Classnotes for October 6

From 0506Topology

#	Week of...	Links (edit)
Fall
1	Sep 12	About, Tue, Thu, Std2Disc
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6	Oct 17	Tue, Thu, HW3
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E	Dec 12	TE2
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20	Feb 27	Tue, Thu, HW9
21	Mar 6	Tue, Thu, HW10
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Study	Apr 17	Office Hours
Exams	Apr 24	Final, PM

Table of contents

1 Review: Topologies on

2 Metric Spaces

3 Scanned Notes

3.1 Notes in JPG

[edit]

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 $π α$ 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 $X α$ has the trivial topology, so does the product.
If each $X α$ 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 $A α$ 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 $X α$ 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.

[edit]

Metric Spaces

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

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

Examples:

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

(insert illustration of the three distance metrics)

Definition:

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

Examples:

(insert illustration of $B 1 (0)$ relative to $d 1$ , $d 2$ and $d 3$ )

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

Proof:

$\forall x\in X, x\in B_{17}(x)$ , so the first property of bases is easily satisfied.
Suppose we $y\in B_{r_1}(x_1)\cap B_{r_2}(x_2)$ . Then $d (y, x 1) < r 1$ and $d (y, x 2) < r 2$ . Set $r = min{r 1 - d (y, x 1), r 2 - d (y, x 2)}$ . 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 $d 1$ , $d 2$ or $d 3$ 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 $B 1 / 2 (x) = {x}$ , 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 $B δ (x)$ and $B δ (y)$ 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 $X k$ 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 $d k$ 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:

$d_k^{\mathrm{new}}$ is a metric.
$d_k^{\mathrm{new}}$ defines the same topology as $d k$ .

(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 $X k$ : only finitely many elements of the product are restricted for $B_\varepsilon((x_k))$ .