27
Jun 15

 Point Set Topology

 A  student was doing miserably on his oral final exam in General Topology (yes, this guy really did give oral finals in topology). Exasperated by the student's abysmal performance up to that point, the professor asked the student "So,what do you know about topology?" The student replied, "I know the definition of a topologist." The professor asked him to state the definition,
expecting to get the old saw about someone who can't tell the difference between a coffee cup and a doughnut. Instead, the student replied: "A topologist is someone who can't tell the difference
between his ass and a hole in the ground, but who can tell the difference between his ass and two holes in the ground. The student passed.
- David Gonda, Yale mathematics student

Basic Topology May 30, 2012 Georgios Dimitroglou Rizell Uppsala University, Sweden  (PG-13)The title might fool you into thinking this is either a strict point set topology course like a thousand others or a hodge podge undergraduate course in topology that covers a blend of classical combinatorial topology and point set topology a la Armstrong. You'd be dead wrong and it would be too bad. This is a very well written and comprehensive course on point set topology that focuses much more-and more appropriately, I think-on the analytic aspects of the subject then the usual book or notes do.. Of particular interest for analysis students are the sections on the topology of normed spaces and the continuity of linear maps on them. Recommended.

Topology Course Lecture Notes by Aisling McCluskey and Brian McMaster ETH Zurich 1997 (PG-13) A very standard set of notes on point set topology. Nicely written and clear,with good examples,but nothing original here and they lack exercises. Good as supplemental reading for a point-set topology course, but not much else.

 

TOPOLOGY: AN INVITATION Lecture notes by Razvan Gelca Texas Tech University  (R) Very inventive, wide ranging and deep set of lectures for a first year graduate course in topology. The homework and problem sets for the exams to accompany the notes can be found here. The author splits the course between point set and algebraic topology and focuses on examples and insights one doesn't usually find in the conventional textbooks like Munkres or Massey. While important results are proven, the focus of the notes are clearly on examples and definitions-many results are left for the student to prove in the exercises.So clearly. a committed set of students that aren't afraid
to labor themselves on most of the work is going to be needed for success in a course based on these notes. That being said, the author builds a very good selection of topics and shows excellent judgment of what should be proven in detail and what's reasonable to leave as a exercise. That being said, clearly these aren't notes intended to be read passively, they're for a serious course for serious graduate students. Recommended for students up for a challenge.

Topology I Tomoo Matsumura Cornell University Fall 2010 Homepage and Course Materials (PG-13) Yes, that Matsumura, the famous topologist that wrote the equally famous book on differential topology. These are the course materials for a standard first year point set topology course based on Munkres;the second link will take you directly to his notes.While they don't deviate much from the book, Matsumura does  develop in more detail several topics and examples that Munkres either leaves to the exercises or skims over, such as  detailed computations of specific fundamental groups of the torus and unit circle. A good supplement for any standard topology course.

Topology Huynh Quang Vu  Vietnam National University(PG) These very extensive and balanced lecture notes cover the essentials of the 3 major classical branches of topology-general, algebraic and differential topology-in a detailed, rigorous and example driven presentation with many pictures, exercises and historical notes. What's striking is how the author shifts gears between the 3 very different branches smoothly without missing a beat. An excellent source all students of topology should familarize themselves with.

Topology Jim L. Brown Clemson University April 29, 2010      (PG-13) Another of Jim Brown's personal lecture notes, these from a first year graduate course in topology he taught at Clemson. These are typed, which makes them a bit more attractive as a study aid. Actually, these notes have a lot more than that going for them. Brown begins with a fairly detailed but mostly standard treatment of point set topology (including a good introduction to topological groups) before taking a left turn by developing the essentials of algebraic topology via
differential forms and the De Rham theorem. It's hard not to love this approach to the subject, where  everything becomes so geometric and calculationally clear- it's also the approach that's best for applications in physics. He finishes with a sheaf theoretic treatment of the Cech cohomology and a brief discussion of the Hodge conjecture. Brown's really to be commended for trying to construct a first year topology course that's genuinely different from most and he does it with great clarity and many examples. And what more could you ask from such a course? Recommended.

Topology  Mariusz Wodzicki University of California at Berkeley December 3 , 2010(R) Ok, this is a supplement to the standard elementary topology course at Berkeley.It turns into one of Wodzicki's acid trip,
super-general notes where everything is done by diagram chasing and categorical constructions, calculations don't exist and most of his examples are abstract concepts that are special cases of even more abstract concepts. In these notes, he gives the zen treatment to point set topology to supplement a more conventional approach. He defines topological spaces  via the  Kurotowski closure axioms and builds the entire network of notions of connectedness, compactness, convergence and continuity via partially ordered sets and the nets and filters defined on them. Most students looking at them with no prior background in advanced mathematics will have their eyes glaze over. But coupled with a standard approach, seeing the material again through this Big Picture Lens would be not only interesting, but quite beneficial to students going on in topology. Recommended for advanced students as a second course or as supplementary material.

