Read-along: Rudin's PoMA: Agenda for first physical meetup

The goal for the meetup is to enable us to work through some of the more difficult proofs together, and to give various non-trivial examples. The agenda will include at least the following:-

Non-trivial examples of metric spaces.
Motivations for compactness and connectedness.
Detailed workthrough of the proofs (by you, not me). [Essentially this is to both force you to work through the proofs, and to allow me to pace the next few weeks readings.]
Exercises 22 to 26 of Rudin.

If we have time I hope to discuss one or more of the following:

(General) topology
Manifolds

Please bring:

Any questions and/or comments. I do not wish to lecture!
Writing implement and paper.

You do not need a copy of the book.

Non-trivial examples of metric spaces.

See Kolmogorov and Fomin, section 5.1, for details. In each case, $ X $ is our set of points and $ \rho : X^2 \to \mathbb{R} $ is a metric. Further, $ n \in \mathbb{N} $ and $ p \in \mathbb{N} $.

	Point set	Metric
1	$ X $ an arbitrary set	$\rho(x,y) = \begin{cases} 0 & x = y \\ 1 & x \neq y \end{cases}$
2	$ X = \mathbb{R} $	$\rho(x,y) = \left\|x-y\right\|$
3	$ X = \mathbb{R}^n $	$\rho_2(x,y) = \left(\sum_{k = 0}^n (x_k - y_k)^2\right)^{1/2}$
4	$ X = \mathbb{R}^n $	$\rho_1(x,y) = \sum_{k = 0}^n \left\|x_k - y_k\right\|$
5	$ X = \mathbb{R}^n $	$\rho_\infty(x,y) = \sup_{1 \leq k \leq n} \left\|x_k - y_k\right\|$
6	$ X = C_{[a,b]} $	$\rho_\infty(x,y) = \sup_{t \in [a,b]} \left\|x(t) - y(t)\right\|$
7	$X = \left\{(x_k) \in \mathbb{R}^\mathbb{N} : \sum_{k = 1}^\infty x_k^2 < \infty \right\}$	$\rho_2(x,y) = \left(\sum_{k = 0}^\infty (x_k - y_k)^2\right)^{1/2}$
8	$ X = C_{[a,b]} $	$\rho_2(x,y) = \left( \int_a^b \thinspace [x(t) - y(t)]^2 \thinspace\mathrm{d}t \right)^{1/2}$
9	$X = \left\{(x_k) \in \mathbb{R}^\mathbb{N} : \exists M > 0, \forall k \in \mathbb{N}, x_k < M \right\}$	$\rho_\infty(x,y) = \sup_{k \in \mathbb{N}} \left\| x_k - y_k \right\|$
10	$ X = \mathbb{R}^n $	$\rho_p(x,y) = \left(\sum_{k = 0}^n (x_k - y_k)^p\right)^{1/p}$
11	$X = \left\{(x_k) \in \mathbb{R}^\mathbb{N} : \sum_{k = 1}^\infty x_k^p < \infty \right\}$	$\rho_p(x,y) = \left(\sum_{k = 0}^\infty (x_k - y_k)^p\right)^{1/p}$

“Three hard theorems”, or three old friends.

We have not yet defined ‘continuity’, and we will not do so for a few weeks. However, the notion of continuity is what motivates the introduction of the concepts of connectedness and compactness.

As proved in MATHS 150 with much difficulty and the use of many $ \varepsilon$s (Spivak chapters 7, 8):

(IVT) If $ f $ is continuous on $ [a,b] $ and $ f(a) \leq x \leq f(b) $ then there is some $ c \in [a,b] $ such that $ f(c) = x $.
If $ f $ is continuous on $ [a,b] $ then there exists $ M > 0 $ such that for all $ x \in [a,b] $, $ f(x) < M $.
(EVT) If $ f $ is continuous on $ [a,b] $ then there is some $ y \in [a,b] $ such that $ f(y) \geq f(x) $ for all $ x \in [a,b] $. (Note that 2 follows from 3.)

More general, and to be proved by us in only a couple of lines: Let $ X $ be a metric space and let $ Y $ be an totally ordered metric space. Let $ f : X \to Y $ be continuous.

(IVT) Suppose $ X $ is connected, and pick $ a, b \in X $. If $ y \in Y $ satisfies $ f(a) \leq y \leq f(b) $ then there exists $ c \in X $ such that $ f(c) = y $. (Rudin 4.23)
(EVT) Suppose $ X $ is compact. Then there exists $ y \in Y $ such that for all $ x \in X $, $ f(y) \geq f(x) $. (Rudin 4.16)

(One can even replace ‘metric space’ with ‘topological space’ in the above! See Munkres theorems 24.3 and 27.4.)

References: A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (trans. Silverman); James Munkres, Topology (second edition); Walter Rudin, Principles of Math. Analysis (third edition); Michael Spivak, Calculus.