Tychonoff’s theorem on arbitrary products of compact spaces

For the proof of Tychonoff’s theorem in my 4th-year module on functional analysis, see the two screencasts at

I am still struggling to achieve good quality video from the webcam I am using at the moment. In my recent screencasts I have taken a desk lamp with me to provide some necessary additional illumination, but I don’t think I have really found the right solution. The picture-in-picture video of me giving the lecture is somewhat jumpy, and the synchronization is a little bit out. At least it is no longer blurry. But perhaps I have reached the limits of what my current tablet laptop can achieve when it comes to video.

On the maths side, I managed to swap between talking about FIP-collections and FIP-families at one point. No difference is intended: I just mean a collection of sets which has the finite intersection property. I call maximal FIP-collections of subsets of X ultrafilters on X in this module. They ARE ultrafilters, but this is not the usual definition: the reader may be puzzled by some of Lemma 2.6 which (essentially) includes the amazing fact that ultrafilters are filters!

In the proof of Lemma 2.6(a) (on extending FIP-collections to form ultrafilters) I forgot to treat the empty chain as a special case. Fortunately, I did note that the partially ordered set was non-empty.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s