Topology and Modern Geometry Anatole Katok Alexey Sossinsky  Penn State University Draft 2010 (PG-13)This is a remarkable book in progress by 2 experts on the title subject, has appeared in various forms and guises online since 2005. As far as I can tell, this link is to the most recent and complete version to
date. I honestly believe when it is finished, it may replace Munkres as the standard first year graduate textbook in topology. Yes, it's that good. Several things are most impressive about the selected content of the work: First, it organized in a very different manner then a traditional topology textbook. Rather then a long thorough
presentation of point set topology followed by the fundamental group and the elements of algebraic topology, concepts are defined and  introduced as soon as the basic machinery that allows them to be defined is given. This results in a completely balanced presentation-analytic and geometric aspects are covered in equal
measure from beginning to end. Manifolds, in particular, are treated early in chapter 1 and used throughout the book.  Secondly, it has an extraordinary number of pictures, examples and exercises, all illuminating and challenging and none are throwaways. It's apparent the notes still need  an enormous amount of polishing to before they can be published in any sense-diagrams are missing, many proofs are incorrect or have gaps and the last 2 chapters are incomplete. (The last flaw is particularly disappointing since chapter 7 is planned as an introduction to simplical and CW complexes!) It seems also that the authors may have abandoned the project in recent years. Even with all these flaws, I strongly encourage all students of topology to seek out these wonderful notes, which will be a huge asset in learning the subject. And hopefully, that will encourage the
authors to complete this stillborn classic, which has the potential to be one of the all time great textbooks in any subject. Most highly recommended.

General Topology Jan Derezinski  Warsaw University January 25, 2006 (PG) Brief and concise treatment of point set topology that's unusual in the completeness and well-organized presentation of the seperation axioms and the various kinds of separated topological spaces. Other then this,though, there's really not much to make these notes stand out from the ton of such notes online- and there are no exercises. Not bad, but nothing worth saving either.

Point-Set Topology Lecture Notes Allen Hatcher Cornell University (PG) By Hatcher's own admission,
these notes don't cover much, just the barest elements. Still, they're extremely clear with many examples and cover all the essentials needed to begin to study algebraic topology from his textbook. More importantly, they have a geometric bent that most point set topology  courses don't have- including an unusually thorough introduction to quotient spaces. Which isn't surprising given Hatcher's background and his intention of providing a warm up to his classic textbook. A good source to have on hand, but don't expect miracles from it.
Recommended.

General Topology  Thomas Baird University of Newfoundland Winter 2011(PG) Excellent set of notes on point-set topology has a fairly standard selection of topics- open and closed sets, metrics, continuity, compactness, etc.-but they are presented in a more modern fashion than most such notes and it strikes a very good balance between examples from analysis and those from geometry. It also contains some unusual touches, like normal spaces and giving the Zariski topology as an example of a non-metrizable topological space.Highly recommended.

Topology Michael Starbird University of Texas June 5, 2008 (PG-13)  This is the first half of a graduate topology sequence Starbird has been teaching for years at U of T that has garnered national praise. It's done via the Moore or Do It Yourself Method of teaching, which was made famous by Ronald Moore in the first half of the last century at this very school. Basically it means the students have to learn and build everything by solving exercises and the teacher has almost no input beyond definitions and theorem statements. I'm not a huge fan of this method in practice for many reasons too lengthy to go into here. That being said, if you enjoy the idea and think your students are up to the task, you might want to give this a try.The people I think could benefit most from these notes are students who have completed a standard point set topology course and
need to test and/or strengthen their command of the subject- especially in preparation for qualifying exams. For those students, these notes will be quite helpful.

A Topology Primer Klaus Wirthmüller Technische Universität Kaiserslautern 2010.pdf 

A Topology Primer Klaus Wirthmüller Technische Universität Kaiserslautern 2010.djvu(PG-13) Both of
these are the same notes, the only difference is the first is in pdf format while the second version is in the more compact djvu format. Whichever you prefer, it's one of the most complete, modern, well written and lucid first year graduate courses on topology you'll find anywhere on the web. Seriously. Wirthmuller defines each concept precisely and gives many examples as well as visual diagrams- and he runs the gamut from the basics of open and closed sets, continuity, connectedness and compactness through quotient spaces and gluing
constructions to simplicial complexes and homotopies (with one of the best introductions to homotopy diagrams you'll find anywhere)  to an introduction to the basic language of categories, functors and commutative diagrams. As German university topology courses traditionally do, it ends at the doorstep of homology theory, where algebraic topology proper is taken to begin. Anyone who studies these notes will be more than adequately prepared for any such follow-up  course. Indeed-I cannot imagine a better source from which to learn topology. Very highly recommended.

Topology: METRIC SPACES ANDREW TULLOCH University Of Sydney (PG) Another very brief set of lecture notes on a single topic of point set topology: metric spaces. Useful for an analysis course or someone
looking for a brief overview, but really nothing you can't get better elsewhere-I'd pass.

