<![CDATA[MKLab - TOPOLOGY]]> https://mklab.gr/ Sun, 13 Sep 2026 16:18:57 +0000 MyBB <![CDATA[Basic Topology [Armstrong]]]> https://mklab.gr/showthread.php?tid=1885 Mon, 07 Sep 2026 22:56:24 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1885 Book:Basic Topology
Author: M. A. Armstrong
Publication date: 5 July 1983 (Springer edition; originally published in 1979)
Publisher: Springer New York
Series:Undergraduate Texts in Mathematics

Basic Topology by M. A. Armstrong is a broad undergraduate introduction to topology that develops the subject through the central idea of topological invariants—properties of spaces that remain unchanged under continuous deformation. The book begins with the foundations of continuity, compactness, connectedness, and quotient or identification spaces, before moving toward more algebraic tools such as the fundamental group. It assumes some familiarity with real analysis, elementary group theory, and linear algebra, making it particularly suitable for advanced undergraduate mathematics students. 

The later chapters introduce triangulations, classification and study of surfaces, simplicial homology, degree theory, the Lefschetz number, knot theory, and covering spaces. This progression allows the reader to see how point-set topology, geometric topology, and algebraic topology fit together. Armstrong emphasizes concrete examples and computation rather than excessive abstraction, and the book contains hundreds of exercises designed to develop problem-solving ability alongside theoretical understanding. 

A major strength of the book is that it gives students an accessible path from elementary ideas about continuous maps and connected spaces to genuinely important algebraic-topological concepts such as $\pi_1(X)$ and homology groups. It is therefore a good first serious topology textbook, especially for readers who want intuition, examples, and exercises before moving to more abstract texts such as Munkres or Hatcher.


BOOK]]> Book:Basic Topology
Author: M. A. Armstrong
Publication date: 5 July 1983 (Springer edition; originally published in 1979)
Publisher: Springer New York
Series:Undergraduate Texts in Mathematics

Basic Topology by M. A. Armstrong is a broad undergraduate introduction to topology that develops the subject through the central idea of topological invariants—properties of spaces that remain unchanged under continuous deformation. The book begins with the foundations of continuity, compactness, connectedness, and quotient or identification spaces, before moving toward more algebraic tools such as the fundamental group. It assumes some familiarity with real analysis, elementary group theory, and linear algebra, making it particularly suitable for advanced undergraduate mathematics students. 

The later chapters introduce triangulations, classification and study of surfaces, simplicial homology, degree theory, the Lefschetz number, knot theory, and covering spaces. This progression allows the reader to see how point-set topology, geometric topology, and algebraic topology fit together. Armstrong emphasizes concrete examples and computation rather than excessive abstraction, and the book contains hundreds of exercises designed to develop problem-solving ability alongside theoretical understanding. 

A major strength of the book is that it gives students an accessible path from elementary ideas about continuous maps and connected spaces to genuinely important algebraic-topological concepts such as $\pi_1(X)$ and homology groups. It is therefore a good first serious topology textbook, especially for readers who want intuition, examples, and exercises before moving to more abstract texts such as Munkres or Hatcher.


BOOK]]> <![CDATA[General Topology [Willard]]]> https://mklab.gr/showthread.php?tid=1884 Mon, 07 Sep 2026 22:52:34 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1884 General Topology
Book:General Topology
Author: Stephen Willard
Original publication: 1970, Addison-Wesley
Dover edition: 2004
Publisher: Dover Publications
ISBN: 978-0-486-43479-7
Field: General / Point-Set Topology


Stephen Willard’s General Topology is a classic, rigorous introduction and reference work in general topology, emphasizing the abstract structures underlying continuity, convergence, compactness, connectedness, and metric spaces. The book develops topology from its basic concepts—topological spaces, open and closed sets, bases, subbases, and continuous mappings—and moves toward more sophisticated topics such as nets and filters, compactness, metrization, complete metric spaces, uniform spaces, and function spaces. Willard treats these subjects systematically, with particular attention to how different notions of convergence and separation interact. 

A distinctive feature is the division between what the publisher describes as “continuous topology” and more geometric aspects of topology. The latter includes substantial discussion of connectedness, local connectedness, topological characterization theorems, and introductory homotopy theory. Rather than restricting itself to standard textbook examples, the book introduces many important and sometimes pathological topological spaces through its exercises. It contains roughly 340 exercises, 27 figures, historical notes, bibliography, and a detailed index, which help make it useful both as a course text and as a long-term reference. 

