20:20. Quick and easy explanation. Just one question: where the uniqueness of topology is proven? Moreover, in the converse implication where the hypothesis of unique topology is used? Thanks in advance.
First of all, thank you so much for making this series! It's just that I have one question... Do we need to check on the finitude of the intersection in the proof for the last proposition? From the two properties to the curl B as a basis, particularly when you were checking the defining features of a topology. Thanks!
You are welcome for the videos. In the last proof I checked that emptyset and X are open, that arbitrary unions of open sets are open, and that intesections of a pair of open sets is open. If intersections of pairs of open sets are open, then also finite intersections of open sets are open, by induction (just start intersecting the sets one at a time.)
Thanks! I understand the confusion. In a general topological space, open sets are primitives (i.e they are part of the data of the space). Hence in that case, we need to prove that the basis criterion holds (a set U is open iff you can find a basis element contained in it). Oftentimes, however, one defines a topology on a space by using a basis (e.g. open intervals on R). In that case the basis criterion holds by definition. But not all topologies are defined in this way. The implication is not trivial because it holds for any basis for a given topology, regardless how that topology was defined.
You forgot to show that the topology is unique in the porosition that starts at 20:20.
Great videos!💪👍🖖
20:20. Quick and easy explanation. Just one question: where the uniqueness of topology is proven? Moreover, in the converse implication where the hypothesis of unique topology is used? Thanks in advance.
Thank you for explaining the proofs patiently
Thanks! Your explaning-tempo is great!
Loved the continuity comment and basis
good videos for learning topology! thanks
First of all, thank you so much for making this series!
It's just that I have one question... Do we need to check on the finitude of the intersection in the proof for the last proposition? From the two properties to the curl B as a basis, particularly when you were checking the defining features of a topology. Thanks!
You are welcome for the videos. In the last proof I checked that emptyset and X are open, that arbitrary unions of open sets are open, and that intesections of a pair of open sets is open. If intersections of pairs of open sets are open, then also finite intersections of open sets are open, by induction (just start intersecting the sets one at a time.)
love your videos! 1 question, on the basis criterion proof ,
Thanks! I understand the confusion. In a general topological space, open sets are primitives (i.e they are part of the data of the space). Hence in that case, we need to prove that the basis criterion holds (a set U is open iff you can find a basis element contained in it).
Oftentimes, however, one defines a topology on a space by using a basis (e.g. open intervals on R). In that case the basis criterion holds by definition. But not all topologies are defined in this way. The implication is not trivial because it holds for any basis for a given topology, regardless how that topology was defined.
Do you have a playlist on set theory?