General Topology John Roe Penn State University(PG-13) A very standard upper level undergraduate/ first year graduate course in point-set topology that follows essentially all the topics in Munkres from topologies to covering spaces, but with Roe's usual lucidity and attention to detail in the presentation.Many examples and welcome insights. A very original and welcome touch by Roe is a very detailed and lengthy section on the topology of normed spaces, with applications to operators and differential equations. Strongly recommended.

Introduction to Set Theory and Topology Ronald C. Freiwald Washington University in St.
Louis Book Draft  2013   (PG-13) What is it with the current rage for topologists to write thier own textbooks on point set topology? It's like all of them spent years in thier younger days using the first edition of Munkres and saying to themselves, "I can do better." and they set out to. I'm sorry to say most of them saying
they can do better then an award winning teacher and researcher at MIT sounds like hubris of the first order and most of thier substandard results bear this out. But not this one.Freiwald has created a very interesting and original point set topology online text here- his central concept is to combine a basic point set topology course with a first undergraduate course in naive set theory. This is an incredibly natural approach and I'm surprised more of the standard textbooks haven't tried it (the only other author I think who's tried it is
Kuratowski). The book is very informative, insightful on the connections of both subjects to not only each other, but to real analysis. The interaction between set theory, point set topology and analysis has been one of the main wellsprings of modern analysis and I  think it's really in that context that the importance of the study of general topological spaces emerges. Freiwald pursues this clearly and comprehensively, with many examples and constructions.His historical scholarship is also impressive, citing many original sources and events in the development of both subjects. The result is one of my favorite online books on the subject and I vehemently hope he continues to make it available for free online as he continues to polish it towards what presumably will be it's final form.Very highly recommended.

Topology Waldemar Schlackow and Helen Lowe Mathematical Institute University of Oxford 2015 (PG-13)  Excellently written and broad set of notes and exercises for a strong introductory UK style topology course that covers both the analytic and geometric side of the elements of classical topology.  Contents 1. Topological spaces  1.1. What is Topology about? 1.2. Definitions and examples 1.3. Closure of a set 1.4. Interior of a set 1.5. Boundary of a set 1.6. Separation axioms 1.7. Subspace of a topological space 1.8. Basis for a topology 1.9. Product of topological spaces 1.10. Disjoint unions 1.11. Connected spaces 2. Compact spaces 2.1. Definition and properties of compact spaces 2.2. Compact spaces and continuous maps 2.3. Sequential compactness 3. Quotient spaces 3.1. Definitions 3.2. Separation axioms 3.3. Quotient maps 4. Simplicial complexes 4.1. Definitions 4.2. Substructures of a simplicial complex 4.3. Elementary  properties of simplicial complexes 5. Surfaces 5.1. Polygons with a complete set of side identifications 5.2. Two lists of surfaces 5.3. Adding handles and crosscaps 5.4. Closed combinatorial surfaces 5.5. The classification theorem 5.6. Distinguishing the surfaces  The course is quite balanced, giving equal time to both the abstract point set topology that's so critical in both classical and modern analysis and the concrete low-dimensional geometric topology that was not only critical to the development of the theory of differentiable manifolds and modern presentations of classical geometry, but acts as the foundation for follow-up courses in algebraic topology and advanced differential geometry. Most such courses favor one side or the other depending on the tastes and goals of the author, but this one attempts to do both and does it quite well. There is also a very nice progression of topics, from the elements of topology to compactness to  connectedness to quotient spaces-which provides a good segway into simplexes and cutting and pasting arguments. There are also many good exercises and pictures. A very solid and balanced first course in topology that will prepare students well for more advanced courses. Highly recommended.

 

Introduction to Topology  Michael Muger Radboud Universiteit Nijmegen 2015 Version

(PG-13) Very solid, comprehensive set of lectures on point set topology from a completely modern point of view. It develops both the analytic and geometric aspects of the subject very completely, although many of the more advanced analytic subjects such as Alaogu's Theorun and normal spaces, are covered as optional topics. The notes are also a very good example of my comments on the index page about the transitory nature of online texts: When I first discovered the 2013 version of these notes, category theory and the axiom of choice were developed in the first chapter and used throughout.  In this year's version (2015), this material has been moved to an appendix and assumed as known throughout. I'm not sure if this is a good assumption.At most universities, this appendix would probably have to be covered in depth first. It does make the presentation more modern and complete than most courses, though.  Other topics usually not presented in such a course that are here are psuedo-metrics, irreducible spaces, quasi-components and geodesic spaces.General convergence (i.e. nets and filters) are covered more completely than any other free source I've seen.  The course ends with an introduction to homotopy theory centered on the fundamental groupoid rather than the fundamental group a la Brown. My one real complaint is these notes are a bit deficient in examples and I'd add more if using them to teach a course. That quibble aside, I was quite impressed with these notes. This a meaty, up-to-date and substantial course on the subject and would make a welcome course text or supplement for a serious first-year topology course at the level of Munkres.When the text is finished, I think it has the potential to compete with Munkres and Lee for the standard text on the subject.  So grab it while it's still free. Highly Recommended.