The style is more theorem–proof oriented and comprehensive than many modern introductory books. It is therefore particularly valuable for someone who already has some mathematical maturity in real analysis or abstract mathematics and wants to understand point-set topology at a deeper level. Topics such as compactness, separation axioms, product spaces, quotient constructions, metrization, and function spaces form much of the conceptual machinery later used in analysis, functional analysis, differential geometry, algebraic topology, and manifold theory. The Dover edition is a reprint of the original 1970 Addison-Wesley text. 

Key takeaways
  • Comprehensive point-set topology: goes considerably beyond a minimal first course.
  • Strong treatment of convergence and compactness: including nets, filters, uniform spaces, and metrization.
  • Excellent reference book: definitions and major classical theorems are developed in a systematic framework.
  • Best suited to mathematically mature readers: especially advanced undergraduates, graduate students, and researchers needing topology as a foundation for other areas. 

BOOK]]> General Topology
Book:General Topology
Author: Stephen Willard
Original publication: 1970, Addison-Wesley
Dover edition: 2004
Publisher: Dover Publications
ISBN: 978-0-486-43479-7
Field: General / Point-Set Topology


Stephen Willard’s General Topology is a classic, rigorous introduction and reference work in general topology, emphasizing the abstract structures underlying continuity, convergence, compactness, connectedness, and metric spaces. The book develops topology from its basic concepts—topological spaces, open and closed sets, bases, subbases, and continuous mappings—and moves toward more sophisticated topics such as nets and filters, compactness, metrization, complete metric spaces, uniform spaces, and function spaces. Willard treats these subjects systematically, with particular attention to how different notions of convergence and separation interact. 

A distinctive feature is the division between what the publisher describes as “continuous topology” and more geometric aspects of topology. The latter includes substantial discussion of connectedness, local connectedness, topological characterization theorems, and introductory homotopy theory. Rather than restricting itself to standard textbook examples, the book introduces many important and sometimes pathological topological spaces through its exercises. It contains roughly 340 exercises, 27 figures, historical notes, bibliography, and a detailed index, which help make it useful both as a course text and as a long-term reference. 

The style is more theorem–proof oriented and comprehensive than many modern introductory books. It is therefore particularly valuable for someone who already has some mathematical maturity in real analysis or abstract mathematics and wants to understand point-set topology at a deeper level. Topics such as compactness, separation axioms, product spaces, quotient constructions, metrization, and function spaces form much of the conceptual machinery later used in analysis, functional analysis, differential geometry, algebraic topology, and manifold theory. The Dover edition is a reprint of the original 1970 Addison-Wesley text. 

Key takeaways
  • Comprehensive point-set topology: goes considerably beyond a minimal first course.
  • Strong treatment of convergence and compactness: including nets, filters, uniform spaces, and metrization.
  • Excellent reference book: definitions and major classical theorems are developed in a systematic framework.
  • Best suited to mathematically mature readers: especially advanced undergraduates, graduate students, and researchers needing topology as a foundation for other areas. 

BOOK]]> <![CDATA[General Topology [Kelley]]]> https://mklab.gr/showthread.php?tid=1883 Mon, 07 Sep 2026 22:49:11 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1883 Author: John L. Kelley
Publication date: 27 June 1975 (Springer edition; originally published in 1955)
Publisher: Springer-Verlag New York
Series:Graduate Texts in Mathematics, Vol. 27
Pages: XIV + 298
ISBN: 978-0-387-90125-1 

Summary
John L. Kelley’s General Topology is a classic graduate-level treatment of general topology, written both as a textbook and as a comprehensive reference, with particular emphasis on the topological ideas needed in modern analysis. Kelley begins with an extensive preliminary chapter covering set theory, relations, functions, orderings, cardinal and ordinal numbers, Cartesian products, and the Hausdorff maximal principle. He then develops the basic language of topological spaces—neighborhoods, closure, interior, boundary, bases and subbases, separation properties, and connectedness—before introducing the powerful theory of Moore–Smith convergence, in which sequences are generalized to directed nets. 

