# Book Recommendation: Topology Without Tears, by Sidney A. Morris

1. Jan 17, 2012

### dalcde

I've came across a book about topolgy, Topology Without Tears by Sidney A. Morris. It can be found here: http://uob-community.ballarat.edu.au/~smorris/topology.htm [Broken]

The explainations are rather clear and an outline of the proof is given before each proof. However, many quite important concepts are contained in the exercises, so you will have to go through all the exercises if you want to learn topology well.

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2. Jan 17, 2012

### micromass

I don't like this book at all. It's far too easy for a topology course. Even Munkres would be a better book.

Where are important concepts like quotient spaces, coproducts, second countable, Hausdorff, completely regular, etc.? Fine, a lot are contained in the exercises. But they don't belong in the exercises. Most of these things are so important to belong in the main text. Putting them in the exercises makes it look like it has mediocre importance, which is not true.

And where are important concepts such as nets or filters?

A topology book should be a little more difficult and have more challenging exercises than this. Otherwise, you're going to suffer greatly in courses like functional analysis or algebraic topology.

Good books are the books by Kelley and Willard. I'm also very charmed by this book: http://www.pdmi.ras.ru/~olegviro/topoman/index.html which basically leaves everything as an exercise.

3. Jan 17, 2012

### dalcde

I agree that this is probably not a good book if you want to study it seriously, but it is a good starter since it is easy enough to understand. I'm reading it because I'm studying on my own and it is the first and only book I found comprehensible. I'll have to read other books to fill in the gaps, though.

4. Jan 17, 2012

I'd recommend Bourbaki, a book by W. Fairchild called Topology which is the most insanely
perfect supplement to Bourbaki in that it offers more elementary exercises & examples
while simultaneously following the pattern Bourbaki follows, Topology by H Schubert for
the fact that it's as if you're having a conversation about Bourbaki's book while reading
it once you've read Bourbaki & Kelley's book for the sets & nets.
It just seems to me that you're going to deal with a lot of petty things like this
when you settle for what are considered easier books so I think the above
recommendations is a virtually guaranteed chance at succeeding through Bourbaki's
book on your own & being able to base your thoughts on something respectable.

Since Willard was recommended you could use Bourbaki & Willard along with 'A General
Topology Workbook' by I.T. Adamson - seeing as these are the exact two books he
recommends to read along with - but I chose not to as it's closer to Willard's book than
to Bourbaki & I found the other, closer to Bourbaki, books I mentioned in the library.
Through painful experience I'd say it's best to use books like topology without tears
when you get stuck. I wish I'd realized all this & given up on the mediocre books sooner
Although if you're not able to deal with, or don't like, books like this then just use whatever works!

5. Jan 17, 2012

### micromass

Did you actually study from Bourbaki?? Did that work out good??
In my experience, the Bourbaki books are an excellent reference, but are not suitable at all for self-studying. I'd be (pleasently) surprised if it actually did work for you!!

6. Jan 17, 2012

### micromass

May I ask you what other books you tried to study and what you did not like about them??
And what mathematical prereqs you already have?? For example, did you already know about metric spaces before studying topology?? Did you already do some real analysis??

In my opinion, it is unwise to start studying topology without at least working through a Spivak level book and without having seen metric spaces somehow.

7. Jan 17, 2012

No MM I'm only working my way through Bourbaki. If you ever have access to a library
try find the books I've recommended & judge for yourself whether a person could read
Bourbaki as their main text, referring to the other two books either after they've digested
Bourbaki's interpretation or when absolutely stuck, based on their content & explanations.
Similarly I'll divulge a secret weapon to conquering Bourbaki's books on set theory & algebra
- R. Godement's Algebra. You can view these fantastic supplementary books as in effect going
to a teacher when you get stuck with Bourbaki.

8. Jan 18, 2012

### dalcde

I think I first studied it here: (not really a book)
http://www.millersville.edu/~bikenaga/topology/topology-notes.html [Broken]

The problem with me is that I'd like to start with some free materials to make sure that I'm capable of understanding the subject or else I'll put the book in the bookshelf and ignore it for the rest of my life, but there aren't many good free materials available.

Back to your questions, I've studied real analysis (didn't actually finish the last part about functions), half way through abstract algebra and a bit of graph theory. I've also studied metric spaces (I found it really important to the understanding because I haven't studied metric spaces yet when I first approached topology, and all those definitions made no sense).

By the way, you said that I should go through a Spivak-level book. Are you referring to the way things and proofs are presented or the exercises or both?

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9. Jan 18, 2012

### micromass

I said to go through a Spivak level book because you have to be acquainted to the way things are done in mathematics. You have to be acquainted to rigorous proofs and the basic style of a math text. Furthermore, you should know all the topics in Spivak before doing topology. You should already know things like the extreme value theorem, the mean value theorem, etc. Why should you know this? Because topology will generalize these things to compactness and connectedness.

I guess topology without tears is a good first encounter to topology, but it is absolutely necessary to go through a more rigorous text later on.

In my opinion, there are two ways to go through before starting with topology. A first way is to study a bit of real analysis. A good real analysis text will already make you acquainted with the idea of an open set, compactness, etc. If you were able to go through a real analysis text, then topology will be a breeze!!

A second way is to do topology first and real analysis next. This has the pro that real analysis will be a lot easier to you (it's quite a daunting topic for newcomers). But a con is that the topology will be unmotivated. This is why I highly recommend the book "metric spaces" by Searcoid. After reading this book, you will have all the motivation you need for tackling topology. Not to mention that the Searcoid book is a joy to read!!

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10. Jan 18, 2012

### dalcde

Totally agreed! Before I studied metric spaces, when I saw topology, all I was thinking was "I thought open sets were ...", based on my knowledge of open sets in the real numbers and it made no sense to me. Now I understand that topology is actually the generalization of open sets in metric spaces.