Graduate Topology  Peter Selick University of Toronto 2008 (PG)

Graduate Topology I-II Peter Selick University of Toronto 2012  (PG)These are the notes for Selick's 2 year long graduate topology course, beginning with point set topology and moving through to classical homotopy theory, homology and cohomology theory.  The notes, like all of Selick's notes, are all business: concise, dry and pictureless; concentrated in the inevitable progression of definitions, theorems and examples. They are cold, matter of fact and bloodless. That being said, they are also well written and extremely thorough, with many examples and exercises for the student to chew on. A very good if unoriginal course and will be quite helpful as a general learning source for any graduate student in topology. Particularly useful for prelim preparation. Recommended.

Topology Kiyoshi Igusa Brandies University Fall 2005(PG-13) These aren't the best notes I've seen-they consist of supplements for a basic point set topology course using Hocking and Young (which as I've said,I'm not a big fan of). Basically,the supplements consist of counterexamples presented in a very clear fashion. Students may find them easier to use and understand then Steen and Seebauch, so they may be useful if not indispensible.

General Topology Richard Barraclugh University of Burmingham MSMYP2(PG-13) Another one of Barraclugh's "back of the napkin chicken scratching" notes. Very brief, concise notes on point set topology. 18 pages. For real. Don't even bother.

Analytic Topology Rolf Suabedissen Course Material University of Oxford 2009 (PG-13) The author of these point-set topology notes opens with a very honest warning to the reader: These are lecture notes, not a textbook. They are not meant to replace going to the lectures. Specifically, they do not contain any motivation for the concepts nor an intuitive explanation. In short, these notes aren't intended to supply a full course on point set topology, they're just intended to give a concise summary that contains just what the author intends for his students to know for the all-important exams at Oxford. And that's more or less what they are-a brute, maximally terse, Satz-Beweis presentation of the basic elements without many details. For those who like working through such skeletal notes, they may be of help-but it's hard to see them being useful to most students except as review.

Topology I Notes Bruce Ikenaga University of Millersville (PG)  Very nice lectures on first year graduate point set topology in postscript format that cover basically the same material as Munkres (the author originally learned the subject from Munkres himself at MIT) ,but he develops the material quite differently-and more enlighteningly,in my opinion-in some ways. A very good supplement to a first year topology course. Recommended.

General Topology Jesper M. Moller Matematisk Institut, Universitetsparken 5(PG-13) This is set of notes on point set theory similar in approach to Munger's above, though not as comprehensive and taking as it's
foundation set theory only. Interestingly, Moller uses Grothendieck universes in order to avoid the Russell Paradox in set theory. I've always thought this approach is basically notational slight-of-hand  to implicitly use classes without the cardinality problems. I'm not sure it avoids the problems and in any event, there are ways of defining proper classes that avoid these issues, so why bother? Still, a very nice set of notes.