The later chapters move into deeper structural topics: product and quotient spaces, embeddings and metrization theorems, compactness, uniform spaces, completeness, and spaces of functions. This approach makes the book especially valuable for analysis and functional analysis, where ordinary metric-space methods are often too restrictive. Rather than treating topology simply as geometry without distances, Kelley emphasizes its role as a general framework for convergence, continuity, compactness, and approximation. The exercises are substantial and often extend the theory rather than merely testing routine understanding.

A notable feature of the book is its level of abstraction. Kelley often formulates results in considerable generality, which makes the text extremely useful as a reference but somewhat demanding for a first introduction to topology. Contemporary reviews already noted this tension between its role as an accessible textbook and as a highly general reference work. Nevertheless, its careful exposition, rigorous proofs, extensive problems, and influence on the modern language of topology have made it one of the enduring classics of the subject. 

Main topics
  • Set theory and mathematical foundations
  • Topological spaces and neighborhoods
  • Closure, interior and boundary
  • Bases and subbases
  • Separation and connectedness
  • Nets and Moore–Smith convergence
  • Product and quotient spaces
  • Embedding and metrization
  • Compact spaces
  • Uniform spaces and completeness
  • Function spaces and convergence of functions 

Key takeaways
  1. Topology provides the natural language of modern analysis. Concepts such as continuity, convergence and compactness can be formulated without relying on a metric.
  2. Nets are central to Kelley’s approach. They generalize sequences and characterize convergence in arbitrary topological spaces.
  3. The book is broader than a typical introductory topology course, covering uniform spaces, function spaces and sophisticated embedding and metrization results.
  4. It remains particularly valuable as a reference for advanced students and mathematicians, although a beginner may find the abstraction demanding. 

Springer — General Topology by John L. Kelley]]> Author: John L. Kelley
Publication date: 27 June 1975 (Springer edition; originally published in 1955)
Publisher: Springer-Verlag New York
Series:Graduate Texts in Mathematics, Vol. 27
Pages: XIV + 298
ISBN: 978-0-387-90125-1 

Summary
John L. Kelley’s General Topology is a classic graduate-level treatment of general topology, written both as a textbook and as a comprehensive reference, with particular emphasis on the topological ideas needed in modern analysis. Kelley begins with an extensive preliminary chapter covering set theory, relations, functions, orderings, cardinal and ordinal numbers, Cartesian products, and the Hausdorff maximal principle. He then develops the basic language of topological spaces—neighborhoods, closure, interior, boundary, bases and subbases, separation properties, and connectedness—before introducing the powerful theory of Moore–Smith convergence, in which sequences are generalized to directed nets. 

The later chapters move into deeper structural topics: product and quotient spaces, embeddings and metrization theorems, compactness, uniform spaces, completeness, and spaces of functions. This approach makes the book especially valuable for analysis and functional analysis, where ordinary metric-space methods are often too restrictive. Rather than treating topology simply as geometry without distances, Kelley emphasizes its role as a general framework for convergence, continuity, compactness, and approximation. The exercises are substantial and often extend the theory rather than merely testing routine understanding.

A notable feature of the book is its level of abstraction. Kelley often formulates results in considerable generality, which makes the text extremely useful as a reference but somewhat demanding for a first introduction to topology. Contemporary reviews already noted this tension between its role as an accessible textbook and as a highly general reference work. Nevertheless, its careful exposition, rigorous proofs, extensive problems, and influence on the modern language of topology have made it one of the enduring classics of the subject. 

Main topics
  • Set theory and mathematical foundations
  • Topological spaces and neighborhoods
  • Closure, interior and boundary
  • Bases and subbases
  • Separation and connectedness
  • Nets and Moore–Smith convergence
  • Product and quotient spaces
  • Embedding and metrization
  • Compact spaces
  • Uniform spaces and completeness
  • Function spaces and convergence of functions 

Key takeaways
  1. Topology provides the natural language of modern analysis. Concepts such as continuity, convergence and compactness can be formulated without relying on a metric.
  2. Nets are central to Kelley’s approach. They generalize sequences and characterize convergence in arbitrary topological spaces.
  3. The book is broader than a typical introductory topology course, covering uniform spaces, function spaces and sophisticated embedding and metrization results.
  4. It remains particularly valuable as a reference for advanced students and mathematicians, although a beginner may find the abstraction demanding. 

