Talk:Topology

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I'm glad to see that someone has started working on a topology textbook. I would like to start making a few additions, and I have a few ideas for the direction I would like to see the book take. I would like to chat / collaborate with anyone else interested in contributing, and I hope that no one takes personal offense to my edits.

Billk 06:24, 30 Jul 2004 (UTC)

• I've made a few clarifications and additions. I'm also interested in discussing how to structure the book, if anyone else interested in editing it is still around, but if not I'll probably go ahead and change things. In particular, I note that the page naming convention seems inadwquate, and I'd like to move a lot of the pages, to a "Topology:" prefix. J Bytheway 20:29, 28 Jan 2005 (UTC)

• You do know about Allen Hatcher's Algebraic Topology book, no? It's a free book, available on his website.

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Contents 1 Zorn's Lemma, AoC in the prereqs? 2 Format 3 Work to be done 4 Proposal for further improvements 5 TOPOLOGY EXPERT 6 Development stages

[edit] Zorn's Lemma, AoC in the prereqs?

Is it really a good idea to put these in the prereqs without qualification? I mean, you can get through a fair part of basic topology without ever needing to invoke ZL or AoC. In the two books I have handy, they introduce AoC and prove Zorn's Lemma and then get to using it. I think it is a similar problem to the use of the word "Monad" in Haskell, it sounds scary, and so people may be offput from _using_ this (lovely) wikibook if they see these things, even though the subjects are not (IMO) super-difficult to grasp. Any thoughts, perhaps just qualifying them with a "For Section Foo, will need Zorn's Lemma and the AoC" or even just shoving them into the text as an Aside? Jfredett (talk) 05:50, 30 October 2009 (UTC)

[edit] Format

Much has been discussed about content, but not about formatting, what should the format of each chapter be? I, given my experience, can deal with a simple, linear "(Theorem-Proof)*" format, others will likely prefer different. As it stands, there seems to be no standard format, some pages (such as the Completeness page) are the TP* style, some are more structured (like the page on the Fundamental Group. I'm happy to do the grunt work of rearranging the material, I just want to know if there is a standard format being used. Perhaps it would be wise to create a "Writing Standards" page somewhere with the format and organization rules organized there? Jfredett (talk) 05:50, 30 October 2009 (UTC)

[edit] Work to be done

Hello, i just saw this book. and it seems to have lots of potential. I think we should list the subjects we would like to see in it, and evaluate them:

• How important they are?
• How diffecult to understanding ?
  • This will mean more examples. I personaly thing that you can never have to many examples.

Here is an initial list of subjects as i see it (ordered as the should appear. ):

1. Metric spaces (the reason in the iterm below), importantcy: high
   1. suequences and convergence importantcy: high
   2. open and closed sets ( For this i think it is perferable to explain about metric spaces first, when it is more intuitive to undersanded and motivated.) importantcy: high
   3. continueity importantcy: high
   4. sub spaces importantcy: low
2. Topological Space. importantcy: high
   1. closed and open sets importantcy: high
   2. The metric induced topology importantcy: medium
   3. Co-finite topology. importantcy: medium
   4. sub spaces importantcy: low
   5. continueity importantcy: high
   6. homermorphism. importantcy: high
      1. definition of a topological proprety, and basic examples.
3. Speration axioms.
4. continueity
5. compactness
6. connectedness
7. ... to be continued (by you ?)

write them (according to importance). Only after covering one subject completly, we will move to the next.

You are invited to contibue to this list. I think that by this way we can consentrate efforts, to make this book more usable, quickly.

If you think this idea is bad, i would like to hear it too.

If no one will object in the next month, i shall consider it as agreeing, and redesign the book to suit this proposed structure.

--Uvgroovy 19:14, 29 April 2006 (UTC)

[edit] Proposal for further improvements

Hello everyone!

I've seen the book and I would like to discuss further improvements to it.

I think that before adding any new materials to it, we should discuss the organization of the book in parts, chapters and sections.

Division of the book in parts

In my opinion the present divison of the book in 5 parts should be kept for now, namely

1. Introduction

2. Point-set topology

3. Algebraic topology

4. Differential topology

5. Help

Contents of each part

1. Introduction

This part should contain introductory material on set theory, basic algebraic structures (group, ring, module), and a survey on real numbers and real functions of a real variable. Everything that we need before starting.

2. Point-set topology

Here I express what I know to be a controversial direction in teaching topology: I believe the general topological space should be presented before the metric space. Reasons for that:

- Abstract distance is not easier to understand than general topology. Take for example the euclidean straight, plane or space with the metric which, given two points in it, associates 0 or 1, whether the points are the same or different. And now explain to a freshman that this distance is so much a distance in each of its particulars as the usual distance and wait for his comments.

- To a certain degree one can present metric independently of topology: definition of metric, sub-space, balls, limited set, Cauchy sequence - none of this concepts needs topology.

