I like to often emphasize that definitions just state a property. Just because the property was stated doesn't tell you a priori how many things satisfy the definition -- it could be 1, 0, 2, or well ... you get the point. So the topology generated by a basis may not exist for some bases, for all you know, until you prove that it always exists.
I have to say, I just passed my first Advanced Calc class and didn’t stumble onto your channel until now! I love the content you do and the fact that its such high production is a real plus. Keep it up man, you really make math digestible for everybody. (Btw In your ‘first year Ph.D. Program’ video I think you mentioned measure theory, as somebody interested in a statistics Ph.D., do you know if this class is common to take?)
Congratulations on getting through analysis! If you have friends coming in behind you, send them my way! Measure theory is integral to statistics. Specifically, probability theory. You’ll definitely be learning a decent amount of measure theory in a statistics PhD. You’ll be concerned with finite measure spaces (because the maximum probability is 1). But most of the theory is the same.
4:43 I had to think about this for a while, but it seems that uniqueness of sequential limits lies strictly between the Hausdorff and T_1 separation axioms. Clearly, sequential limits are unique in every Hausdorff space. But the converse is not true -- take the cocountable topology on an uncountable set. Every two nonempty open sets meet, but also every convergent sequence is eventually constant. If a space is not T_1, then there exist distinct points p and q such that every open set including p must also include q. This implies that every sequence converging to q must also converge to p. Thus, every space in which sequential limits are unique must be T_1. The converse can be seen not to be true by considering the cofinite topology on an infinite set. Since every singleton is closed, the space is T_1, but any sequence of distinct elements converges to every point. It is true though, that being Hausdorff is equivalent to the uniqueness of limits of nets.
Hmm… maybe I didn’t give this as much thought as I should have. Thanks for taking the time to write this up. It’s been a while since I really took a serious look at topology. I will trim out that statement when I get a chance.
You can manufacture such a topology pretty easily. Basically, you can take the reals and for any open set that is not empty, union it with (0,1). This will give you a collection of sets that are closed under arbitrary unions and finite intersections, and you have the empty set and the set of all reals. Then the sequence 1/n converges to every element in (0,1).
