TOPOLOGY CHECKLIST - SPRING 2010 The list below serves as an indication of what we have covered in our course on topology. (It was written in a hurry, so there is a high risk of some mistake being made somewhere. When in doubt, consult the book or ask me.) Short introduction: The most fundamental objects we have looked at are metric and topological spaces, as well as the functions that respect this structure, namely continuous functions. We have studied several properties of these spaces: completeness, countability and separation properties, compactness, connectedness. Much of the course has been about how these properties relate to one another and how they transfer between spaces that are somehow related to one another: I.e., are they preserved if you form product spaces (e.g. Tychonoff s theorem), if you pass to a subspace (properties of the subspace topology) or when you pass to the image or inverse image of a continuous function (e.g. the image of a connected space is connected). Another aspect of this course has been about how these properties manifest themselves in specific settings which are prominent throughout mathematics: E.g., theorems on Banach spaces (the Principle of Uniform boundedness which is also called the Banach-Steinhaus theorem) or on spaces of continuous functions (Stone-Weierstrass theorem). The oral exam is about checking whether you are familiar with the definitions of the course, if you are able to state both basic and important theorems and results (some of these may come from the exercises), and if you are able to say something sensible about their proofs. More precisely: You will do a good job on the oral exam if you are able to provide precise definitions, give outlines for how to prove the theorem(s) we discuss, and explain examples that are pointed out to you. To do a great job, you should be prepared to give more detailed proofs here and there, and be able to point to examples yourself. Metric spaces: 1. The Basics What is the definition of a metric, a metric space, an open ball, and an open set? Why is an open ball open? Why is the interior of a set open? What are adherence points, convergent sequences, closed sets and closures? What are equivalent formulations for closed sets in metric spaces? (we spent some time on this...) What are DeMorgan s laws for open and closed sets? What are metric subspaces and what does it mean to be open in a subspace? What are the various ways to define continuity for functions, and why are they equivalent? (we spent some time on this...) What is a homeomorphism? What is a Lipschitz continuous function? (we talked about this when we did the contraction principle) Topological spaces: What is the definition of a topological space and their open sets? Why are metric spaces topological spaces? (you should probably make a list of some of the topological spaces we have seen) What is a metrizable topology? 1
2 TOPOLOGY CHECKLIST - SPRING 2010 What is a neighbourhood of a point, and what are interior points (why is the interior of a set open?). What are closed sets, closures and adherence points? DeMorgan s laws for closed sets? Why are limits of sequences less important than in metric spaces (see e.g. exercise 2.1.12 concerning the co-countable topology). Definition of the subspace/relative topology. Open and closed sets in the relative topology. What is a base for a topology? (what is a subbase for a topology?) How do you check that a family of subsets is a base for some topology? How do you check that a family of subsets is a base for a certain topology? How do we define continuity for functions on topological spaces? How can you check if a function is continuous using the basis of a topology? What is a homeomorphism of topological spaces? What does it mean that a property is topological? Can you make a list of which of the properties below are topological, and which are not? (what about being bounded, or totally bounded?) What does it mean for two metrics to be equivalent? Topologically equivalent? (if you are able to formulate the notions of equivalences, but you are not sure which is called what, this is totally acceptable as the names seem to change from book to book) Important definitions/concepts: 2. Completeness What is a Cauchy sequences? How do Cauchy sequences relate to convergent sequences? What is a complete metric space? When is a subset of a metric space dense? When is it nowhere dense? Important special case: R is complete. (we never looked at a proof for this) The rationals and irrationals are dense in R. (this you should be able to prove) The rationals are countable, while the irrationals are uncountable. (this seems to have confused some of you) Important theorems: Baire Category Theorem and its corollary (we spent a lot of time on the proof of this theorem...). Contraction principle (we also spent some time on this one... and the proof is quite intuitive.). Important special case: Complete normed spaces. Can you define a norm, a linear operator, and Banach spaces? What is the connection between boundedness and continuity of linear operators? B(X, Y ) is a Banach space if X, Y is normed and Y is Banach. Important theorem: The principle of uniform boundedness (aka: Banach-Steinhaus theorem) The family of bounded and continuous functions on any set forms a complete space in the uniform metric (we covered this when we proved Tietze s extension theorem).
