For the final example about the connected set that is not path connected, I was having trouble figuring out the detail in the extra notes about the existence of the minimum value a. I managed to figured out a proof which might help others understand. The claim is that there is a minimum a such that g(a)=(0,y). Notice that any value t which satisfies g(t)=(0,y) equivalently satisfies (fg)(t)=0, where f is the projection function along the first coordinate. Hence the set of such t is exactly the set of zeroes for the composite function fg. We know the projection function is continuous and g is continuous by assumption, so fg is also continuous. An important fact to know is that the zeroes of a continuous real function form a closed set. This set is a subset of [0,1], so it is bounded below. This and the fact that it’s closed implies that the set of zeroes contains its infimum. We take this infimum to be the value of a.

Great quality as usual. Very much look forward to the discussion on compactness after you're done with locally connectedness!

These videos are so awesome i really hope this playlist reaches even more interesting topics like homotopy someday 😁

Wondering if these morphing spaces are used by orian to track down the power of money counter accumulation on the black markets.

great video

Keep going mate ❤️

@mariusfurter

2 жыл бұрын

Thanks, I will :)

These are so so so helpful! Do you have a video on group topology?

@mariusfurter

Жыл бұрын

Unfortunately, I don't have any videos covering topological groups yet.