If M = [0, 1], the function f(x) = 0 if x is irrational & f(p/q) = 1/q if p & q are coprime integers is an example of one such f.
I have tried coming up with some iteration of 1/n, since I know the interior is empty and it'd bounded, but I can't seem to get infinite limit points.
Problem in full is in title. In my attempt at this, I tried to first think of a set with an infinite number limit points. To this end, I got E = Union of {i + 1\n | n = 1,2,...} from i = 1 to infinity. So E' (the set of all limit points of E) is both infinite and countable. Yay! However, I don't think E is compact. I tried changing my original idea, but everything I come up with has a non-infinite E'. Any thoughts?
Thanks!
> Definitions: By an open cover of a set E in a metric space X, we mean a collection {G_a} of open sets of X such that E is a subset of Union G_a for each a in A.
> A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. More explicitly, the requirement is that if {G_a} is an open cover of K, then there are finitely many indices a_1,...,a_n such that K is a subset of G_a_1 Union ... Union G_a_n.
(I've learned that there are often many different definitions for the same term, so I thought I'd put these here just in case these have different definitions as well).
Thanks!
Bonus question for the DMs: How to you deal/get around with your players doing something absurd with the recent UA?
35 seems very low given the amount of cards with multiple options.
This is a gift for a father. He loves flashlights and gadgets. Hit me with your best options. Budget irrelevant !
Thanks!
Across many accounts - to the point where I post a comment, it says wait ten minutes, I wait an hour and it still says wait ten minutes. No matter what subreddit I try and comment on, I have to wait ten minutes between ANY comments, ANYWHERE.
What the fuck is going on?
I don't see why the Hausdorff property is needed.
Suppose X is limit point compact and first countable. Let xi be a sequence. If xi repeats any value infinitely many times, then trivially it has a convergent subsequence, so assume every value appears only finitely many times. Then xi takes on infinitely many different values and has a limit point x by assumption. x has a nested neighborhood basis Bk by assumption.
Choose a subsequence such that xnk is contained in Bk for all k. This sequence converges to x.
Edit:
I found a PDF that confirms this theorem 9.
I'm going through Lee's Topological Manifolds, and for whatever reason he states/proves the less general theorem as mentioned in the title and uses the Hausdorff property in the proof (which only serves to make the proof longer and somewhat confusing). I'm wondering if there's any reason for this, or maybe I'm missing something.
According to a recent paper, a spinning neutron star having magnetic poles near to its equator can induce strong magnetic repulsion force to another compact magnetic body regardless of polar orientations of these bodies. This interaction when balanced with a conventional magnetic or gravitational pull, might result in a binary without the requirement of orbital mechanism in certain conditions. Another feature of this repulsive interaction is its exceptional short range which is inversely proportional to the seventh power of the distance.
Earlier this year I got a letter from Wells Fargo essentially lowering my credit card limit to a few hundred bucks, which is basically nothing. Then this week I got a letter from them that they are removing a Personal Line of Credit option because they are "discontinuing our Personal and Portfolio Line of Credit Products."
Now I used the credit card rarely, it accounted for a <10% available credit for me but to reduce it to a few hundred bucks after a decade? Makes no sense.
The LOC I'd dive into on and off again but it's been available to me for about 8 years. During the Pandemic I know many banks tightened up their credit availability and declined plenty of applicants but nothing ever affected my existing credit. If there was a time to reel in credit availability, it would have been in 2020.
Until GME? I'm thinking, we had a world wide market crash in 2020 and nothing like this happened.
I'm just wondering if any other apes are having older accounts or minor accounts closed out. Smooth brain me feels like they are reeling in their over extended credit to users because of future market issues.