TOPOLOGY CHECKLIST - SPRING 2010 3 Countability: 3. Countability and separation When is a space: Separable? First countable? Second countable? Lindelöf (i.e. satisfies conclusion of Theorem 2.4.3)? Can you show that a compact metric space is separable? Can you show that second countability implies the other three countability axioms (we never proved that it implies the Lindelöf property... so you shouldn t worry too much about that part...) Separation: When is a space: T 1? T 2 (Hausdorff)? T 3? T 4? Can you make a simple drawing illustrating these four axioms? When is a space normal and/or regular? Important results (we spent some time on all of these...): Urysohn s lemma (including Lemma 2.5.2). Tietze s extension theorem. The Weierstrass theorem and the Stone-Weierstrass theorem (real case). A sequence in a Hausdorff space converges to at most one point. In metric spaces: 4. Compactness What is compactness? What is sequential compactness? When is a space totally bounded? Why is every compact metric space separable? Continuous functions f : X Y, where X, Y metric spaces and X compact are uniformly continuous. Important theorems (metric spaces): Compactness, sequential compactness and being complete and totally bounded are all equivalent. (in particular, the proof for sequential compactness implies compactness indicated in exercise set 4-2. This replaces the more complicated argument in the book.) Heine-Borel theorem (without proof). In topological spaces: Be aware that the equivalences which exist the metric space setting are not always true topological spaces (unfortunately, we didn t have the time to look at examples). Why is a finite union of compacts compact? Why is a compact set always closed in Hausdorff spaces? (in class we saw an example of why this is not true in general.)
4 TOPOLOGY CHECKLIST - SPRING 2010 When is an arbitrary union of compact sets compact? (we commented on this in class...) Why is a closed set in a compact space compact? Important theorem (topological spaces) X compact and Hausdorff implies X normal. (we spent some time on this...) Local compactness: What is the definition of a locally compact space? Can state the properties of the one-point compactification of a locally compact Hausdorff space, and outline the proof of its existence? What is the one-point compactification of R? How does it relate to the one point compactification of a circle with one point missing? Do you know of any other compactification of R? A continuous real valued function on a compact space attains its maximum and minimum. Path connectedness: 5. Connectedness Definition of path connectedness and path components (why does this give an equivalence relation?). Connectedness: Definition of connectedness (there are several ways to formulate this) and connected component. When is a union of connected sets connected? What is a connected component? What are the connected subsets of R? Continuous image of connected is connected. Path connected sets are connected (we spent some time on this...). Connected components are closed. Connected sets need not be path connected. Every interval in R is homeomorphic to one of the intervals [0, 1], (0, 1), [0, 1) or (0, 1]. (we did this in class) Metric spaces: 6. Product spaces Reasonable metrics give product topology (Theorem 1.4.1). Topological spaces (finite products): Definition of product topology on a finite product of topological spaces. Projections π j and why they are continuous in the product topology. Why are the projections π j open mappings? (What does this mean? Also, this is used extensively in exercise 2.10.4) A function f with image in a product space is continuous if and only if π j f is continuous for every projection. What do we mean when we say that the product topology is the smallest topology making all π j continuous?
TOPOLOGY CHECKLIST - SPRING 2010 5 Topological spaces (infinite products) Definition of an infinite product space. Definition of the subbase for the product topology of such a space. (what is a subbase?) The axiom of choice, and why is it important when we discuss infinite product spaces (and why is the axiom of choice a subject of debate among mathematicians?). A function f with image in a product space is continuous if and only if π j f is continuous for every projection. What is a partially ordered set, a totally ordered set, a maximal element and an upper bound? What is Zorn s lemma? What is its relation to the axiom of choice? Alexander s subbase theorem and Tychonoff s theorem. (we spent some time on these...) What is the box topology? Why do we prefer the product topology over the box topology? (we considered the function f(t) = (t, t, t, t,...) in class) What properties are preserved when taking finite products? Products of closed sets are closed. Products of Hausdorff spaces are Hausdorff. For metric spaces we looked at the statement "if X, Y are compact spaces, then X Y is compact. Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden E-mail address: janfreol@maths.lth.se