Classnotes for October 6

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Fall
1 Sep 12 About, Tue, Thu, Std2Disc
2 Sep 19 Tue, Thu, HW1, 14 Sets
3 Sep 26 Tue, Thu, Photo
4 Oct 3 Tue, Thu
5 Oct 10 HW2, Tue, Thu
6 Oct 17 Tue, Thu, HW3
7 Oct 24 Mon, Tue, Thu
8 Oct 31 Tue, Thu, HW4
9 Nov 7 TE1, Tue, Thu
10 Nov 14 Tue, Thu, HW5
11 Nov 21 Tue, Thu
12 Nov 28 Tue, Thu, HW6
13 Dec 5 Tue, Thu
E Dec 12 TE2
Spring
14 Jan 9 Tue, IT83, Thu, HW7
15 Jan 16 Tue, Thu
16 Jan 23 Tue, HW8, Thu
17 Jan 30 Tue, Thu
18 Feb 6 TE3, Tue, Thu
19 Feb 13 Tue, Thu
R Feb 20
20 Feb 27 Tue, Thu, HW9
21 Mar 6 Tue, Thu, HW10
22 Mar 13 Tue, Thu
23 Mar 20 Tue, Thu, HW11
24 Mar 27 Tue, Thu
25 Apr 3 Tue, Thu, HW12
26 Apr 10 Tue, Thu
Study Apr 17 Office Hours
Exams Apr 24 Final, PM

Review: Topologies on $\prod X_\alpha$

We have defined two topologies on the inifinite product $\prod X_\alpha$ which generalize the topology on $X\times Y$:

• The box topology: we generalize the basis to $\mathcal B_{\mathrm{box}}=\left\{\prod U_\alpha:U_\alpha\ \hbox{open in}\ X_\alpha\right\}$.
• The product topology: we generalize the requirements to the following.
1. Each projection function πα must be continuous.
2. If $f_\alpha:Z\to X_\alpha$ is continuous for each α, so is $\prod f_\alpha$.

A basis for the product topology is $\mathcal B_{\mathrm{cyl}}=\left\{\prod U_\alpha:U_\alpha\ \hbox{open in}\ X_\alpha,\ \hbox{and for all but finitely many}\ \alpha,\ U_\alpha=X_\alpha\right\}$.

As the course progresses, we will discover that it makes the most sense to use the cylinder topology. For now, we know that both topologies behave nicely in the following ways: (someone please check that these are all correct, then remove this worrynote)

• If each Xα has the trivial topology, so does the product.
• If each Xα is Hausdorff, so is the product.
• If $A_\alpha\subset X_\alpha$ for each α, the topology induced by taking the subspace topology on each Aα and then taking the product topology $\prod A_\alpha$ is the same as the topology induced by taking the product topology on $\prod X_\alpha$ and then taking the subspace topology for $\prod A_\alpha\subset\prod X_\alpha$.
• If each Xα has the discrete topology, then the box topology on the product is also the discrete topology. However, we don't know that the cylinder topology is also the discrete topology.

We want more examples of products on which the two topologies are different. This is part of the motivation for introducing metric spaces.

Metric Spaces

Definition: A metric d on X is a function $d:X\times X\to\mathbb R$ such that

1. d(x,y) = d(y,x) (d is symmetric)
2. $d(x,y)+d(y,z)\ge d(x,z)$ (the triangle inequality)
3. $d(x,y)\ge 0$; equality holds iff x = y.

Examples:

1. $\mathbb R^n$, $d_1(x,y)=\sqrt{\sum(x_i-y_i)^2}$
2. $\mathbb R^n$, $d_2(x,y)=\sum|x_i-y_i|$
3. $\mathbb R^n$, $d_3(x,y)=\max_i\left|x_i-y_i\right|$

(insert illustration of the three distance metrics)

Definition:

Suppose $x\in X$ and X has metric d. Then we define Br(x) = {y:d(x,y) < r}.

Examples:

(insert illustration of B1(0) relative to d1, d2 and d3)

Claim: $\{B_r(x):r>0,x\in X\}$ is a basis for a topology.

Proof:

1. $\forall x\in X, x\in B_{17}(x)$, so the first property of bases is easily satisfied.
2. Suppose we $y\in B_{r_1}(x_1)\cap B_{r_2}(x_2)$. Then d(y,x1) < r1 and d(y,x2) < r2. Set r = min{r1d(y,x1),r2d(y,x2)}. Then it can be shown using the triangle inequality that $y\in B_r(y)\subset B_{r_1}(x_1)\cap B_{r_2}(y_2)$, satisfying the second condition.

So we have a way of getting a topology from a metric.

More Examples:

• The topology intuced on $\mathbb R^n$ by d1, d2 or d3 is the standard topology.
• Let X be an arbitrary set, and set d(x,x) = 0 and d(x,y) = 1 whenever $x\not=y$. Then B1 / 2(x) = {x}, so the topology induced on X is the discrete topology.

Definition: A topological space is metrizable if its topology is induced by some metric.

Claim: The discrete topology is always metrizable. (See most recent example for the proof.)

Claim: If X is metrizable, X is Hausdorff.

Proof:

Suppose $x\not=y$. Then d(x,y) > 0. Let δ = d(x,y) / 2. Then Bδ(x) and Bδ(y) are disjoint open sets containing x and y, respectively.

More Examples:

Let $f,g:\mathbb R\to\mathbb R$.

• $d_4(f,g)=\sqrt{\int{(f-g)^2}}$
• $d_5(f,g)=\int{|f-g|}$
• $d_6(f,g)=\sup{|f-g|}$

These defines metrics on appropriate function spaces.

Theorem 1: If Xk is metrizable for $k\in\mathbb N$, then $\left(\prod X_k\right)_{\mathrm{cyl}}$ is metrizable. (good news)

Theorem 2: $\left(\mathbb R^{\mathbb N}\right)_{\mathrm{box}}$ is not metrizable. (bad news... but who cares about the box topology?)

Theorem 3 $\mathbb R^{\mathbb R}$ is not metrizable in either the box topology or the cylinder topology. (really bad news)

Start of Proof for Theorem 1:

Suppose $(X_k,d_k)_{k\in\mathbb N}$ are metric spaces.

Lemma: Without loss of generality, each dk is bounded: in fact, for $x,y\in X_k$, $d_k(x,y)\le 1$.

Proof:

Define $d_k^{\mathrm{new}}(x,y)=\min\{1,d_k(x,y)\}$.

Exercise:

1. $d_k^{\mathrm{new}}$ is a metric.
2. $d_k^{\mathrm{new}}$ defines the same topology as dk.

(end proof of lemma)

For $(x_k),(y_k)\in\prod X_k$, define $d((x_k),(y_k))=\sup_k\frac{1}{k}d_k(x_k,y_k)$.

Claim: d induces the cylinder topology on $\prod X_k$.

Proof:

$B_\varepsilon((x_k))$ is open relative to the cylinder topology: for $k>\frac{1}{\varepsilon}$, $\frac{1}{k}\varepsilon>1$, so $B_\varepsilon((x_k))$ imposes no restrictions on Xk: only finitely many elements of the product are restricted for $B_\varepsilon((x_k))$.

(TODO: fix up this proof)

Scanned Notes

Notes in JPG

Handwritten notes by Annat Koren
Handwritten notes by Annat Koren
Handwritten notes by Annat Koren
Handwritten notes by Annat Koren
Handwritten notes by Annat Koren