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Very Short Introductions #622
Topology: A Very Short Introduction
How is a subway map different from other maps? What makes a knot knotted? What makes the Möbius strip one-sided? These are questions of topology, the mathematical study of properties preserved by twisting or stretching objects. In the 20th century topology became as broad and fundamental as algebra and geometry, with important implications for science, especially physics.
In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eye-opening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field.
ABOUT THE The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eye-opening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field.
ABOUT THE The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
176 pages, Paperback
Published February 1, 2020
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225 people want to read
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Displaying 1 - 17 of 17 reviews
July 22, 2021
More than a hundred years ago when Heisenberg discovered that quantum mechanical observables could be written within arrays, he was jubilant for a while. Why just for a while? Because he soon discovered that all of it was based on a mathematical model wherein X times Y did not equal Y times X. It was mind boggling, physics had known nothing like it at the time. When Max Born received a report of this, he was troubled too. Not just because the mathematics seemed strange, but also because it seemed vaguely familiar. Soon, Born realised that Heisenberg had used matrix multiplication and linear algebra unknowingly. When told so, Heisenberg was massively annoyed. "I do not even know what a matrix is" he lamented. And so it was with me when I realised that I will need at least a preliminary understanding of topology if I want to probe further into the general theory of relativity, quantum field theory or fluid mechanics.
"But I don't even know what topology is." Well, not much anyway.
I feel like physicists and physics students are forever fated to circle around mathematics, stalk it and frequently scream with equal parts enthusiasm and fear "That! That looks useful! Give me that."
In hindsight, I should have tried Studying topology much earlier given that I'm interested in theoretical physics and that my partner has a masters in pure mathematics and is especially interested in algebraic topology. But well, I tend to ignore things until they slap me in the face. Now I know what you're thinking. I should be reading an actual textbook. But sheesh, I'm swamped with other textbooks and studies right now, so I don't have the mental capacity for a topology textbook. I tried to sate my curiosity temporarily with this.
It was honestly quite disappointing, because I'm interested in differential topology (which is of most value to physics right now). But this book mostly covers geometric topology which I find very boring. But well, beggars can't be choosers I suppose.
"But I don't even know what topology is." Well, not much anyway.
I feel like physicists and physics students are forever fated to circle around mathematics, stalk it and frequently scream with equal parts enthusiasm and fear "That! That looks useful! Give me that."
In hindsight, I should have tried Studying topology much earlier given that I'm interested in theoretical physics and that my partner has a masters in pure mathematics and is especially interested in algebraic topology. But well, I tend to ignore things until they slap me in the face. Now I know what you're thinking. I should be reading an actual textbook. But sheesh, I'm swamped with other textbooks and studies right now, so I don't have the mental capacity for a topology textbook. I tried to sate my curiosity temporarily with this.
It was honestly quite disappointing, because I'm interested in differential topology (which is of most value to physics right now). But this book mostly covers geometric topology which I find very boring. But well, beggars can't be choosers I suppose.
January 5, 2020
It is an accessible read, but not a trivial one. As somebody with no previous studies of topology, I felt that the book managed to convey a good image of the relevance and the beauty of the field. A reader with little mathematical experience could probably follow this book if they are willing to put in the work, but following the concepts and demonstrations will demand some effort even from those who already have some exposition to mathematics. This is not the author's fault, as he clearly presents what one must know to face the challenges posed by the book, but it's still a lot of content to cover within the limits of a Very Short Introduction. But even if one only engages lightly with the book, it still seems to provide a nice panoramic view of topology.
February 22, 2021
An introductory book that gives a look at topology: what it is, what is can be used for and some work being done in topology. The first chapter goes in gently by looking at Euler's formula for polygons and showing how it applies to polygons in general. Later chapters rapidly become very mathematical and probably requires some level of mathematical education to appreciate properly, even if you have to skim through some mathematical relationships to get at the heart of topology.
Chapter One gives an introduction to the study of topology, which is concerned with the relationship of shapes, connections and relative positions of objects. It then introduces Euler's formula, which relates the number of vertices (V), edges (E) and faces (F) of objects into a mathematical formula and shows that for standard, three-dimensional shapes, V - E + F = 2 always holds. The chapter then goes on to give a readable proof for why this equation is true of such shapes. Then, using the formula, it demonstrates why there can are only five Platonic solids in three dimensions.
