these videos are so amazing i really hope more people get to see them, they have really nice explanations and this has been an amazing way to learn topology!
Yes, there are two possible definitions. For all practical purposes, they are equivalent. If you have an open cover containing X, you can get one equal to X by intersecting it with X. Conversely, every open cover that equals X also contains X. In proofs one usually goes back and forth between both options, depending on what's most convenient.
I am following your videos to understand Topology for the first time. Very good explanations. My question: In a discrete space, why is every subset open? It's a question also from the first lecture, I didn't understand. Why can't we find a closed subset? Ex: a set of points.
In a discrete space every subset is open by definition, in particular singleton subsets are open. Conversely, if all singleton subsets are open, then every subset is open since one can express an arbitrary subset as the union over singletons (its elements). Hence a space is discrete if and only if all singleton subsets are open. By definition, a subset is closed if its complement is open. Hence in a discrete space, every subset is closed as well (since every subset, in particular its complement, is open)
I have recently started a PhD so my focus is currently elsewhere. I hope to be able to record and release the next topology video within the coming weeks.
these videos are so amazing i really hope more people get to see them, they have really nice explanations and this has been an amazing way to learn topology!
the textbook I'm using says the open cover only needs to contain the space X. Not that it needs to equal it.
Yes, there are two possible definitions. For all practical purposes, they are equivalent. If you have an open cover containing X, you can get one equal to X by intersecting it with X. Conversely, every open cover that equals X also contains X. In proofs one usually goes back and forth between both options, depending on what's most convenient.
I am looking forward to your new videos
I am following your videos to understand Topology for the first time. Very good explanations.
My question:
In a discrete space, why is every subset open? It's a question also from the first lecture, I didn't understand. Why can't we find a closed subset? Ex: a set of points.
In a discrete space every subset is open by definition, in particular singleton subsets are open. Conversely, if all singleton subsets are open, then every subset is open since one can express an arbitrary subset as the union over singletons (its elements). Hence a space is discrete if and only if all singleton subsets are open.
By definition, a subset is closed if its complement is open. Hence in a discrete space, every subset is closed as well (since every subset, in particular its complement, is open)
Nice explanation! Thanks!👍
are there more videos on the way?
I have recently started a PhD so my focus is currently elsewhere. I hope to be able to record and release the next topology video within the coming weeks.
If you don't mind me asking, what app do you use for writing and recording?
I'm using GoodNotes on iPad. The recording is just the default screen capture on iPad followed by some cropping during editing.