The material included does not comprehensively cover every topic of
mathematical economics; instead, I have collected the principal results and
issues from (1) the central topics of the foundation of general equilibrium
analysis, and (2) what seems highly mathematical and abstract but crucial
from a methodological viewpoint in the social sciences. The former includes
fixed-point theorems for multivalued mappings (Chapter 2), the existence
of Nash and generalized Nash equilibria (Chapter 3), market equilibrium
(Gale–Nikaido–Debreu) theorems (Chapter 4), and general equilibrium
theory with non-ordered preferences and infinite dimensional commodity
spaces (Chapter 5). The latter includes homological (algebraic) methods
in fixed-point arguments (Chapter 6), a homological type of index theory
(Chapter 7), and axiomatic set theory with mathematical logic as the most
fundamental method to describe objects in the social sciences (Chapter 9).