Spivak's Calculus as a Prerequisite for General Topology

[CaptainAmerica17]

High school student here...

Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a prerequisite for general topology. Someone recommended that I work my way through Spivak's calculus. I'm just wondering if that could be considered a thorough enough intro to analysis that I could start on topology afterward. Also, does Spivak's calculus cover metric spaces?
(I'm still kind of new to PF, so I wasn't sure what prefix to put on my post.)

 

[fresh_42]

... which from what I've read is a prerequisite for general topology.

It is not.

It even can be a handicap. Calculus is all about metric spaces, $\mathbb{R}^n$ and $\mathbb{C}^n$ to be exact. So these are a subset of all metric spaces which are a (small) subset of all topological spaces. This means, given calculus, you always have some examples at hand, which e.g. tell you what an open set could look like or a continuous function. However, students tend to think that all topological spaces look like them, which is by no means true at all. So it can in fact be a disadvantage to automatically compare topological concepts to real vector spaces like the plane.

If you are interested in topology, there is little needed to start with. A little paperback titled "Topology" or "Introduction to Topology" will already do. The more advanced fields of topology will probably need some preparations, which depend on the direction. Knot theory is topology and doesn't need calculus. Functional analysis is a lot of topology and does need calculus. But to start with the basic concepts of topology, you will only need to know what a set is - and the empty set. That's all you need to get started.

Personally, I do not like such recommendations as given to you. They only discourage students without any necessity. They could even be translated by: "your too young / stupid to learn topology" which is straight away bu...

 

[CaptainAmerica17]

It is not.

It even can be a handicap. Calculus is all about metric spaces, $\mathbb{R}^n$ and $\mathbb{C}^n$ to be exact. So these are a subset of all metric spaces which are a (small) subset of all topological spaces. This means, given calculus, you always have some examples at hand, which e.g. tell you what an open set could look like or a continuous function. However, students tend to think that all topological spaces look like them, which is by no means true at all. So it can in fact be a disadvantage to automatically compare topological concepts to real vector spaces like the plane.

If you are interested in topology, there is little needed to start with. A little paperback titled "Topology" or "Introduction to Topology" will already do. The more advanced fields of topology will probably need some preparations, which depend on the direction. Knot theory is topology and doesn't need calculus. Functional analysis is a lot of topology and does need calculus. But to start with the basic concepts of topology, you will only need to know what a set is - and the empty set. That's all you need to get started.

Personally, I do not like such recommendations as given to you. They only discourage students without any necessity. They could even be translated by: "your too young / stupid to learn topology" which is straight away bu...

Wow! Thanks for the thorough reply. I guess there's no harm in starting a little earlier than anticipated then :D

 

[fresh_42]

Wow! Thanks for the thorough reply. I guess there's no harm in starting a little earlier than anticipated then :D

And in case you get stuck, e.g. with the concept of a pre-image of a function, just come on over and ask. If it is a basic question about understandings, you can post it here: https://www.physicsforums.com/forums/topology-and-analysis.228/ and if it is an exercise or some special examples, you should use our homework forums https://www.physicsforums.com/forums/precalculus-mathematics-homework.155/ or https://www.physicsforums.com/forums/calculus-and-beyond-homework.156/. Just make sure you fill out the (automatically inserted) template, esp. part 3 which is a measure for us to distinguish serious students from lazybones who search for an opportunity to get their homework done.

 

[CaptainAmerica17]

And in case you get stuck, e.g. with the concept of a pre-image of a function, just come on over and ask. If it is a basic question about understandings, you can post it here: https://www.physicsforums.com/forums/topology-and-analysis.228/ and if it is an exercise or some special examples, you should use our homework forums https://www.physicsforums.com/forums/precalculus-mathematics-homework.155/ or https://www.physicsforums.com/forums/calculus-and-beyond-homework.156/. Just make sure you fill out the (automatically inserted) template, esp. part 3 which is a measure for us to distinguish serious students from lazybones who search for an opportunity to get their homework done.

Ok, thanks for the heads-up. You have been very helpful :D

 

[dgambh]

I would suggest to get learn a bit of set theory and how to proof things about sets, some topology textbooks have an introduction chapter about this, but usually is very brief.
Also, it depends on the person. But I have found reading some popular textbooks to understand what topology is about, the big picture I guess. Particularly, I loved Euler's Gem: The Polyhedron Formula and the Birth of Topology. Also, Armstrong's book Basic Topology has really good content, however is very disorganized compared to most textbooks.

