1. Sets, Real Numbers, and Axioms (集合，实数和公理)

1.1 What is a set? – Introduction to (naive) Set Theory (集合论初步-朴素集合论)

1.2 Several examples of sets – Natural Numbers & Integers, Rational Numbers, and Real numbers (自然数与整数，有理数和实数)

1.3 How to organize sets? – Binary Relation, Function, and Order (二元关系，函数和序)

1.4 Go beyond the imagination – The size of a set, finite and infinite cardinality (集合的大小，有限和无限)

1.5 Sometimes things are not that obvious – Zorn’s lemma and Axion of choice (佐恩引理与选择公理)

1.6 Sequences and Cauchy sequences

1.7 More about $\mathbb{R}$?

2. Point set topology (点集拓扑)

2.1 What is topology? – Open sets, closed sets, basis, and separation axioms.

2.2 Compactness.

2.3(a) Continuity.

2.3(b) Continuity of real valued functions.

2.4 Several special topologies: Subspace topology, Product topology, and order topology.

2.5 Metric topology.

3*. Algebra

3.1 Group.

3.2 Ring.

3.3 Fields.

4. Linear Algebra

4.1 Structure of Linear spaces – basis, dimension, and linear transformation.

4.2

References:

1. Wikibooks and Wikipedia.

2. Number, sets and axioms, A. G. Hamilton, Cambridge University Press 1982.

3. Real Analysis – Modern Techniques and Their Applications (second edition), Gerald B. Folland, John Wiley & Sons, Inc. 1999.