TOPOLOGY VOLUME 01: Foundations of Metric and Topological Spaces (Bachelor's Degree in Mathematics)

Are you looking for a topology textbook that explains why the theory works, not just how to apply definitions?

Topology Vol. 01 — Foundations of Metric and Topological Spaces provides a rigorous and carefully motivated introduction to the core concepts of modern topology. Starting from metric spaces and distance functions, this volume guides the reader step by step toward the abstract framework of general topology based on open and closed sets.

Rather than presenting topology as a collection of isolated definitions, this book emphasizes structure, intuition, and logical development. Historical motivation, precise proofs, and well-chosen examples from functional analysis, differential geometry, and mathematical physics are integrated throughout, ensuring both depth and applicability.

What you will learn in this volume Metric and Normed Spaces
A systematic axiomatic treatment of metrics, including symmetry, identity, and the triangle inequality. Classical examples such as Euclidean spaces, LpL^pLp norms, and the supremum norm are explored, together with applications to spaces of functions. Topics include convergence of sequences, completeness, and homeomorphisms between normed spaces.

Pseudometric Spaces
A fundamental generalization allowing distinct points to have zero distance, essential for quotient constructions and equivalence classes. The induced topology is studied with applications to measurable function spaces (LpL^pLp), fractal structures, and singular measures.

Continuous Functions
Continuity is presented from two complementary perspectives: the classical ε\varepsilonε–δ\deltaδ definition and the topological approach via preimages of open sets. Uniform continuity on compact spaces is analyzed in detail, highlighting its role in global stability.

Open and Closed Sets
The axioms governing unions and intersections, along with the construction of bases and subbases, are developed as the foundation of topological reasoning. This chapter establishes the language used throughout all later areas of topology.

Connected Sets
The concept of connectedness as non-separability into disjoint open sets is introduced and analyzed through classical and advanced examples, paving the way for applications in homotopy theory and algebraic topology.

Solved and Proposed Exercises
Over 200 carefully structured problems, including fully worked solutions, guided strategies, and advanced challenges that connect topology with physics, engineering, and data modeling.

Why this book stands out Rigorous mathematical proofs with clear logical structure

Strong pedagogical focus with examples and counterexamples

Clean layout and reader-friendly typography

A solid foundation for advanced topics developed in Volume II

Ideal for Undergraduate and graduate students in Mathematics, Physics, Engineering, and Applied Sciences; instructors of Analysis and Topology; researchers and professionals who use topological methods in theory or applications.

Series: Bachelor’s Degree in Mathematics
Author: Helbert Justo Luque Zevallos
Origin: Arequipa, Peru

Strengthen your understanding of the foundations of topology and build the conceptual tools required for advanced mathematical study.