Springer — General Topology by John L. Kelley]]> <![CDATA[Topology [Munkres]]]> https://mklab.gr/showthread.php?tid=1882 Mon, 07 Sep 2026 22:45:30 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1882 Book name:Topology
Author: James R. Munkres
First publication: 1974
Publisher: Pearson / Prentice Hall
Edition commonly used today: 2nd Edition
Field: General Topology & Algebraic Topology

Quote:
Professor  James R. Munkres passed away on July 30 20026
An obituary for him is here OBITUARY

James R. Munkres’ Topology is one of the standard introductions to modern topology. The book is divided into two major parts. The first develops general (point-set) topology, beginning with sets, functions, topological spaces, bases and product topologies before treating continuity, connectedness, compactness, countability and separation axioms. It then moves to more advanced topics such as the Tychonoff theorem, metrization theorems, paracompactness, complete metric spaces, function spaces, Baire spaces, and dimension theory. The presentation is rigorous and theorem–proof oriented, but it includes numerous examples designed to clarify abstract definitions.

The second part provides an introduction to algebraic topology, centered on the fundamental group and covering spaces. Important topics include homotopy, the Seifert–van Kampen theorem, separation results such as the Jordan curve theorem, classification of compact surfaces, classification of covering spaces, and applications of topology to group theory. One of the book's strengths is that the two parts can essentially serve as separate semester courses, making it a bridge from elementary real analysis and metric spaces to more advanced algebraic topology. Pearson describes it as suitable for senior undergraduate or first-year graduate courses. 

Key takeaways
  • Foundational text: especially strong for mastering rigorous point-set topology.
  • Broad coverage: progresses from basic topological spaces to fundamental groups and covering spaces.
  • Proof-intensive: excellent preparation for graduate mathematics and texts such as Hatcher's Algebraic Topology.
  • Best suited to: students already comfortable with proofs, sets, functions and basic real analysis.
  • Mathematical significance: concepts such as compactness, connectedness and continuity developed here underpin analysis, geometry, differential topology and algebraic topology.

BOOK]]> Book name:Topology
Author: James R. Munkres
First publication: 1974
Publisher: Pearson / Prentice Hall
Edition commonly used today: 2nd Edition
Field: General Topology & Algebraic Topology

Quote:
Professor  James R. Munkres passed away on July 30 20026
An obituary for him is here OBITUARY

James R. Munkres’ Topology is one of the standard introductions to modern topology. The book is divided into two major parts. The first develops general (point-set) topology, beginning with sets, functions, topological spaces, bases and product topologies before treating continuity, connectedness, compactness, countability and separation axioms. It then moves to more advanced topics such as the Tychonoff theorem, metrization theorems, paracompactness, complete metric spaces, function spaces, Baire spaces, and dimension theory. The presentation is rigorous and theorem–proof oriented, but it includes numerous examples designed to clarify abstract definitions.

The second part provides an introduction to algebraic topology, centered on the fundamental group and covering spaces. Important topics include homotopy, the Seifert–van Kampen theorem, separation results such as the Jordan curve theorem, classification of compact surfaces, classification of covering spaces, and applications of topology to group theory. One of the book's strengths is that the two parts can essentially serve as separate semester courses, making it a bridge from elementary real analysis and metric spaces to more advanced algebraic topology. Pearson describes it as suitable for senior undergraduate or first-year graduate courses. 

Key takeaways
  • Foundational text: especially strong for mastering rigorous point-set topology.
  • Broad coverage: progresses from basic topological spaces to fundamental groups and covering spaces.
  • Proof-intensive: excellent preparation for graduate mathematics and texts such as Hatcher's Algebraic Topology.
  • Best suited to: students already comfortable with proofs, sets, functions and basic real analysis.
  • Mathematical significance: concepts such as compactness, connectedness and continuity developed here underpin analysis, geometry, differential topology and algebraic topology.

BOOK]]> <![CDATA[Essential Topology [Crossley]]]> https://mklab.gr/showthread.php?tid=1678 Mon, 17 Aug 2026 20:08:42 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1678 Author: Martin D. Crossley
Publication date: 2005
Publisher: Springer London
Series:Springer Undergraduate Mathematics Series

Martin D. Crossley’s Essential Topology is a compact introduction designed to take students from the elementary ideas of continuity and topological spaces toward some of the central concepts of modern algebraic topology. Rather than developing point-set topology in exhaustive generality, Crossley emphasizes geometric intuition, examples, and the reasons particular concepts matter. The early chapters introduce continuous functions, topological spaces and fundamental properties such as connectedness, compactness and the Hausdorff condition. The book then develops constructions involving subspaces, products and quotient spaces, creating the foundation needed for the more algebraic material that follows. 

