- #1
forget_f1
- 11
- 0
Construct a compact set of real numbers whose limit points form a
countable set.
countable set.
Yeah, I kind of remembered that a bit late...forget_f1 said:Note: A single point has no limit point, since
a limit point of a set A is a point p such that for any neighborhood of p
(ie Ball(p,r) , where p is the origin and r=radius can take any value >0)
there exists a q≠p where q belongs in B(p,r) and q belongs to A.
arildno said:Yeah, I kind of remembered that a bit late...
Finite sets are countable.
A compact set in mathematics is a set that is closed and bounded. This means that the set contains all of its limit points and can be enclosed by a finite radius. In other words, a compact set is a set that is not "missing" any of its points and can be contained within a finite region.
A compact set with countable limit points means that the set has a finite or infinite number of limit points, but can be put into a one-to-one correspondence with the natural numbers. This means that the set has a countable number of limit points.
A compact set of real numbers can have countable limit points if it has a finite or infinite number of limit points that can be put into a one-to-one correspondence with the natural numbers. This means that the set must have a finite or infinite number of limit points that can be counted using the natural numbers.
Constructing a compact set with countable limit points is important in mathematics because it allows for a better understanding of the structure and properties of compact sets. Additionally, many important theorems and results in analysis and topology involve compact sets with countable limit points.
Some examples of compact sets with countable limit points in the real numbers include the set of natural numbers, the set of rational numbers, and the Cantor set. These sets have a finite or infinite number of limit points that can be put into a one-to-one correspondence with the natural numbers.