Classnotes for September 29
|#||Week of...||Links (edit)|
|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|
|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|
|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|
Infinite Product Spaces
Let be a collection of topological spaces.
Then we define the product .
Note: The statement "If then ." 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.
The Box Topology
We use as a basis for the product topology in . So we'll use the basis to generate a topology on . We'll call this topology the box topology.
In fact, this is the wrong choice of topology, as we'll find out.
The Tychonoff Topology
Theorem: There is a unique topology (the Tychonoff topology, or the cylinder topology) on such that the following conditions are satisfied:
- For each , is continuous. (πβ is the projection function mapping .)
- If is continuous for each , then is continuous. ( maps .)
Uniqueness: same as for : see the notes on the Product Topology.
Each πβ is continuous, so is open if Uβ is open in Xβ.
Finite intersections of open sets are open, so if we have such that is open in , then must be open.
So select the basis .
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 .
. This is open since it is a finite intersection of open sets.
Scanned and Recorded Lecture Notes
Notes in JPG, by Annat Koren
Notes in PDF
Available here (http://katlas.math.toronto.edu/0506-Topology/index.php?title=Image:3-2-09-29.pdf).
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
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 , where the removed point is circled (or squared) in red below:
The chess scene above: Kasparov vs. Deep Blue, game 1. =)