Introduction  to Topology Renzo Cavalier et al. Colorado State University 2007     (PG-13) A comprehensive,
unique and intriguingly constructed set of notes from Cavalier's intensive course he gave while visiting the University of Michigan-apparently it was pasted together from the lecture notes taken by 23 students in his course, who all get co-authorship credit on the first page!  Contents 1 Topology 1.1 Metric Spaces 1.2 Open Sets
(in a metric space) 1.3 Closed Sets (in a metric space) 1.4 Topological  spaces 1.5 Closed Sets (Revisited)  1.6 Continuity 1.7 Homeomorphisms  1.8 Homeomorphism Examples 1.9 Theorems On Homeomorphism 1.10 Homeomorphisms Between Letters of Alphabet 1.10.1 Topological Invariants 1.10.2 Vertices 1.10.3 Holes 1.11 Classification of Letters 1.11.1 The curious case of the “Q”  1.12 Topological Invariants 1.12.1 Hausdorff Property 1.12.2 Compactness Property 1.12.3 Connectedness and Path Connectedness Properties 2 Making New Spaces From Old 2.1 Cartesian Products of Spaces 2.2 The Product Topology 2.3 Properties of Product Spaces 2.4 Identification Spaces 2.5 Group Actions and Quotient Spaces 3 First Topological Invariants 3.1 Introduction 3.2 Compactness 3.2.1 Preliminary Ideas 3.2.2 The Notion of Compactness3.3 Some Theorems on compactness 3.4 Hausdorff Spaces 3.5 T1 Spaces 3.6 Compactification 3.6.1 Motivation 3.6.2 One-Point Compactification
3.6.3 Theorems 3.6.4 Examples 3.7 Connectedness 3.7.1 Introduction 3.7.2 Connectedness 3.7.3 Path-Connectedness 4 Surfaces 4.1 Surfaces 4.2 The Projective Plane 4.2.1 RP2 as lines in R3 or a sphere with antipodal points identified 4.2.2 The Projective Plane as a Quotient Space of the Sphere  4.2.3 The Projective Plane as an identification space of a disc 4.2.4 Non-Orientability of the Projective Plane  4.3 Polygons  4.3.1 Bigons 4.3.2 Rectangles 4.3.3 Working with and simplifying polygons 4.4 Orientability 4.4.1 Definition 4.4.2
Applications To Common Surfaces 4.5 Euler Characteristic 4.5.1 Requirements 4.5.2 Computation 4.5.3 Usefulness 4.5.4 Use in identification polygons 4.6 Connected Sums 4.6.1 Definition4.6.2 Well-definedness 4.6.3 Examples 4.6.4 RP2#T= RP2#RP2#RP2 4.6.5 Associativity 4.6.6 Effect on Euler Characteristic4.7 Classification Theorem 4.7.1 Equivalent definitions 4.7.2 Proof 5 Homotopy and the Fundamental Group 5.1 Homotopy of functions 5.2 The Fundamental Group 5.2.1 Free Groups 5.2.2 Graphic Representation of Free
Group 5.2.3 Presentation Of A Group 5.2.4 The Fundamental Group 5.3 Homotopy Equivalence between Spaces 5.3.1 Homeomorphism vs. Homotopy Equivalence 5.3.2 Equivalence Relation 5.3.3 On the usefulness of Homotopy Equivalence 5.3.4 Simple-Connectedness and Contractible spaces 5.4 Retractions 5.4.1 Examples of Retractions 5.5 Computing the Fundamental Groups of Surfaces: The Seifert-Van Kampen Theorem
5.5.1 Examples 5.6 Covering Spaces  5.6.1 Lifting  Apparently, he has continued to use these notes as a reference for his first year graduate course in topology at Colorado State. The overall choice of topics is fairly standard and seems to be based largely on Munkres.  However,  the manner in which the topics are
presented is very unusual and enlightening indeed. What's really special and effective here is that since the notes were constructed by students, the presentation is highly unorthodox in many regards when compared to how this material is usually presented in the standard sources written by experts. For example, many of the explicit examples given are those usually left as exercises, such as the explicit homeomorphism sending
the real line into [0,1] and mapping a square into a circle in the plane. Other original examples present in the notes are a discussion of loops as being homeomorphic to "paths without holes",  the importance of
n-verticies as topological invariants without explicit discussion of simplexes and the "classification of the letters"-i.e. a careful discussion of the letters of the alphabet as closed curves with 3-vertices. (I'm sure students who read  Chapter 0 of Allen Hatcher's  sometimes confusingly informal Algebraic Topology would have found studying this example concurrently with it immensely clarifying! )    The presentation is also careful while still being quite intuitive- there are many, many informative hand and computer drawn pictures in addition to the equally numerous examples. Interestingly, some major theorems are left without proof. This makes sense given the notes were compiled by students-proofs can be looked up in textbooks, but illuminating
discussions of examples and concepts are really of more help to those learning for the first time and would be more helpful in notes then mechanical proofs. Ok, those are the positives of these notes. Sadly, because it was put together by amateurs-clearly talented amateurs, but amateurs nevertheless-who haven't really mastered
the material themselves yet, there are quite a few drawbacks to the notes. First of all, the clarity of the notes is quite inconsistent throughout. For example, the section on compactness is quite detailed with full proofs and is careful with many wonderful pictures, but the section on the product topology, while well-written, it lacks detail and has a major error in it. The definition of the product topology is that it is the topology on X x Y,
where X and Y are topological spaces, generated by a basis of the Cartesian product of the basis sets of each space. This is simply false if we're talking about a product of infinite spaces since discontinuous maps can have continuous projections in this version of the product topology. A counter-example can be found here.    I'm sure there are many other errors throughout, but that was the most glaring.  Still, overall, the notes are a wonderfully informative and detailed source for this material-I'd love to see the professor go carefully through it one day and get all the bugs out of it. If he does, we would have a truly
superior set of notes for a first course on topology. As it is, for all it's flaws, it's well worth checking out. Highly recommended.

