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MA30055 Introduction to Topology, Spring 2017

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MA30055 Introduction to Topology, Spring 2017 MA30055 Introduction to Topology, Spring 2017 G.K. Sankaran February 6, 2018 What this course is about This course is about topology, which is really not a subject you have yet seen. The prerequisites tell you that you need metric spaces: that's true (just about) but misleading, because it suggests that it is yet another analysis course. It isn't an analysis course. There is barely an " in sight and we shan't differentiate anything in earnest. The ideas and patterns of thinking often (not always) resemble things you have seen in algebra; but it's not an algebra course either. It is often said that a topologist is a person who cannot distinguish between a doughnut and a coffee cup, because they both have one hole. Then you are told that topology is about things that don't change when you stretch or bend spaces. That's true, at last partly, but it isn't the mechanism we use for thinking about topology. A lot of the time we use, quite simply, sets: topology is rather fundamental in the same way that set theory is. De Morgan's rules will turn up from time to time. The way we present topology in this course uses the language of categories, which is another way of looking at the fundamentals of mathematics. Part of the idea is to give a (very light) introduction to category theory, but in a context. Category theory is hugely general (almost everything is a category), but topology is a good place to encounter it, as well as being historically the place it came from. This has one odd consequence. Pretty much every pure mathematics course you have done so far (Analysis I was perhaps an exception) started with some familiar concrete things and gradually became more abstract. This course does the opposite. We shall begin with the general idea of a topological space, which allows for some fairly wild animals; then we shall gradually impose conditions making things less exotic, and eventually we shall end up talking about surfaces, doughnuts and coffee cups. The course is in four sections. In Section 1 we shall see some very general definitions and basic properties. In Section 2 we shall meet ways to construct topological spaces that have interesting properties. In Section 3 we shall see 1 various special conditions that a topological space may have, and in Section 4 we shall look at surfaces and (cheating only once) classify them, i.e. give a complete list of all of them. Books There are lots of books on topology, easily spotted by the word \Topology" (and frequently nothing else) in the title. Some of these are on algebraic topology, which is a different thing (we see just a hint of it near the end of this course), but then they say so. What we are doing is sometimes called \general topology" or \point-set topology". I am not following any book, but two which some people find helpful, at least for parts of the course, are Topology by J.R. Munkres and Basic topology by M.A. Armstrong. The standard reference book is generally General topology by J.L. Kelley: this is a bit more usable than standard references often are. My own lecture notes are extremely brief, being just reminders to myself. These are more extended notes, based on what I actually did in 2017. Local information There is a web page for the course at people.bath.ac.uk/masgks/Topology I will link to that from the Moodle page, but I probably shan't use the Moodle page very much directly. The course structure is simple and Moodle, invaluable for big complicated courses, is too heavy a tool for this one. 1 Topological spaces Definitions and examples 1.1 The basic idea of the definition of topological space is this: when you do metric spaces, after a while you find you stop writing the metric quite so often, and instead work with open sets a lot. So why not jettison the metric altogether, and simply say from the start which ones the open sets are? Maybe that is more general? (It is, of course.) How much survives of metric space theory if you do do that? 1.2 Recall that for any set S, the power set of S is P(S) := fA j A ⊆ Sg, the set of all subsets of S (thus A 2 P(S) () A ⊆ S). You can also think of it as the set of all functions from S to a 2-point set such as f0; 1g (for which reason set theorists sometimes call it 2S): the set A corresponds to the indicator function IA, which takes the value 1 on A and 0 elsewhere. Definition 1.3 A topological space X is a set equipped with a collection TX ⊆ P(X) of subsets (of X) satisfying the following properties: 2 (a) X 2 TX ; (b) ? 2 TX ; S (c) if Uα 2 TX 8 α 2 A (some index set), then α2A Uα 2 TX ; (d) if U; V 2 TX then U \ V 2 TX . The collection T is called the topology of the topological space. 1.4 Things to note: • I am abusing notation already. I really ought to use a different symbol such as jXj for the set X (\just a set") and the topological space X (\a set with a chosen topology"). After all, I could easily find two different subsets T and T 0 of P(X), both satisfying (a){(d), and then I wouldn't want to have to call both the topological spaces X. But very often the set X is a special one { R, for instance { that has a \usual" topology, and I don't want to have to keep writing jRj when I'm not thinking about the topology. So I will use X for both the topological space and the set. If I need to be careful about it, as will happen sometimes, I will write jXj to denote just the set, and (X; T ) to denote the topological space. • Although (d) only tells us that the intersection of two open sets is open, a trivial induction shows that then any finite intersection of open sets is open, and we could have made the axiom say that if we had preferred. • The four axioms should be memorised by repeating to yourself the words \all, nothing, arbitrary unions, finite intersections". Terminology 1.5 The set jXj is called the underlying set of X and its elements are called the points of X { my abuse of notation allows me to write x 2 X instead of x 2 jXj, which is just as well. The collection TX , or T if we all know perfectly well what X is and don't need reminding, is called the topology of X (or \on jXj"). A subset U ⊆ jXj is said to be open, or an open subset of X, if and only if U 2 TX . Notice that whether a subset is open or not depends on the topology, not just jXj { in fact, the topology is nothing but a list of all the open subsets. 1.6 As a general set-theory rule, an empty collection of subsets of a set X has union ? and intersection X. Think of this as a convention if you like, or argue that x 2 S Y exactly when it belongs to a Y 2 , and there Y 2? ? aren't any such Y so that never happens. so if we were really determined, we could ditch axioms (a) and (b) and make do with (c) and (a slight modification of) (d). But there are no prizes for having fewer axioms than anybody else. Nevertheless, these rules are convenient. 3 Example 1.7 Any metric space can be made into a topological space X by defining the topology TX to be the set of all open subsets with respect to the metric d. That is, if U ⊆ X then U 2 TX if and only if 8x 2 U 9δ > 0 such that fx0 2 X j d(x; x0) < δg ⊆ U: TX is called the topology induced by the metric d or the metric topology. 1.8 The example in (1.7) is the basic motivating example for the definition, and the axioms in (1.3) are what they are because that is what open sets in metric spaces do. Nevertheless, topological spaces are a lot more general than metric spaces: we shall see many examples of topological spaces that are interesting and important and definitely not metric spaces. Definition 1.9 Let X, X0 be topological spaces with the same underlying 0 set jXj = jX j. The topology TX0 is said to be finer or stronger than TX if 0 TX ⊆ TX0 (that is, X has more open subsets); TX is then said to be coarser or weaker than TX0 (that is X has fewer open subsets). (This is a case where I need the notation jXj because I am changing the topology while leaving the set alone, so using the same name for the set with or without the topology won't do.) Example 1.10 (i) The strongest topology on a set S is called the discrete topology: every subset is open and the topology is P(S). (ii) The weakest topology on a set S is called the trivial topology or the indiscrete topology: the only open subsets are ? and S. Please note: (in)discrete, not (in)discreet. Only mathematicians ever use the word \indiscrete".
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