Yeah, that sounds cool.
For 21, I would also show that all metric spaces are 1c (but I think you were already going to do that
)
If 25 is to short, maybe you can add this two theorems:
1) A set G in a first countable space is open iff every x in G and for every x_n --> x, x_n is eventually in G.
2) A set F in a first countable space is closed iff, whenever a sequence in F is converging to x, then x is in F.
And if the video is still to short, you can maybe add that such a space is called sequential.
For 27, maybe you can say a bit about ultranets and there existance.
I'm really looking forward to next week, I've never decently learned about nets so I really want to know what you have to say about it.