lines of unit 1. In addition we introduce the concept of compactness and its effects on continuity of functions.
Unit 5 – Topology
The structures of topological spaces are studied along side those of metric spaces. However the main difference here is that the axioms that define a metric space are dependent on the concept of distance whereas in the axioms that define a topological space the concept of distance is absent. In particular the study of the twin concepts of convergence and continuity brings out this difference very well. Finally a look at different topologies like product or quotient topology endowed on a set is essential in this unit.
Unit 6/7 – Measure Theory
In this unit we start with the study of both the Lebesgue outer measure and the real line before we look at the Lebesgue measurable subsets of the real line. A look at the sigma algebra of subsets of a given underlying set gives rise to measurable space on which we can also study a class of functions called measurable functions. A part from the Lebesgue measure we also study an abstract measure leading to an abstract measure space on which we introduce an abstract integral. Finally a brief comparison of the Lebesgue integral and the well known Riemann integral is also essential in this unit.
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