Some elementary mathematical concepts and facts related to some areas of mathematical analysis — set theory, topological spaces, Euclidean spaces, measurable structures, and linear topological spaces — are given in this preliminary chapter. After introducing various concepts in each of these topics, mathematical properties and facts are stated as “mathematical property” or “mathematical fact.” The distinction between them is somewhat arbitrary, and yet the term “facts” tends to be used when they are more or less immediate or straightforward compared to “properties.” Some of them are classified as “Mathematical Theorem” or “Mathematical Proposition,” in which case proofs are given. In various parts of this book they are used or needed in the development of a theory presented.