Chapter Two looks at surfaces in general. Starting from a square plane, its edges are then deformed and glued together in certain procedures to give rise to shapes like a torus. It also shows how the value found by applying Euler's formula changes for different kind of surfaces. However, Euler's formula is not sufficient as the introduction of one-sided surfaces (the Möbius strip and Klein bottle) show. This leads to the Classification Theorem which is used to classify two-sided and one-sided surfaces. Complex numbers are then briefly introduced, leading to Riemann Surfaces, used to represent surfaces in higher dimensions.
Chapter Three looks at Continuous Functions and the issues with coming up with a definition for them that is mathematically rigorous. This chapter is probably 'heavy going' for non-mathematicians but gives an idea of why some 'common sense' definitions may not be rigorous enough for the needs of mathematicians.
Chapter Four continues the mathematical theme by looking at defining metrics and distances between functions, leading on to sets and subsets and connectivity. All of which would come to bear on more advanced topics on topology.
Chapter Five builds on the previous two chapters by covering different kinds of topology. From geometric topology introduced in Chapter One, other kinds of topology are introduced here: differential topology (dealing with curves), the 'hairy ball theorem' which deals with vectors on topological surfaces, and so on.
Chapter Six looks at the topic of knots (and unknots), showing how topology deals with how to define whether a loop is a knot and what kind of knot it is.
Chapter One gives an introduction to the study of topology, which is concerned with the relationship of shapes, connections and relative positions of objects. It then introduces Euler's formula, which relates the number of vertices (V), edges (E) and faces (F) of objects into a mathematical formula and shows that for standard, three-dimensional shapes, V - E + F = 2 always holds. The chapter then goes on to give a readable proof for why this equation is true of such shapes. Then, using the formula, it demonstrates why there can are only five Platonic solids in three dimensions.
Chapter Two looks at surfaces in general. Starting from a square plane, its edges are then deformed and glued together in certain procedures to give rise to shapes like a torus. It also shows how the value found by applying Euler's formula changes for different kind of surfaces. However, Euler's formula is not sufficient as the introduction of one-sided surfaces (the Möbius strip and Klein bottle) show. This leads to the Classification Theorem which is used to classify two-sided and one-sided surfaces. Complex numbers are then briefly introduced, leading to Riemann Surfaces, used to represent surfaces in higher dimensions.
Chapter Three looks at Continuous Functions and the issues with coming up with a definition for them that is mathematically rigorous. This chapter is probably 'heavy going' for non-mathematicians but gives an idea of why some 'common sense' definitions may not be rigorous enough for the needs of mathematicians.
Chapter Four continues the mathematical theme by looking at defining metrics and distances between functions, leading on to sets and subsets and connectivity. All of which would come to bear on more advanced topics on topology.
Chapter Five builds on the previous two chapters by covering different kinds of topology. From geometric topology introduced in Chapter One, other kinds of topology are introduced here: differential topology (dealing with curves), the 'hairy ball theorem' which deals with vectors on topological surfaces, and so on.
Chapter Six looks at the topic of knots (and unknots), showing how topology deals with how to define whether a loop is a knot and what kind of knot it is.
June 22, 2020
Topology is, in my estimation, one of the most fascinating subfields of mathematics. Even though I personally never gravitated towards it, I find that the topics and questions that it explores are very intriguing. This is in large measure due to the fact that these questions seem very intuitive to state and oftentimes even more easy to visualize. Nonetheless, many of them are not so easy to answer. Classification theorems in particular are very hard to get, and we apparently still don't have them for certain very familiar objects.
This very short introduction is an incredibly thorough, yet very dense book. It is in my estimation perhaps the most technically difficult books in this series. This is partly due to the nature of the subject, but I believe that the author could have left many fascinating, yet difficult, discussions out, or rather replaced them with some more generally accessible material. Nonetheless, it has been valuable to at least get a glimpse of how some very fundamental theorems in topology can be proven. For me the most interesting tidbit of information in the book was the fact that there is still no constructive algorithm by which we can prove that two knots are equivalent.
Overall, this is a very well written book, but it's not a light reading by any stretch of imagination.
This very short introduction is an incredibly thorough, yet very dense book. It is in my estimation perhaps the most technically difficult books in this series. This is partly due to the nature of the subject, but I believe that the author could have left many fascinating, yet difficult, discussions out, or rather replaced them with some more generally accessible material. Nonetheless, it has been valuable to at least get a glimpse of how some very fundamental theorems in topology can be proven. For me the most interesting tidbit of information in the book was the fact that there is still no constructive algorithm by which we can prove that two knots are equivalent.
Overall, this is a very well written book, but it's not a light reading by any stretch of imagination.
