# 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
http://wirksworthii.nottingham.ac.uk/webcast/maths/G14FUN-09-10/FUN-090210/
and
http://wirksworthii.nottingham.ac.uk/webcast/maths/G14FUN-09-10/FUN-120210a/

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.