Introductory notes in topology Stephen Semmes Rice University ( PG-13) Highly detailed, intensive point-set topology notes emphasizing the analytic properties of general spaces. A complete set of contents is
in order here to give the reader a good indication of the range and slant of the notes: Contents 1 Topological spaces 1.1 Neighborhoods 2 Other topologies on R 3 Closed sets3.1 Interiors of sets 4 Metric spaces 4.1 Other metrics 5 The real numbers 5.1 Additional properties 5.2 Diameters of bounded subsets of metric spaces 6 The extended real numbers 7 Relatively open sets 7.1 Additional remarks 8 Convergent sequences8.1 Monotone sequences 8.2 Cauchy sequences 9 The local countability condition 9.1 Subsequences 9.2 Sequentially closed sets 10 Local bases 11 Nets 11.1 Sub-limits 12 Uniqueness of limits 13 Regularity 13.1 Subspaces 14 An example 14.1 Topologies andsubspaces 15 Countable sets 15.1 The axiom of choice 15.2 Strong limit points 16 Bases 16.1 Sub-bases 16.2 Totally bounded sets 17 More examples 18 Stronger topologies 18.1 Completely Hausdorff spaces 19 Normality 19.1 Some remarks about subspaces 19.2 Another separation condition 20 Continuous mappings 20.1 Simple examples 20.2 Sequentially continuous mappings 21 The product topology 21.1 Countable products 21.2 Arbitrary products 22 Subsets of metric spaces 22.1 The Baire category theorem 22.2 Sequences of open sets 23 Open sets in R 23.1 Collections of open sets 24 Compactness 24.1 A class of examples 25 Properties of compact sets 25.1 Disjoint compact sets 25.2 The limit point property 26
Lindel¨of ’s theorem 26.1 The Lindelof property 26.2 Applications to metric spaces 27 Continuity and compactness 27.1 The extreme value theorem 27.2 Semicontinuity 28 Characterizations of compactness 28.1
Countable and  sequential compactness 28.2 Some variants 29 Products of compact sets 29.1 Sequences of subsequences 29.2 Compactness of closed intervals 30 Filters 30.1 Nets and filters 30.2 Mappings and
filters 31 Refinements 31.1 Another characterization of compactness 32 Ultrafilters 32.1 Connections with set theory 32.2 Tychonoff’s theorem 33 Continuous  real-valued functions 34 Compositions and inverses
34.1 Compact spaces 35 Local compactness 36 Localized separation conditions 37 s-Compactness 38 Topological manifolds 38.1 Unions of bases 39 s-Compactness and normality 40 Separating points 40.1 Some
examples 41 Urysohn’s lemma 41.1 Complete regularity 41.2 R{0} 42 Countable bases 43 One-point compactification 43.1 Supports of continuous functions 44 Connectedness 44.1 Connected topological spaces
44.2 Connected sets 44.3 Other properties 44.4 Pathwise connectedness 44.5 Connected components 44.6 Localconnectedness 44.7 Local pathwise connectedness 45 A little set theory 45.1 Mappings and their properties 45.2 One-to-one correspondences 45.3 One-to-one mappings 45.4 Some basic properties 45.5 Bases of topological spaces 45.6 Exponentials 45.7 Properties of exponentials 45.8 Countable dense sets 45.9 Additional properties of exponentials 46 Some more set theory 46.1 Zorn’s lemma 46.2 Hausdorff’s maximality principle 46.3 Choice functions 46.4 Comparing sets 46.5 Well-ordered sets 46.6 Comparing well-ordered sets 46.7 Well-ordered subsets 47 Some additional topics 47.1 Products of finite sets 47.2 The Cantor set 47.3 Quotient spaces and mappings 47.4 Homotopic paths 47.5 The fundamental group 47.6 Topological groups 47.7 Topological vector spaces 47.8 Uniform spaces 47.9 The supremum norm 47.10 Uniform continuity 47.11 The supremum metric 47.12 An approximation theorem 47.13 Infinitely many variables Appendix A Additional homework assignments A.1 Limit points A.2 Dense open sets A.3 Combining topologies A.4 Complements of countable sets A.5 Combining topologies again A.6 Combining topologieson R A.7 Connections with product topologies References What should be clear to the reader scanning them is that Semmes strongly emphasizes the classical abstract topology with connections to set theory and real and functional analysis-algebraic and geometric topology is virtually nonexistent  There are some topics of geometry topology in the exercises-such as loops and the Brower hairy ball theorem-but the main lecture notes are virtually devoid of them. There are no commutative diagrams, classification of compact surfaces or simplex theory. The author chooses one major aspect of topology and sticks to that playbook. The result is a set of notes that will benefit mostly graduate students in analysis courses.The focus on analytic aspects of topology allows him to go much deeper into them
then the usual first course in topology usually allows. Most of the examples-and there are many-are drawn from real analysis and set theory. The choice of examples is quite interesting as Semmes uses many of them
to present more specialized aspects of topology that are of great importance in modern analysis, such as locally constant spaces, the Lindof property and theorem and semi-continuity. There is an unusually deep presentation of general convergence i.e. nets and filters.The notes are somewhat dense but clear, so they're probably better suited for graduate students.  For students and teachers who want to emphasize
the connections to analysis, these notes will be a real gift. For working topologists who wonder where the commutative diagrams, knots and the Van Kampen theorem are, they will most definitely not be for you.
But for students and teachers who are looking to prepare students for advanced graduate analysis courses via topology, you couldn't ask for better. Highly recommended.