 

[CaptainAmerica17]

I would suggest to get learn a bit of set theory and how to proof things about sets, some topology textbooks have an introduction chapter about this, but usually is very brief.
Also, it depends on the person. But I have found reading some popular textbooks to understand what topology is about, the big picture I guess. Particularly, I loved Euler's Gem: The Polyhedron Formula and the Birth of Topology. Also, Armstrong's book Basic Topology has really good content, however is very disorganized compared to most textbooks.

I've done some very basic set theory proofs (de Morgan's law etc.). At this point, I'm not sure what would be considered "enough" to move forward. It's sort of one of the reasons I've been a little stagnated.

 

[fresh_42]

I've done some very basic set theory proofs (de Morgan's law etc.). At this point, I'm not sure what would be considered "enough" to move forward. It's sort of one of the reasons I've been a little stagnated.

If you know what unions of sets, intersections, a power set, and the empty set are, you can start to read a book on topology. You will find most basic definitions on Wikipedia, so even if you do not know some terms, just look them up on Wikipedia and you will know within minutes. To understand the proofs, just get used to ask yourself "what if not?".

E.g. "Every prime number greater than two is odd." - "What if not?" - "If not, then there would be an even prime number greater than two. But this number would be divisible by two, so it cannot be prime. Therefore all primes greater than two have to be odd."

This is basically how proofs are built, resp. how they should be read.

 

[dgambh]

I agree with fresh. I'll add functions and families of sets to the list of things you need to know.

George Simmons book, for example has a first chapter reviewing a lot about sets. I think other books do the same.
Also, take a look at this. I think this link is legal since is from steklov institute.
http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf
That's a fun book to try to follow. They give the definitions and you have to work the examples and rprove the theorems yourself. Hopefully this is useful.

 

[CaptainAmerica17]

I agree with fresh. I'll add functions and families of sets to the list of things you need to know.

George Simmons book, for example has a first chapter reviewing a lot about sets. I think other books do the same.
Also, take a look at this. I think this link is legal since is from steklov institute.
http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf
That's a fun book to try to follow. They give the definitions and you have to work the examples and rprove the theorems yourself. Hopefully this is useful.

Thank you for the link. I was just searching for some PDF's when you replied. I appreciate all of the extra problems to build understanding.

 

[fresh_42]

I appreciate all of the extra problems to build understanding.

First Google hit on "Introduction to Topology": http://www.math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf

One thing before you start. It might be new to you how those books are written. Don't be surprised or frustrated. It usually always takes a while to get accustomed to. It is different from how school books are written, so many students have problems with it at the beginning. You should always have pen and paper at hand to make some sketches to illustrate what you read, resp. scribble some examples. The notes above start with metric spaces. This is for the reason to have simple examples at hand later in the notes.

 

[CaptainAmerica17]

First Google hit on "Introduction to Topology": http://www.math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf

One thing before you start. It might be new to you how those books are written. Don't be surprised or frustrated. It usually always takes a while to get accustomed to. It is different from how school books are written, so many students have problems with it at the beginning. You should always have pen and paper at hand to make some sketches to illustrate what you read, resp. scribble some examples. The notes above start with metric spaces. This is for the reason to have simple examples at hand later in the notes.

Thanks! The first section is on metric spaces too :D

 

[lavinia]

High school student here...

Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a prerequisite for general topology. Someone recommended that I work my way through Spivak's calculus. I'm just wondering if that could be considered a thorough enough intro to analysis that I could start on topology afterward. Also, does Spivak's calculus cover metric spaces?
(I'm still kind of new to PF, so I wasn't sure what prefix to put on my post.)

In my opinion Topology is too broad a subject to take on its own. Things go easier if there is a goal that requires some topological ideas. What are you interested in?

I personally think that Complex Analysis is a good venue for learning the various ideas of continuity of functions and of mapping properties of functions. You would also learn about the topology of the Riemann sphere - perhaps the easiest manifold to understand.

 

[mathwonk]

topology is about the geometric consequences of continuity. thus you might begin with learning the definition of continuity. this is something that often occurs first in analysis or calculus. that might be one reason some people think analysis naturally occurs before topology. I would suggest learning the epsilon - delta definition of continuity in a metric space, and then learn what opoena nd closed sets are, and then try to prove that function is epsilon -delta continuousn if and only if the inverse image of every open set is open. then you have translated the definition iof continuity into a statement that only uses "topological" concepts, namely open sets.

Then learn the intermediate value theorem, and then learn the meaning of connectivity, i.e. connected sets. Try to prove that a subset of the real line is connected if and only if it is an interval. Then prove that a continuous map always takes connected sets to connected image sets, and see how that relates to the intermediate value theorem.

