Classnotes for September 29

From 0506Topology

#	Week of...	Links (edit)
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

Classnotes for September 29

Table of contents

1 Infinite Product Spaces

1.1 The Box Topology
1.2 The Tychonoff Topology

2 Scanned and Recorded Lecture Notes

2.1 Notes in JPG, by Annat Koren
2.2 Notes in PDF
2.3 AUDIO Of Entire Lecture

3 Interesting Challenge

[edit]

Infinite Product Spaces

Let $\left\{X_\alpha\right\}_{\alpha\in I}$ be a collection of topological spaces.

Then we define the product $\prod_{\alpha\in I}X_\alpha=\left\{\left(x_\alpha\right)_{\alpha\in I}:x_\alpha\in X_\alpha\right\}=\left\{\left(x:I\to\bigcup_{\alpha\in I}X_\alpha\right):x(\alpha)\in X_\alpha\right\}$ .

Note: The statement "If $\forall\alpha\in I\ X_\alpha\not=\varnothing$ then $\prod_{\alpha\in I}\not=\varnothing$ ." is equivalent to the axiom of choice in set theory, and it's nicer for things not to depend on that axiom (someone insert something more enlightening here). However, for the particular infinite product spaces we'll be using, the axiom of choice is not required to show that they're non-empty; the axiom of choice is only needed for the general case. And even if they are empty, everything we say will still be true.

Examples:

$\{0,1\}^{\mathbb N}=\prod_{n\in\mathbb N}\{0,1\}=\{\hbox{strings of the form}\ 00110101\cdots\}$
$\mathbb R^{\mathbb N}=\prod_{n\in \mathbb N}\mathbb R=\{a_1a_2a_3\cdots:a_i\in\mathbb R\}$
$\mathbb R^{\mathbb R}=\prod_{\mathbb R}\mathbb R=\{f:\mathbb R\to\mathbb R,\ f\ \hbox{not necessarily continuous}\}$

[edit]

The Box Topology

We use $\mathcal B=\{U\times V:U\ \hbox{open in}\ X,\ V\ \hbox{open in}\ Y\}$ as a basis for the product topology in $X\times Y$ . So we'll use the basis $\mathcal B_{\mathrm{box}}=\left\{\prod_{\alpha\in I}U_\alpha:U_\alpha\ \hbox{open in}\ X_\alpha\ \hbox{for each}\ \alpha\right\}$ to generate a topology on $\prod_{\alpha\in I}X_\alpha$ . We'll call this topology the box topology.

In fact, this is the wrong choice of topology, as we'll find out.

[edit]

The Tychonoff Topology

Theorem: There is a unique topology (the Tychonoff topology, or the cylinder topology) on $\prod_{\alpha\in I}X_\alpha$ such that the following conditions are satisfied:

For each $\beta\in I$ , $\pi_\beta:\prod_{\alpha\in I}\to X_\beta$ is continuous. ( $π β$ is the projection function mapping $\left(x_\alpha\right)\mapsto x_\beta$ .)
If $f_\alpha:Z\to X_\alpha$ is continuous for each $\alpha\in I$ , then $\prod_{\alpha\in I}f_\alpha:Z\to\prod_{\alpha\in I}X_\alpha$ is continuous. ( $\prod_{\alpha\in I}f_\alpha$ maps $z\mapsto\left(f_\alpha(z)\right)_{\alpha\in I}$ .)

Proof:

Uniqueness: same as for $X\times Y$ : see the notes on the Product Topology.

Existence:

Each $π β$ is continuous, so $\pi_\beta^{-1}\left(U_\beta\right)$ is open if $U β$ is open in $X β$ .

Finite intersections of open sets are open, so if we have $U_{\beta_1},U_{\beta_2},\ldots,U_{\beta_n}$ such that $U_{\beta_i}$ is open in $X_{\beta_i}$ , then $\bigcap_{i=1}^n\pi_{\beta_i}^{-1}\left(U_{\beta_i}\right)$ must be open.

So select the basis $\mathcal B=\left\{\bigcap_{i=1}^n\pi_{\beta_i}^{-1}\left(U_{\beta_i}\right):\beta_i\in I\ \hbox{and}\ U_{\beta_i}\ \hbox{open in}\ X_{\beta_i}\ \hbox{for}\ i=1,\ldots,n\right\}$ .

Motivation for the word cylinder: a cylinder is a set of points where two coordinates are constrained but the third is not. This is a bit like an open set in this basis.

Proof that this satisfies condition 2 of the theorem:

A typical basic open set is $\bigcap_{i=1}^n\pi_{\beta_i}^{-1}\left(U_{\beta_i}\right)$ .

$\left(\prod_{\alpha\in I}f_\alpha\right)^{-1}\left[\bigcap_{i=1}^n\pi_{\beta_i}^{-1}\left(U_{\beta_i}\right)\right]$ $=\left\{z:\left(\prod_{\alpha\in I}f_\alpha\right)(z)\in\bigcap_{i=1}^n\pi_{\beta_i}^{-1}\left(U_{\beta_i}\right)\right\}$ $=\left\{z:f_{\beta_1}(z)\in U_{\beta_1},\ldots,f_{\beta_n}(z)\in U_{\beta_n}\right\}$ $=\left\{z:f_{\beta_1}(z)\in U_{\beta_1}\right\}\cap\cdots\cap\left\{z:f_{\beta_n}(z)\in U_{\beta_n}\right\}$ $=\bigcap_{i=1}^nf_{\beta_i}^{-1}\left(U_{\beta_i}\right)$ . This is open since it is a finite intersection of open sets.

[edit]

Scanned and Recorded Lecture Notes

[edit]

Notes in JPG, by Annat Koren

Handwritten notes by Annat Koren

[edit]

Notes in PDF

Available here (http://katlas.math.toronto.edu/0506-Topology/index.php?title=Image:3-2-09-29.pdf).

[edit]

AUDIO Of Entire Lecture

First Half of Lecture: Media:Sep2905_1stHalf.ogg.

Second Half of Lecture: Media:Sep2905_2ndHalf.ogg.

NOTE: If you are having difficulty playing the .OGG file format, here is a link to free .OGG players, for all platforms: http://wiki.xiph.org/index.php/VorbisSoftwarePlayers

[edit]

Interesting Challenge

Let us start with a graphical challenge. Recall $Σ$ , the Seifert surface of the trefoil from Classnotes for September 15:

Can you draw the picture on the left

on a standardly drawn punctured torus? How about, on $Σ$ ? Notice that boundaries should go to boundaries, so the entire boundary of your drawn surface should be red at the end. If you like chess better, do the same with the right-most picture. (BTW, who played this game?)

Note that these pictures can be thought of as living on $T^2-\{pt\}=({\mathbb R}^2-{\mathbb Z}^2)/{\mathbb Z}^2$ , where the removed point is circled (or squared) in red below: