Find all the limit points of the set a frac12n frac15m m in...

Transcript

00:01 So in this question, we're asked to determine the set of limit points of the interval 0 to 1 in the finite complement topology on r.

00:21 So remember that the finite complement topology, the open sets, so open sets are sets that have a finite complement, which means that the closed sets are the finite sets.

00:48 So open sets aren't just infinite sets.

00:52 They're a bit more than that, right? like, you can have an infinite set that also has an infinite complement, right? like, the interval zero to infinity is infinite, but it also has, you know, as a complement negative infinity to zero.

01:07 So anyways, it can't just be infinite.

01:10 They have to be sort of so infinite that they're compliment that's fine.

01:13 And the closed sets are personally the finite sets.

01:16 So we want to find the limit points, right? so our goal is to find the limit points of 0 1.

01:30 And remember a limit point is a point such that every neighborhood that you can draw around that point contains some other point in the set in question.

01:46 So if you think about this, let's just think for a second.

01:53 So say that the point x is in 0 .1, right? now, a neighborhood is just some set that contains an open set that contains the point.

02:10 Right.

02:11 So let v be a neighborhood of x.

02:21 Is there exists some open set u such that x is an element of u, which is a subset of v.

02:30 And for you to be open in this topology, since u is open, you has a finite complement...

Question

Find all the limit points of the set $A = \{\frac{1}{2^n} + \frac{1}{5^m} : m \in \mathbb{N} \text{ & } n \in \mathbb{N}\}$. Discuss whether A is open or closed in $\mathbb{R}$.

Question

Find all the limit points of the set $A = \{\frac{1}{2^n} + \frac{1}{5^m} : m \in \mathbb{N} \text{ & } n \in \mathbb{N}\}$. Discuss whether A is open or closed in $\mathbb{R}$.

Please give Ace some feedback