The second half moves relatively quickly into homotopy, the Euler number, homotopy groups, the fundamental group, simplicial homology and singular homology. This progression is one of the book's main strengths: students see how algebraic objects such as groups can be attached to topological spaces and then used to distinguish spaces that may otherwise appear difficult to compare. The journey includes memorable results such as the Hairy Ball theorem, while the extensive use of examples keeps the abstract machinery connected to geometric problems. Crossley deliberately omits or abbreviates some traditional topics in order to reach homotopy and homology sooner. 

Overall, Essential Topology is particularly suitable for a second-year undergraduate mathematics student who wants a relatively direct route from elementary topology into algebraic topology. Springer describes it as requiring essentially familiarity with continuity and basic algebra and containing enough material for two semester-long courses. Its concise approach makes it attractive for self-study, although a reader wanting a highly comprehensive treatment of point-set topology may eventually want a more extensive companion text. The Mathematical Association of America similarly praised the book for balancing abstract definitions with geometric intuition and for its clear, streamlined treatment. 

 Key Takeaways
  • Direct route to algebraic topology: The book gets from basic topology to homotopy and homology unusually quickly.
  • Geometric motivation: Definitions and theorems are supported by examples intended to explain why the concepts are useful.
  • Broad undergraduate coverage: Fundamental groups, homotopy groups, Euler number, simplicial homology and singular homology are all introduced.
  • Good bridge to advanced study: It is especially useful for readers preparing for more specialized courses or texts in algebraic topology. 

Essential Topology — Springer

Essential Topology — Goodreads]]> Author: Martin D. Crossley
Publication date: 2005
Publisher: Springer London
Series:Springer Undergraduate Mathematics Series

Martin D. Crossley’s Essential Topology is a compact introduction designed to take students from the elementary ideas of continuity and topological spaces toward some of the central concepts of modern algebraic topology. Rather than developing point-set topology in exhaustive generality, Crossley emphasizes geometric intuition, examples, and the reasons particular concepts matter. The early chapters introduce continuous functions, topological spaces and fundamental properties such as connectedness, compactness and the Hausdorff condition. The book then develops constructions involving subspaces, products and quotient spaces, creating the foundation needed for the more algebraic material that follows. 

The second half moves relatively quickly into homotopy, the Euler number, homotopy groups, the fundamental group, simplicial homology and singular homology. This progression is one of the book's main strengths: students see how algebraic objects such as groups can be attached to topological spaces and then used to distinguish spaces that may otherwise appear difficult to compare. The journey includes memorable results such as the Hairy Ball theorem, while the extensive use of examples keeps the abstract machinery connected to geometric problems. Crossley deliberately omits or abbreviates some traditional topics in order to reach homotopy and homology sooner. 

Overall, Essential Topology is particularly suitable for a second-year undergraduate mathematics student who wants a relatively direct route from elementary topology into algebraic topology. Springer describes it as requiring essentially familiarity with continuity and basic algebra and containing enough material for two semester-long courses. Its concise approach makes it attractive for self-study, although a reader wanting a highly comprehensive treatment of point-set topology may eventually want a more extensive companion text. The Mathematical Association of America similarly praised the book for balancing abstract definitions with geometric intuition and for its clear, streamlined treatment. 

 Key Takeaways
  • Direct route to algebraic topology: The book gets from basic topology to homotopy and homology unusually quickly.
  • Geometric motivation: Definitions and theorems are supported by examples intended to explain why the concepts are useful.
  • Broad undergraduate coverage: Fundamental groups, homotopy groups, Euler number, simplicial homology and singular homology are all introduced.
  • Good bridge to advanced study: It is especially useful for readers preparing for more specialized courses or texts in algebraic topology. 