Introduction to Topology David Mond University of Warwick Spring 2013 (PG-13)  David Mond is a very popular teacher and researcher in topology at the University of Warwick and looking at these notes, one
can see why. These are his notes for the undergraduate/first year graduate topology course at Warwick  They are comprehensive and emphasize more the geometric over the analytic aspects of topology, as well as being literately and clearly written.Mond uses modern notation right off the bat-commutative diagrams are introduced in the very first section and categories and functors are introduced soon after.While algebraic topology per se is not introduced, all the foundational elements of modern topology-simplexes, elementary homotopy theory and the fundamental group are discussed in great detail. There are many pictures and examples,including many detailed constructions of the famous quotient spaces of Euclidean space, such as the torus and cone, as well as their fundamental groups.There are also many interesting casual historical comments, many uncommonly known. For example, Mond comments on how the Poincare conjecture was originally formulated by Poincare in speculation of whether or not the Newtonian, Euclidean geometry of the universe that he knew was homeomorphic to the 3-sphere.  There are many fine topology courses online, but Mond's is one of the very best to prepare students for serious courses in algebraic topology and geometric topology. Very highly recommended! Contents 1 Introduction 1.1 Conventions 2 Topology versus Metric Spaces
2.1 Subspaces .2.2 Homeomorphism 2.3 Overview of the fundamental group 3 Examples and Constructions 4 The fundamental group 4.1 Calculation of 1(S1; x0) 4.2 How much does 1(X; x0) depend on the choice of x0?4.3 The degree of a map S1 ! S1 and the Fundamental Theorem of Algebra 4.4 Winding numbers 4.5 Induced homomorphisms 4.6 Categories and Functors . 4.7 Homotopy Invariance 4.8 The fundamental group of a product 4.9 The fundamental group of a union 4.10 Applications of van Kampen's Theorem 5 Covering Spaces 5.0.1 The degree of a covering map 5.0.2 Path lifting and Homotopy Lifting 5.0.3 Applications of Unique Homotopy Lifting 5.0.4 Solving the lifting problem 5.1 Classi cation and Construction of Covering Spaces 5.1.1 Universal coverings 5.1.2 Deck transformations 5.1.3 Further developments 6 Surfaces 6.1 Orientation and orientability 6.2 Triangulations 6.3 Sketch proof of the classification theorem 6.4 Appendix: Xg as a quotient of a regular 4g-gon 6.5 Gaps

General Topology Notes Jack Porter University of Kansas August 2009    (PG-13) Another intensive course in point set topology emphasizing the analytic aspects. Porter's notes are quite similar in approach, intent and content to Semmes' notes above. Like Semmes above, this course assumes a strong analysis course in metric spaces. There is considerable overlap between the 2 lecture notes, with some minor but significant differences. First of all, Porter's notes aren't as dense and comprehensive as Semmes, so they'll probably be found more accessible by upper level undergraduates.They also contain many more informative pictures then Semmes.  Secondly, Porter's notes are considerably more terse-many more results are left as exercises for the student. This helps move things along and probably makes them more amenable to students who are active learners, but it does make them less useful as a complete course reference. Lastly, Porter's notes have a number of examples that aren't present in Semmes, such as paracompact spaces and the Arens space. Although both are fine sources on point set topology, I think students will find Porter somewhat more readable and useful for coursework and review. Highly recommended. Contents 1 Review of Metric Spaces 2 Topological Spaces - The First Steps 3 The Building Blocks 4 Two Major Tools 5 Separation Axioms 6 Paracompactness 7 Cardinal Invariants 8 Compactness 9 Connectedness 10 Compactifications and Extensions 11 Metric Spaces 12 Function Spaces 13 Appendix A - Basic Set Theory 14 Appendix B - Review Items

Topology John Rognes University of Oslo November 29th 2010     (PG-13) Another comprehensive set of topology lecture notes based on The All-Hallowed Green Book.  Based, actually, is an understatement. In many ways, Rognes' notes are almost a word-for-word copied compendium of selected results from the book  interspersed with his own insights and commentary. While most of the main theorems, definitions and proofs are drawn straight from the book, there is more than enough original material to make them worthwhile to study. Rognes' original commentary is mostly designed to either a) add examples not in the book or b) clarify certain points that Munkres either states brusquely or leaves as an exercise. Some of the more exotic topics, like the separation axioms and normal spaces, are referred to the text. This allows him to focus on the main topics in more detail. The sections on the fundamental group and manifolds are more modern and detailed then the presentation in the text, which will be very helpful for students that go on to algebraic topology.  The result is a very complete and readable introduction to point set topology that functions very nicely as a free alternative to Munkres (although it would have to be supplemented with more exercises) or even better, as an outstanding supplementary reading guide to Munkres that will clarify many points and assist in studying from that legendary text.  Either way, it makes a very nice source for students in such courses. Highly recommended.