January 19, 2025
He who lacks a why can bear almost no how.
Notes
Focus of topology is shape, connections and relative position rather than geometry’s angles, distance and area.
E, F, T, Y are all homeomorphic to the topologist, different from {C,G,I,J,S,U,V,W,L,M,N,Z) in that they have a T-junction. Other groupings are DO, KX, AR, B, PQ, H.
Mathematical proof for why there are only 5 platonic solids starts with Euler’s V-E+F=2, deconstruct into 2D shapes (square for cube) to add n (number of edges making up a face, Cube=4) and m (number of edges that meet in a vertex, Cube=3) to get 1/ + 1/n needs to be >½. So the only 5 combinations are {m,n} = 3,3 Tetrahedron; 3,4 Cube; 4,3 Octahedron; 3,5 Dodecahedron; 5,3 Icosahedron (20 faces). But doesn’t necessarily mean only 1 Platonic solid per combination, or that a combination must have a solid.
4,4; 3,6; 6;3 are solutions if we permit E to be infinite. These ‘solutions’ correspond to tessellations of the plane where four squares meet at a vertex, where three regular hexagons meet at a vertex (as with honeycombs), and where six equilateral triangles meet at a vertex.
the cube and octahedron, and likewise the dodecahedron and icosahedron are dual to one another—the midpoints of the faces of a cube make an octahedron, and vice versa;
Football, combination of pentagons and hexagons, is a truncated icosahedron - planing down the icosahedron, around each vertex, the five edges meeting there to form pentagon (centered around the vertex), and hexagon (space between vertices). 12 Pentagons (equal to original number of vertices) and 20 Hexagons (original faces).
Any surface on which possible to reverse the sense of an orientation loop is non-orientable - Mobius strip. Surface with an inside and an outside is orientable.
Euler number not enough to know the shape. Torus (orientable) and Klein bottle (non orientable) both have E=0.
The complex version of y=x^2 if we include its point at infinity is topologically a sphere, a surface called Riemann surface.
Rigorous definition of continuity of function f at point a: for any positive e, there is a positive d such that difference in outputs between f(x), f(e) is less than e when difference in inputs between x and a is less than d. (So even though an exponential function might jump insanely between 1000 and 1001, the d will just be set accordingly tightly).
Given a set M, then any collection of sets T in M which satisfies the following: for any collection of sets in T, then their union is also in T; for any finite collection sets in T, then their intersection is also in T; M is in T; the empty set is in T. is called a topology on M and a set M, with a topology of sets , is known as a topological space. Small topology, the trivial topology just includes M and empty-set. Largest topology, discrete topology, contains extreme separation, every single set.
a sequence xn, in a metric space M, converges to a limit a in M if any ball B(a, r) contains a tail of the sequence; this means that from some term onwards, all remaining terms of the sequence are inside B(a, r). Using this definition, we can see why the sequence converges to 0. The ball B(0, r) contains all terms xn of the sequence where n > 1/r, which is a tail of the sequence.
If C is a set in a metric space M, then it is closed if whenever a sequence of points in C converges then the limit is also in C. Closed sets might be thought of as those sets that contain all their boundary points.
A metric space in which the Bolzano–Weierstrass theorem holds, so that all sequences in the space have a selection that converges to a limit in the space, is called compact.
A point of a connected space which, when removed, disconnects the space is called a cut point.
An important first result about smooth functions is that the gradient of the function is zero at any maximum or minimum. This result is known as Fermat’s theorem and these maxima and minima are known as critical values and the corresponding inputs as critical points.
Number of maxima + number of minima - number of saddles = Euler number of the surface. In this way the Euler number is a global constraint as to what features smooth functions may have on a closed surface.
the hairy ball theorem says that you can’t comb a hairy ball flat without creating a tuft or cow-lick where the hair refuses to lie flat. a metaphor for a tangent vector field which might be easiest thought of as a fluid flowing on a surface: for any fluid flow on a sphere, there will be one or more points where the fluid is still and unmoving, the way in the eye of a storm the wind is calm. Such points, where the fluid is still, are called singularities.
Poincaré’s theorem:The indices of the singularities of a vector field on a closed surface add up to its Euler number.
A knot is a circle. If two knots are equivalent, one knot and the space around it can be continuously deformed into another - ambient isotropy.
Any 3D knot can be unknotted in 4D: use extra dimension to make any under-crossing into an over-crossing; take the under-part of the knot smoothly into the future, raise it above the over-part of the knot which only exists in the present, and then smoothly return the knot from the future to the present, now as an over-part
Notes
Focus of topology is shape, connections and relative position rather than geometry’s angles, distance and area.