Essential Topology — Springer

Essential Topology — Goodreads]]> <![CDATA[Topology [Jänich]]]> https://mklab.gr/showthread.php?tid=1655 Mon, 17 Aug 2026 19:01:26 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1655 Topology
Author: Klaus Jänich
Translator: Silvio Levy
Series:Undergraduate Texts in Mathematics
English edition: Springer, 1995

Klaus Jänich’s Topology is a compact and unusually intuitive introduction to topology. Rather than presenting the subject primarily through a long sequence of definitions and formal theorems, Jänich emphasizes the geometric ideas and motivations behind topological concepts. Beginning with fundamental notions of topological spaces, the book proceeds through topological vector spaces, quotient topology and completion of metric spaces before introducing homotopy. It then develops countability axioms, CW-complexes, continuous functions, covering spaces and the Tychonoff theorem. A final section on set theory is contributed by Theodor Bröcker.

What distinguishes Jänich’s treatment is its visual and conceptual approach. The book is relatively short, yet attempts to communicate the central ideas of topology through motivation, geometric intuition and illustrations rather than encyclopedic coverage. Springer describes the later German editions as unconventional, highly understandable, extensively illustrated and motivated while still providing formally rigorous arguments. 

This makes it especially attractive as a conceptual introduction or companion to a more comprehensive textbook. Readers looking for a traditional exercise-heavy course text may find it less suitable: the Goodreads page itself includes a reader review noting its emphasis on exposition and overview rather than exhaustive proofs and exercises. 

Key Takeaways
  • Intuition first: Jänich concentrates on understanding why topological concepts are introduced, not merely memorizing their formal definitions.
  • Broad for its size: despite being roughly 200 pages, it reaches from basic topology through homotopy, CW-complexes, covering spaces and the Tychonoff theorem. 
  • Highly visual: illustrations play an important role in communicating the geometric meaning of abstract ideas; the eighth German edition contains around 182 illustrations. 
  • Best as a conceptual introduction or companion: students wanting extensive problem sets and a systematic reference may prefer something like Munkres alongside it.

Overall: ★★★★½ — A distinctive, concise and visually motivated introduction to topology. Jänich is particularly good for readers who want to develop an intuition for what topology is really about before—or alongside—studying the subject from a more exhaustive textbook.


Goodreads — Topology by Klaus Jänich
Springer — Topologie by Klaus Jänich]]> Topology
Author: Klaus Jänich
Translator: Silvio Levy
Series:Undergraduate Texts in Mathematics
English edition: Springer, 1995

Klaus Jänich’s Topology is a compact and unusually intuitive introduction to topology. Rather than presenting the subject primarily through a long sequence of definitions and formal theorems, Jänich emphasizes the geometric ideas and motivations behind topological concepts. Beginning with fundamental notions of topological spaces, the book proceeds through topological vector spaces, quotient topology and completion of metric spaces before introducing homotopy. It then develops countability axioms, CW-complexes, continuous functions, covering spaces and the Tychonoff theorem. A final section on set theory is contributed by Theodor Bröcker.

What distinguishes Jänich’s treatment is its visual and conceptual approach. The book is relatively short, yet attempts to communicate the central ideas of topology through motivation, geometric intuition and illustrations rather than encyclopedic coverage. Springer describes the later German editions as unconventional, highly understandable, extensively illustrated and motivated while still providing formally rigorous arguments. 

This makes it especially attractive as a conceptual introduction or companion to a more comprehensive textbook. Readers looking for a traditional exercise-heavy course text may find it less suitable: the Goodreads page itself includes a reader review noting its emphasis on exposition and overview rather than exhaustive proofs and exercises. 

Key Takeaways
  • Intuition first: Jänich concentrates on understanding why topological concepts are introduced, not merely memorizing their formal definitions.
  • Broad for its size: despite being roughly 200 pages, it reaches from basic topology through homotopy, CW-complexes, covering spaces and the Tychonoff theorem. 
  • Highly visual: illustrations play an important role in communicating the geometric meaning of abstract ideas; the eighth German edition contains around 182 illustrations. 
  • Best as a conceptual introduction or companion: students wanting extensive problem sets and a systematic reference may prefer something like Munkres alongside it.

Overall: ★★★★½ — A distinctive, concise and visually motivated introduction to topology. Jänich is particularly good for readers who want to develop an intuition for what topology is really about before—or alongside—studying the subject from a more exhaustive textbook.


