Wow man, this is great stuff!! I actually teach topology at university and I'm surely going to recommend this to all my students. I'm looking forward to your new videos.
Anyway, here are some things I'd like to mention
- I think you have a nice balance between theory and examples. I also like that there are many counterexamples. It's really nice... Some more examples of topologies would be appreciated tho, since the concept really many good examples to be understood.
- There are some very small mistakes somewhere. For example, in video 2, at 6:13 you forgot the square (but it isn't really that important). And in video 7, something weird happens at 6:49: the video just jumps back in time and repeats itself. But its all minor.
- I think you should also have defined closed sets, interiors or closures for metric spaces. It might be to abstract if you suddenly introduce it for topological spaces. On the other hand, the metric piece is already very long, so I don't really know...
- I think the prood that open balls are open would have been quite instructive. But that's just me.
Overall, you did a great job. Is there something I can help you with in making these videos? I'd really like to contribute somehow (: