Your lecture notes will cover all the material (except for those results assigned for self-study)
in the syllabus. The occasional chapter references
below are to a more extensive treatment of the topic in question and indicate the primary source of the
material presented in the lectures.
The natural numbers, Peano's axioms, mathematical induction, Peano arithmetic
Aspects of the theory of sets, the axioms of specification and union, De Morgan's laws
Two-fold cartesian products, relations and functions, equivalence relations
The integers: the definition/construction of the set of integers and integer arithmetic
The rational numbers: the definition/construction of the set of rationals and rational arithmetic, the rationals as a field
Ordered sets, the "usual order" on the rationals, ordered fields, the least upper bound property
(The treatment of the above topics follows, although selectively, that of Chapters 1–4 of
Tao's Analysis 1)
The least upper bound property, the definition/meaning of the system of real numbers
Dedekind cuts, construction of the real line (Chapter 1: Appendix of Rudin's Principles)
The Archimedean property of the real line
Metric spaces, open and closed sets in metric spaces and associated concepts, the closure of a set
Open and closed sets relative to a metric subspace
Compact sets in a metric space (Chapter 2 of Rudin's Principles)
The characterisation of compact subsets of Euclidean spaces
Countable and uncountable sets
Sequences and convergence
Subsequences, subsequential limits
Extracting convergent subsequences and the role of compactness
Cauchy sequences, the definition of completeness
Sufficient conditions for completeness
Topics listed up to this point comprise the syllabus of the mid-term examination. They will
also be a part of the syllabus of the final examination.