The Bend in the River

Read-along: Rudin's PoMA: Weeks 4 and 5 (22nd April to 5th May)

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Goal: The goal for the next two weeks is to learn some topology.

My plan is to spend the next two weeks on the rest of the topology chapter. This material is difficult: it is essentially a generalisation of the first week or so of the calculus half of MATHS 253, in such a way that we examine precisely which properties of our space (in 253, $ \mathbb{R}^3 $; now, metric spaces; and in the future - but not in this text - topological spaces) are needed in order to build up analysis.

Reading: The majority of chapter 2 of Rudin: more precisely, paragraphs 2.14 to the end. The chapter is scanned here. A terse list of theorems is here.

Important notes.

• Rudin defines a neighbourhood of a point $ p $ to be a set of the form $$ N_r(p) := \{q \in X : d(p, q) < r\} $$

  for some $ r \in (0,\infty) $. The definitions I tend to use are as follows:

  • The ball of radius $ r $ about $ p $ is the set $$ B_r(p) := \{q \in \mathbb{R}^k : d(p, q) < r\}. $$
  • A point $ p $ is an interior point of $ E \subseteq X $ if there exists $ r \in (0,\infty) $ such that $ B_r(p) \subseteq X $.
  • A set $ U \subseteq X $ is open if each $ p \in U $ is an interior point of $ U $. Equivalently^{[prove this]}, $ U $ is open if $ U $ is a union of balls.
  • A neighbourhood of $ p \in X $ is an open set $ U \ni p $.
• Rudin uses $ d $ to denote a metric (i.e. a function from $ X\times X \to \mathbb{R} $ satisfying the three conditions of 2.15). Alternative notations for the distance from $ p $ to $ q $ include $ \rho $ (i.e. $ \rho(p,q) := d(p,q) $), or $ \left|PQ\right| $ (this latter usually in classical geometry).

The following theorems are the most important:

• 2.24, 2.27, 2.28, 2.30, 2.33, 2.34, 2.35. These are the ‘basic’ topological concepts. (Note: Rudin does not define the term ‘topology’, despite using it for the title of the chapter. A topological space is, philosophically, the most general kind of object on which it makes sense to define continuous functions; and the topology of the space is the set of information which encodes this continuous behaviour. In particular, a metric space has a natural topology on it.)
• 2.37, 2.41, 2.42. These are the important analysis results. They are results which we will use numerous times
• Note: theorem 2.39 is often called the Cantor intersection theorem; the preliminary theorem, 2.38, is the nested intervals lemma. Both are important.

You should also spend some time with the Cantor set (2.44). (This set is a rather instructional counterexample for some nice, but false, conjectures: e.g. ‘every compact set in $ \mathbb{R}^n $ is a finite union of closed intervals’.)

Compactness can be a difficult concept to understand; in $ \mathbb{R}^n $, compactness is equivalent to being closed and bounded (this is the content of the Heine-Borel theorem, 2.41), but in general the definition is that given in 2.32. Some motivation for the concept can be found here. See also these questions on MSE: 1, 2.

Please also read my comments below; in particular, if you do not find yourself struggling with the material from this week onwards, it likely means that you do not understand it.

Suggested problems: I am not setting down any recommended problems in particular. However, it will be to your advantage to do (by which I mean, attempt) as many problems from this chapter as possible.

It has come to my attention that some of the people following these readings have been doing so without doing any problems, or at least doing only a few. I would like to stress two points:

1. I do not care in the slightest whether or not you do any problems. I am not running a lecture course. I will not be setting a final exam. Your success or failure is not, philosophically speaking, my concern.
2. However, if you do not do any problems then you will not understand the material. This is a fact. It is all very well and good to say ‘I have always understood mathematics quite well by just listening in lectures and reading passively’. However, as you pass from stage II into some real mathematics papers (320, 332, etc.) you must eventually learn that the only way to learn any mathematics is to do some mathematics. Some of you will not believe me, and you may even get to the end of Rudin having passively read all his proofs, thought `Yes, I could have written that’; and you will have been cheated out of learning analysis. The best way to read Rudin, as I have said a couple of times now, is to read up until he states a theorem; then, closing the book, one should write down the proof oneself. (In particular, when I list the ‘important theorems’ above, some are theorems whose proofs you should be able to reproduce - not by memorisation, but by understanding.)

Next: We will discuss this when we meet in person at some point over the next few weeks.