E, F, T, Y are all homeomorphic to the topologist, different from {C,G,I,J,S,U,V,W,L,M,N,Z) in that they have a T-junction. Other groupings are DO, KX, AR, B, PQ, H.
Mathematical proof for why there are only 5 platonic solids starts with Euler’s V-E+F=2, deconstruct into 2D shapes (square for cube) to add n (number of edges making up a face, Cube=4) and m (number of edges that meet in a vertex, Cube=3) to get 1/ + 1/n needs to be >½. So the only 5 combinations are {m,n} = 3,3 Tetrahedron; 3,4 Cube; 4,3 Octahedron; 3,5 Dodecahedron; 5,3 Icosahedron (20 faces). But doesn’t necessarily mean only 1 Platonic solid per combination, or that a combination must have a solid.
4,4; 3,6; 6;3 are solutions if we permit E to be infinite. These ‘solutions’ correspond to tessellations of the plane where four squares meet at a vertex, where three regular hexagons meet at a vertex (as with honeycombs), and where six equilateral triangles meet at a vertex.
the cube and octahedron, and likewise the dodecahedron and icosahedron are dual to one another—the midpoints of the faces of a cube make an octahedron, and vice versa;
Football, combination of pentagons and hexagons, is a truncated icosahedron - planing down the icosahedron, around each vertex, the five edges meeting there to form pentagon (centered around the vertex), and hexagon (space between vertices). 12 Pentagons (equal to original number of vertices) and 20 Hexagons (original faces).
Any surface on which possible to reverse the sense of an orientation loop is non-orientable - Mobius strip. Surface with an inside and an outside is orientable.
Euler number not enough to know the shape. Torus (orientable) and Klein bottle (non orientable) both have E=0.
The complex version of y=x^2 if we include its point at infinity is topologically a sphere, a surface called Riemann surface.
Rigorous definition of continuity of function f at point a: for any positive e, there is a positive d such that difference in outputs between f(x), f(e) is less than e when difference in inputs between x and a is less than d. (So even though an exponential function might jump insanely between 1000 and 1001, the d will just be set accordingly tightly).
Given a set M, then any collection of sets T in M which satisfies the following: for any collection of sets in T, then their union is also in T; for any finite collection sets in T, then their intersection is also in T; M is in T; the empty set is in T. is called a topology on M and a set M, with a topology of sets , is known as a topological space. Small topology, the trivial topology just includes M and empty-set. Largest topology, discrete topology, contains extreme separation, every single set.
a sequence xn, in a metric space M, converges to a limit a in M if any ball B(a, r) contains a tail of the sequence; this means that from some term onwards, all remaining terms of the sequence are inside B(a, r). Using this definition, we can see why the sequence converges to 0. The ball B(0, r) contains all terms xn of the sequence where n > 1/r, which is a tail of the sequence.
If C is a set in a metric space M, then it is closed if whenever a sequence of points in C converges then the limit is also in C. Closed sets might be thought of as those sets that contain all their boundary points.
A metric space in which the Bolzano–Weierstrass theorem holds, so that all sequences in the space have a selection that converges to a limit in the space, is called compact.
A point of a connected space which, when removed, disconnects the space is called a cut point.
An important first result about smooth functions is that the gradient of the function is zero at any maximum or minimum. This result is known as Fermat’s theorem and these maxima and minima are known as critical values and the corresponding inputs as critical points.
Number of maxima + number of minima - number of saddles = Euler number of the surface. In this way the Euler number is a global constraint as to what features smooth functions may have on a closed surface.
the hairy ball theorem says that you can’t comb a hairy ball flat without creating a tuft or cow-lick where the hair refuses to lie flat. a metaphor for a tangent vector field which might be easiest thought of as a fluid flowing on a surface: for any fluid flow on a sphere, there will be one or more points where the fluid is still and unmoving, the way in the eye of a storm the wind is calm. Such points, where the fluid is still, are called singularities.
Poincaré’s theorem:The indices of the singularities of a vector field on a closed surface add up to its Euler number.
A knot is a circle. If two knots are equivalent, one knot and the space around it can be continuously deformed into another - ambient isotropy.
