Chapter 10

प्रस्तावना (Preface)

The preface introduces the book 'वास्तविक विश्लेषण एवं दूरीक समष्टि' (Real Analysis and Metric Spaces), which is designed according to the curriculum proposed by Vardhman Mahaveer Open University, Kota. The book is intended for B.Sc. Part II Mathematics, specifically for the first question paper. The language of the book is kept simple, engaging, and easy to understand, with appropriate use of English terminology alongside Hindi. The authors have referred to standard texts to ensure factual accuracy, and express gratitude to the original authors. The book aims to guide students not only for regular study but also for competitive exams. The preface emphasizes the importance of mathematical rigor and clarity, and assures students that the content is structured to facilitate effective learning.

• Book is structured for B.Sc. Part II Mathematics (First Paper).
• Language is simple, engaging, and includes English terminology.
• Authors have used standard references for accuracy.
• Useful for regular study and competitive examinations.
• Emphasis on mathematical rigor and clarity.
• 📌 वास्तविक विश्लेषण (Real Analysis): The study of real numbers, their properties, and related mathematical concepts.
• 📌 दूरीक समष्टि (Metric Spaces): Mathematical structures that generalize the notion of distance.

इकाई 1 : वास्तविक संख्या निकाय (Real Numbers System) इकाई की रूपरेखा

This section outlines the structure of Unit 1, which focuses on the Real Number System. The unit is divided into several subtopics, each addressing a fundamental aspect of real numbers and their properties. The topics include objectives, introduction, field axioms, order axioms, boundedness, completeness, properties of real numbers (such as Archimedean property and density), modulus, summary, terminology, self-assessment questions, and exercises. The detailed breakdown ensures that students will systematically learn the algebraic and order structure of real numbers, their completeness, and related properties. The outline also mentions the axiomatic approach, which is central to understanding real numbers rigorously.

• Unit covers the Real Number System and its properties.
• Includes field and order axioms, completeness, boundedness, and modulus.
• Axiomatic approach is emphasized.
• Contains summary, terminology, self-assessment, and exercises.
• 📌 Field: A set with two operations (addition and multiplication) satisfying certain axioms.
• 📌 Order: A relation that allows comparison between elements.
• 📌 Boundedness: Property of a set having upper and lower bounds.