- But when one comes to the matters of closeness, convergence, adherence, continuity, there's nothing why one should present it firstly on a metric space, they are just topological aspects which metric spaces possess among others.

Now, I believe one should gather in different bags what is hard to comprehend and what is a piece of cake though necessary (an unecessary piece of cake shouldn't even be mentioned). The concept of topology is necessary but a piece of cake. Such are all the basics one can say about it: sub-space, closure, neighbourhood, interior and so on.

I believe that compactness and connectedness are hard bones in topology. That, and their applications in other mathematical branches, demand they should be gathered in a single chapter for mind-order's sake as for easing the work of analists, geometers and so on, who know topology well but just need a quick view of the bone. On the other hand all the basic topological constructions such as cartesian product, quotient space, space summing and glueing are nothing but technical instruments, constantly being used in every branch of mathematics needing topology, but uninteresting for itself (unless one's in a cathegory theory discussion).

I also believe one should take a short look to "general countability" and "general uniformity" before realizing that metric has all these propierties.

Thus I propose that part 2 should be organized in chapters and sections as follows

1. Topology

1.1. topology (including definition, open and closed sets, sub-space, base and generated topology, neighbourhood, interior, exterior, border, closure, density)

1.2. separation properties (including Hausdorff topology and other separation axioms)

1.3. limit and continuity (including concepts of convergence, adherence, continuity in a point and in a set, homeomorphism)

1.4. topological constructions (including quotient, product, sum, wedge-sum, glueing and others)

2. Compactness and connectedness

2.1. compactness (including definition, results on compactness and closeness, Thychonoff's Theorem, local compactness, Alexandroff's compactification, normal space)

2.2. connectedness (including connectedess, path-connectedness and local connectedness)

3. Countability, uniformity and metric

3.1. countability (including the first and second axioms of countability, sequential closure)

3.2. uniformity (including Cauchy property, uniform continuity, completness)

3.3. metric (including definition, ball, limitated set, uniform structure of metric, topology of metric)

3.4. countability and uniformity of metric (how sequences suffice in metric to describe limit and continuity, completness in metric spaces Lipschitz property, Banach's fixed point theorem)

3.5. Compactness in a metric space (how in a metric space behaves compactness)

4. Topology and metric of the euclidean space (including inner product, norm, compactness and connectedness in euclidean space)

5. Sequences of functions (What can one say about sequences of functions between topological or metric spaces, pointwise convergence versus uniform convergence)

3. Algebraic topology, 4. Differential topology, 5. Help

The contens of these chapters isn't so important until chapter 1 be in an advanced state.

Signed

--AnneKent 10:17, 17 September 2007 (UTC)

I would say that although metric spaces are not easier to understand, an understanding of the nature of topological spaces are based upon an understanding of what metric spaces are. Regards, A 04:13, 26 November 2007 (UTC)

[edit] TOPOLOGY EXPERT

Dear all,

I am intending to vastly improve this book. I have so far created three sections on the comb space, linear continua, and on locally connected spaces. I have noticed that some people have added exercises to some chapters in this book which is great! However, some of the exercies are far too easy or far to boring. If you are going to include an exercise such as, 'check whether the following collection of subsets of X is a base', it would be appropriate to include one question on this but not anymore. Such exercises are far too repetitive and are not generally challenging for the more interested reader. Therefore, I recommend that you try to add more interesting exercises. Of course, adding any sort of exercise is great! In future, I am looking to completely improve the two chapters on general topology to make it readable.

Topology Expert (talk) 04:28, 8 June 2008 (UTC)

I think that what needs work is the differential topology section, and more on manifolds, and so on.--131.243.31.197 (talk) 02:13, 31 July 2008 (UTC)

In addition, I have found that I may have added some stuff that was in the realm of functional analysis. For example, I have added the Arzela-Ascoli Theorem to the completeness page. I see that on the functional analysis book, then mention this theorem, but don't prove it. I think that some of this functional analysis stuff may be moved to the functional analysis book, but I find it troubling about the manner in which this should be done. After all, this would have to either assume that one has already read this topology book, or some topology material would have to be copied over to the Functional Analysis book. Regards, 131.243.31.197 (talk) 23:33, 4 August 2008 (UTC)

You are right that there is this basic question: whether or not a background in general topology (not algebraic topology or differential topology) is prerequisite for functional analysis, a background that is little more than passing acquaintance. The current Functional Analysis is written based on the assumption that the answer to the question is affirmative. Since the weak topology play an important role in functional analysis, I don't think anyone can go far without experience with non-first-countable spaces. This could change, of course, in the future. -- Taku (talk) 14:14, 17 August 2008 (UTC)

[edit] Development stages

I would suggest that someone more authoritative on topology (I'm haven't had any formal education in the subject yet) should go over the chapters and add development stages to each chapter, so as to get an overview of the content and quality of each chapter.

This would help identify areas that need improvement and completion and would turn the attention to those areas.

--Jerome Baum (talk) 18:22, 24 November 2009 (UTC)