A confession: always looking for the easy way out, I used to think that general topology made the intermediate value theorem trivial, since it is so easy to prove the continuous image of a connected set is connected. I totally missed the fact that proving an interval is connected is the hard part.

One useful technical fact: try to prove that a set S is connected if and only if every continuous map S-->{0,1} is constant.

 

[fresh_42]

One useful technical fact: try to prove that a set S is connected if and only if every continuous map S-->{0,1} is constant.

What do you mean? If we have $S:=\{\,(x,\sin(x))\,|\,0 \leq x \leq \pi/2\,\}$ and a map $\pi \, : \,S \longrightarrow [0,1]$ given by $\pi((x,\sin(x)))=\sin^2(x)$ then $S$ is connected, $\pi$ is continuous, but not constant.

 

[mathwonk]

It takes very good vision, but if you look closely you will see I used braces, not closed brackets, so my set {0,1} is not an interval, but a set composed of two distinct points. In words, a set S is connected iff every continuous map from S to a 2 point set (with the discrete topology) is constant.

 

[fresh_42]

It takes very good vision, but if you look closely you will see I used braces, not closed brackets, so my set {0,1} is not an interval, but a set composed of two distinct points. In words, a set S is connected iff every continuous map from S to a 2 point set (with the discrete topology) is constant.

Thanks and sorry, I really hadn't realized it. Good vision is indeed an issue for me ...

 

[mathwonk]

well those are really tiny undistinguished brackets. yours are much clearer somehow. also i just got cataract surgery last week and it helps me some. i should have used words, since they express the idea better.

anyway i like that characterization of connectivity since it makes all proofs of standard properties rather easy. e.g. the image of a connected set is connected since if S-->T is surjective, any non constant map from T to a two point set would by composition yield a non constant map from S to that same set.

 

[CaptainAmerica17]

In my opinion Topology is too broad a subject to take on its own. Things go easier if there is a goal that requires some topological ideas. What are you interested in?

I personally think that Complex Analysis is a good venue for learning the various ideas of continuity of functions and of mapping properties of functions. You would also learn about the topology of the Riemann sphere - perhaps the easiest manifold to understand.

Well, what sparked my interest in topology was learning about the Mobius strip. I would like to understand it's properties in mathematical terms, and eventually prove things about manifolds etc.

 

[CaptainAmerica17]

topology is about the geometric consequences of continuity. thus you might begin with learning the definition of continuity. this is something that often occurs first in analysis or calculus. that might be one reason some people think analysis naturally occurs before topology. I would suggest learning the epsilon - delta definition of continuity in a metric space, and then learn what opoena nd closed sets are, and then try to prove that function is epsilon -delta continuousn if and only if the inverse image of every open set is open. then you have translated the definition iof continuity into a statement that only uses "topological" concepts, namely open sets.

Then learn the intermediate value theorem, and then learn the meaning of connectivity, i.e. connected sets. Try to prove that a subset of the real line is connected if and only if it is an interval. Then prove that a continuous map always takes connected sets to connected image sets, and see how that relates to the intermediate value theorem.

A confession: always looking for the easy way out, I used to think that general topology made the intermediate value theorem trivial, since it is so easy to prove the continuous image of a connected set is connected. I totally missed the fact that proving an interval is connected is the hard part.

One useful technical fact: try to prove that a set S is connected if and only if every continuous map S-->{0,1} is constant.

I learned the definition of continuity in my calculus class. Of course, not very in-depth, so I'll definitely look into more. Admittedly, I don't really know the importance of the intermediate value theorem. We really haven't made any use for it in class besides learning the definition.

 

[fresh_42]

Well, what sparked my interest in topology was learning about the Mobius strip. I would like to understand it's properties in mathematical terms, and eventually prove things about manifolds etc.

... in which case some knowledge of abstract algebra is recommendable. Simplified: manifolds need calculus, geometric objects need abstract and linear algebra, and to cut a möbius strip along its surface is fun and needs scissors. That is what @mathwonk has meant by "topology is broad". At least I thought that he said it, although I cannot find it at the moment. What I have written above and the reference I gave was about basic set theoretical topology: the basic tools. When it comes to other areas, you will quickly find yourself in the need to understand more tools like differentiability, covering groups and similar.

 

[lavinia]

Well, what sparked my interest in topology was learning about the Mobius strip. I would like to understand it's properties in mathematical terms, and eventually prove things about manifolds etc.

Try The Shape of Space by Jeffrey Weeks. It covers all surfaces including the Mobius band and other much wierder surfaces such as the Klein bottle and the Projective plane. It is a good book written for laymen, intuitive, and well written.