Any 3D knot can be unknotted in 4D: use extra dimension to make any under-crossing into an over-crossing; take the under-part of the knot smoothly into the future, raise it above the over-part of the knot which only exists in the present, and then smoothly return the knot from the future to the present, now as an over-part
September 28, 2025
Topology is often called "rubber sheet geometry" because it includes the study of invariance under stretching and twisting, but that's only a small part of it. It might be more accurately though less poetically described as the study of connectedness and adjacency. In any case, the ideas behind topology are fascinating, initially a little counterintuitive, but it is not so hard to comprehend the basics and get comfortable with them. This book requires no more than high school level geometry, algebra and set theory as a background.
But, sigh, following the discussion in a math book isn't enough to really learn its subject matter. For math to sink in, unless you have a level of facility that I have not been blessed with, you have to actually do it. You need to work through some problems and simple proofs on your own to appreciate the deeper logic of it and to attain enough skill in the area of study that its foundational concepts become intuitive. Then when the more complex ideas come, they will stand on a firm base so that they too will make sense. Sadly, I'm too lazy in my casual reading to do that. That means that I am doomed to the most superficial kind of understanding, so inevitably I begin to have more trouble following the arguments as they move past the basics. I think that what I need to do to go beyond the most simplistic understanding of topology is to take a class in it, where I would have a teacher who could guide, explain and give me assignments to build my skills.
But, sigh, following the discussion in a math book isn't enough to really learn its subject matter. For math to sink in, unless you have a level of facility that I have not been blessed with, you have to actually do it. You need to work through some problems and simple proofs on your own to appreciate the deeper logic of it and to attain enough skill in the area of study that its foundational concepts become intuitive. Then when the more complex ideas come, they will stand on a firm base so that they too will make sense. Sadly, I'm too lazy in my casual reading to do that. That means that I am doomed to the most superficial kind of understanding, so inevitably I begin to have more trouble following the arguments as they move past the basics. I think that what I need to do to go beyond the most simplistic understanding of topology is to take a class in it, where I would have a teacher who could guide, explain and give me assignments to build my skills.
October 11, 2020
This is not an easy book for the lay reader. It starts gently with a discussion of the Euler characteristic – during which Riemann surfaces are introduced without much warning. When it deals with the classification of metric spaces, I really start to struggle to follow all the intricacies (admittedly by trade I am merely a medical doctor). The last several pages on Jones polynomial are totally inscrutable to me. Overall it is a story on the classification of surfaces/spaces and identification of equivalents and invariants. The book does an excellent job in explaining how topology is inextricably interconnected with other branches of mathematics such as analysis and algebra. This is the strength of the book and the reader should be able to appreciate the underlying nature of mathematics which is the description of reality in multiple but logically consistent ways. Three stars.
September 5, 2021
Easy read covering wide range of topics in topology. I used it to get an overview of the field before embarking on more serious self-study of a “real” textbook. Would have been 5 stars but felt author lost the some of the thread when going into continuity and metric spaces, to me an unnecessary detour.
January 2, 2024
This book is definitely a short introduction to topology, but it does assume the reader has a fair bit of mathematical experience. If you’re looking to learn about the history of topology and what I entails without diving into the proofs and equations, this isn’t really for you.
Either way, covers a lot of ground in a short book.
Either way, covers a lot of ground in a short book.
May 10, 2025
A head “connected” to body listening to audio version and going in loops trying to imagine and see the pictures when not using the book for pictures will be in a knot or unknot ? .. that’s the question 🤪
Not the right book for audio version for sure .
Not the right book for audio version for sure .
October 25, 2020
Good introduction
It introduces the motivation and key results in a simple manner without 2:much mathematical sophistication. I read it aloud on kindle
It introduces the motivation and key results in a simple manner without 2:much mathematical sophistication. I read it aloud on kindle
November 24, 2021
Thanks. I now know I hate Topology.
April 22, 2023
Won't blow your mind, but a nice, high-level introduction to the subject
February 27, 2025
An an enjoyable book. The AVSI series is highly reccomended by me
March 2, 2025
Pretty technical, and probably better as a companion to math course than casual reading.
December 14, 2025
Gave up on the audiobook because of abundant references to "figure x in the attached PDF". This book is simply not doable by audiobook, nothing against the author.
December 3, 2024
Exactly what it says on the cover, though this book has a very narrow target demographic. You will probably only enjoy this book if you barely know anything about topology but are still actively interested in math. I fit this description when I first read this book in high school, but I didn't get anything out of rereading it now that I've been well exposed to university-level math. I was also disappointed that there were no exercises given to the reader. I know this is supposed to be a Very Short Introduction and not a textbook, but to me, exercises feel necessary in a pure math book.
Displaying 1 - 17 of 17 reviews