Goodreads — Topology by Klaus Jänich
Springer — Topologie by Klaus Jänich]]> <![CDATA[A Topological Picturebook [Francis]]]> https://mklab.gr/showthread.php?tid=1344 Sun, 26 Jul 2026 02:17:46 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1344 [Image: A_Topological_Picturebook.jpg]
A Topological Picturebook 
BY [George K. Francis]

Summary

A Topological Picturebook, written by mathematician George K. Francis and published by Springer in 1987, is a celebrated text on mathematical visualization within low-dimensional topology. Designed to bridge the gap between abstract geometric concepts and intuitive visual art, the book primarily focuses on manual hand-drawing techniques—such as ink work, cross-hatching, shading, and graphical perspective—to represent complex curved surfaces and three-dimensional spaces. 

Following foundational chapters on surface outlines, cusps, and self-intersections, the narrative unfolds through a series of rich visual "picture stories" that illustrate advanced topological phenomena. These case studies visually unpack intriguing mathematical constructs, including optical illusions like the Penrose triangle, immersions of the projective plane such as Boy's and Roman surfaces, sphere eversion, braid groups, knot theory, and the Hopf fibration. 

Recommended by the Mathematical Association of America for undergraduate mathematics libraries, the work is widely acclaimed by reviewers as both an evocative drawing manual for geometry enthusiasts and an inspiring demonstration of how concrete visual examples can deepen our understanding of intricate abstract mathematics.


BOOK]]> [Image: A_Topological_Picturebook.jpg]
A Topological Picturebook 
BY [George K. Francis]

Summary

A Topological Picturebook, written by mathematician George K. Francis and published by Springer in 1987, is a celebrated text on mathematical visualization within low-dimensional topology. Designed to bridge the gap between abstract geometric concepts and intuitive visual art, the book primarily focuses on manual hand-drawing techniques—such as ink work, cross-hatching, shading, and graphical perspective—to represent complex curved surfaces and three-dimensional spaces. 

Following foundational chapters on surface outlines, cusps, and self-intersections, the narrative unfolds through a series of rich visual "picture stories" that illustrate advanced topological phenomena. These case studies visually unpack intriguing mathematical constructs, including optical illusions like the Penrose triangle, immersions of the projective plane such as Boy's and Roman surfaces, sphere eversion, braid groups, knot theory, and the Hopf fibration. 

Recommended by the Mathematical Association of America for undergraduate mathematics libraries, the work is widely acclaimed by reviewers as both an evocative drawing manual for geometry enthusiasts and an inspiring demonstration of how concrete visual examples can deepen our understanding of intricate abstract mathematics.


BOOK]]> <![CDATA[Counterexamples in Topology [Steen]]]> https://mklab.gr/showthread.php?tid=1318 Sun, 26 Jul 2026 00:12:55 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1318 Counterexamples in Topology  
BY Lynn Arthur Steen

Summary

Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach Jr. is a classic reference book that demonstrates the power of examples in understanding abstract mathematical concepts. Instead of focusing mainly on proofs, it presents more than 140 carefully chosen topological spaces that show why certain assumptions in theorems are necessary and why one property does not always imply another.  

The book begins with basic terminology and general topology concepts, then explores examples related to separation axioms, compactness, connectedness, countability, and metrizability. Each example is analyzed as a complete object, with diagrams, charts, and explanations that reveal its important properties.  A major goal of the book is to train mathematicians to think creatively by constructing and studying unusual spaces that challenge intuition. 

It is especially valuable for undergraduate and graduate students because it turns abstract definitions into concrete experiences and helps explain the subtle relationships between topological ideas. Overall, the book shows that counterexamples are not just exceptions but essential tools for discovering the limits of mathematical theories.


BOOK]]> Counterexamples in Topology  
BY Lynn Arthur Steen

Summary

Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach Jr. is a classic reference book that demonstrates the power of examples in understanding abstract mathematical concepts. Instead of focusing mainly on proofs, it presents more than 140 carefully chosen topological spaces that show why certain assumptions in theorems are necessary and why one property does not always imply another.  

The book begins with basic terminology and general topology concepts, then explores examples related to separation axioms, compactness, connectedness, countability, and metrizability. Each example is analyzed as a complete object, with diagrams, charts, and explanations that reveal its important properties.  A major goal of the book is to train mathematicians to think creatively by constructing and studying unusual spaces that challenge intuition. 

It is especially valuable for undergraduate and graduate students because it turns abstract definitions into concrete experiences and helps explain the subtle relationships between topological ideas. Overall, the book shows that counterexamples are not just exceptions but essential tools for discovering the limits of mathematical theories.


BOOK]]>