 

[mathwonk]

in reference again to the characterization of connectivity given in post #16, it makes also the proof of another, otherwise tricky to me, property of connected sets easy again, namely: the closure of a coonected subset is again connected. i.e. if you have a continuous map of the closure of S to a 2 point set, and the restriction to S is constant, then also the original map must be constant since it must map the closure of S into the closure of the image of S. I think you will find this version of connectivity makes every property easier to prove than going back to the usual definition does.

And the intermediate value theorem is a great first example of what topology is good for: it makes proving things have solutions very easy, even when finding solutions is hard. Just take any equation that you cannot readily solve, like X^7 + X^3 + 1 = 0. Since X= -1, gives -1 here and X = 1 gives 3, there must be a solution with X somewhere between -1 and 1, even if we don't know exactly where! and by plugging in more numbers in that range we can get an arbitrarily close approximation to a solution.

The principle here is if the border of our domain encloses a solution then the interior of our region must contain a solution. In dimension 2 this becomes, if a continuous map of the disc into the plane wraps its boundary around ), then some point inside the disc must map to 0. Thus we want to learn to define winding numbers...Topology is about using deformations to prove existence theorems, by deforming to an easier case and proving that does not change the answer to the problem.

 

[CaptainAmerica17]

in reference again to the characterization of connectivity given in post #16, it makes also the proof of another, otherwise tricky to me, property of connected sets easy again, namely: the closure of a coonected subset is again connected. i.e. if you have a continuous map of the closure of S to a 2 point set, and the restriction to S is constant, then also the original map must be constant since it must map the closure of S into the closure of the image of S. I think you will find this version of connectivity makes every property easier to prove than going back to the usual definition does.

And the intermediate value theorem is a great first example of what topology is good for: it makes proving things have solutions very easy, even when finding solutions is hard. Just take any equation that you cannot readily solve, like X^7 + X^3 + 1 = 0. Since X= -1, gives -1 here and X = 1 gives 3, there must be a solution with X somewhere between -1 and 1, even if we don't know exactly where! and by plugging in more numbers in that range we can get an arbitrarily close approximation to a solution.

The principle here is if the border of our domain encloses a solution then the interior of our region must contain a solution. In dimension 2 this becomes, if a continuous map of the disc into the plane wraps its boundary around ), then some point inside the disc must map to 0. Thus we want to learn to define winding numbers...Topology is about using deformations to prove existence theorems, by deforming to an easier case and proving that does not change the answer to the problem.

That makes a lot of sense. That's one thing about proofs I'm still learning - utilizing EVERYTHING I know to logically come up with a solution. Even the IVT :P

 

[mathwonk]

the theorems that are most fundamental in calculus arte the ones usually not proved, and they are all based on topological ideas. These are the intermediate value theorem and the max/min value theorem. these are based on the topological ideas of connectedness and compactness, respectively. After giving the definitions, one uses the least upper bopund property of the real numbers to prove that a subset of the reals is connected precisely hen it is an interval, and is comopact precisely when it is both closed and bounded. it ius easy to prove that the continuous image of a connected set is connected and the continuous image of a compact set is compact. hence the continuous image of an interval is an interval (IVT), and the continuous image of a closed bounded interval is a closed bounded interval (MMVT).

Don't get caught learning only the abstract topological ideas though, since the purely topological part about images of connected and compact sets being connected and compact is the easy part. The hard is proving that, in the reals, connected sets are intervals and that closed bounded intervals are compact. So don't just read a point set topology book that does everything abstractly and leaves out all the important special cases involving real numbers.

 

[CaptainAmerica17]

the theorems that are most fundamental in calculus arte the ones usually not proved, and they are all based on topological ideas. These are the intermediate value theorem and the max/min value theorem. these are based on the topological ideas of connectedness and compactness, respectively. After giving the definitions, one uses the least upper bopund property of the real numbers to prove that a subset of the reals is connected precisely hen it is an interval, and is comopact precisely when it is both closed and bounded. it ius easy to prove that the continuous image of a connected set is connected and the continuous image of a compact set is compact. hence the continuous image of an interval is an interval (IVT), and the continuous image of a closed bounded interval is a closed bounded interval (MMVT).

Don't get caught learning only the abstract topological ideas though, since the purely topological part about images of connected and compact sets being connected and compact is the easy part. The hard is proving that, in the reals, connected sets are intervals and that closed bounded intervals are compact. So don't just read a point set topology book that does everything abstractly and leaves out all the important special cases involving real numbers.

Noted. I just found out about the relationship between real numbers and the IVT. It's really cool and definitely sheds some light on this stuff you guys have been talking about :D