  • GENERAL TOPOLOGY An Introduction J.B. Hart Tennesee State University August 8, 2013(PG) Yet another course on abstract point set topology. These are more terse then the others, with many results shunted to the exercises, including the examples, although there are quite a few spelled out in some detail.  They're also a bit more anally careful than the previous entries like Semmes and Rognes- spending some time breaking down basic set theory and logic. There's basically no difference between what's covered in these notes and the more exhaustive sources on point set topology around the web
    we've already commented on except in style-these notes, as I said, are less detailed and more exercise driven. And if you like that kind of mathematical text, it's done fairly well if routinely. So if you like
    that kind of approach, then by all means. Personally, I'd prefer Porter or Rognes.
  • Topology  Curtis McMullen  Harvard University Fall 2013(PG-13) Another one of McMullen's masterly
    lecture notes, these for an intensive one semester topology course for the super-undergraduates at Harvard. As one would expect from McMullen, they are very broad, covering topics in both point set and
    algebraic topology and emphasizing the fully modern geometric aspects  of the subject. The course presumes strong courses in undergraduate analysis and abstract algebra as background. The notes are very concise and move at Warp Factor 9-as one would expect they'd need to complete its intense trajectory. Shockingly, despite that and the enormous volume of material they cover in such a minimalist fashion, McMullen remains extremely clear and manage to fit in many good examples, pictures and insights. There's also a freight car of exercises that range in difficulty from easy to impossible. He also
    presents many topics not usually covered in such a course-note in particular the inclusion of a discussion of the Galois theory of covering spaces, usually a topic for advanced graduate seminars! If you're looking for a challenge, you can't do better then this whirlwind tour of both classical and modern topology by a master. But you've been warned- it's for serious students only!   Contents 1 Introduction 2 Background in set theory 3 Topology 4 Connected spaces 5 Compact spaces 6 Metric spaces 7 Normal spaces 8 Algebraic topology and homotopy theory 9 Categories and paths 10 Path lifting and covering spaces 11 Global topology: applications 12 Quotients, gluing and simplicial complexes 13 Galois theory of covering spaces 14 Free groups and graphs 5 Group presentations, amalgamation and gluing
  • POINT SET TOPOLOGY MICHAEL STRAYER from the lectures of Vladimer Uspensky Ohio University Fall 2013   (PG-13)Very detailed yet concise "Russian" style point set topology notes, something I've always been a big fan of. Vladimer Uspensky, who delivered the original lectures from which Strayer compiled these notes, was trained at the once famous Moscow State University. legendary for it's topology and geometry courses-particularly in point set topology, which mid-20th century Russian mathematicians were quite fascinated with-which trained such luminaries as S. Novikov, M. Kontsevich, V. Voevodsky, G. Perelman, I. Shafarevich, D. Fuchs, and M. Postnikov. The influence and excellence
    of Uspensky's alma mater shines through here.  The notes are superbly organized, which is what allows them to cover such an enormous range of topics while still remaining relatively concise. There are quite literally tons of examples, many good and unusual pictures of point-set objects such as the bow tie space.While commutative diagrams aren't extensively used , they are used in places where they are quite informative. For example, a commutative diagram is used to simplify the proof that the set of all
    ultrafilters on a discrete countable space admits the Stone-Cech  compactification. There are some minor quibbles that might pose a problem for the beginner. Firstly,definitions are sometimes not given in
    the usual manner. which may or may not assist the student in their assimilation. For example, a topological space Y is defined as compact if for any topological space X, f : X x Y → X is a closed mapping. This may or may not cause a student's eyes to glaze  over, while some may find it clearer then the usual "covering" definition. Also, major results are sometimes shunted to the exercises.Still, those quibbles aside, this is an outstanding set of lecture notes on point set topology that will prepare the diligent
    student well for advanced courses. One of the best. Very highly recommended.
  • General Topology Richard Williamson Norwegian University of Science 2013    (PG-13) Another impressive and lucid set of point set topology lecture notes, one with broader range then most of the others here. The emphasis of these notes is clearly geometric, mostly focusing on simplexes, curves and surfaces in Euclidean spaces while not neglecting the more analytic aspects. There are 2 strengths to these notes over most others. Firstly, there are an enormous collection of worked examples, all of them accompanied by detailed diagrams and explanations and covering many important special cases and counterexamples that are usually either not included or shunted to the exercises. The diagrams
    are methodically constructed and beautifully illustrated-they will be of enormous assistance to the beginner. The other strength is the inclusion of  elementary presentations of many unusual topics, such as paracompactness, the coproduct topology, basic knot theory and the Jones polynomial, the classification of surfaces via ?-complexes and much more. There is also a host of excellent exercises, none too difficult. All in all, these notes are one of the most informative and pleasant free sources freely available.For students and teachers who love the subject and wish to learn a very modern and vast treatment of it, you can't get better then these. Very highly recommended.
  • Metric and Topological Spaces T. W. Korner University of Cambridge 2014  (PG)  Another set of notes by the Cambridge master, these on the elements of point set topology. The notes are typical Korner: deep,
    literately and often amusingly written and brusquely concise without being dry. They don't cover a wide range of topics and don't really aim to:just the guts needed for more advanced courses in analysis and
    topology-and for students not looking for more then this, they'll be a perfect choice. As one would expect from a master analyst, they focus unapologetically on the analytic aspects of the subject.  Also in
    typical Cambridge fashion, there are many, many exercises,ranging from simple proofs to open-ended challenging questions, often formulated in Korner's quirkily humorous manner. In short, like most everything written by Korner, they're immensely informative and a joy to read.  Contents 1 Preface 2 What is a metric? 3 Examples of metric spaces 4 Continuity and open sets for metric spaces 5 Closed sets for metric spaces 6 Topological spaces 7 Interior and closure 8 More on topological structures 9 Hausdorff spaces 10 Compactness11 Products of compact spaces 12 Compactness in metric spaces 13 Connectedness 14 The language of neighbourhoods 15 Final remarks and books 16 Exercises 17 More exercises 18 Some hints 19 Some proofs 20 Executive summary