1. ## Limit points-Topology

Take Real Set, R, in the cofinite topology.

• Show that every sequence, whereno element repeats more than fini tely times, converges to every point in the real set.

• Find a sequence, which has a range that is infinite, which converges only to 0.
• Find a sequence, with infinite range, which does not have a limit.

2. Originally Posted by Andreamet
Take Real Set, R, in the cofinite topology.

(1) Show that every sequence, whereno element repeats more than fini tely times, converges to every point in the real set.

(2) Find a sequence, which has a range that is infinite, which converges only to 0.
(3) Find a sequence, with infinite range, which does not have a limit.
Let $\displaystyle \{x_{n}\} n=1,2, .., \infty$ be a sequence in R

For (1), every open set containing a point in $\displaystyle \{x_{n}\}$ should contain all but finite elements of $\displaystyle \{x_{n}\}$, since an open set in a cofinite topology is defined as X\O is finite, where X is a set with a topological space T and O is an open set in a cofinite topology.
For an arbitrary open set O containing a point $\displaystyle \{x_{n}\}$ should contain all memebers of $\displaystyle x_{n}$ for n>=N where N is a positive integer in a cofinite topology. That means, every sequence with distinct points (including elements repeated finite times) converges to every point in the real set in a cofinite topology.

For (2), {1,1/2,1/3,...,1/n}, n=1,2,..,$\displaystyle \infty$.
For (3), {1,2,3,....,n}, n=1,2,..., $\displaystyle \infty$.

3. Thanks! However, For (2), the range of the sequence is finite so that is not a good example

4. Originally Posted by Andreamet
Thanks! However, For (2), the range of the sequence is finite so that is not a good example
What is the problem with #2?
The range of that sequence is $\displaystyle \{ \tfrac{1}{n} | n\geq 1\}$.
This is an infinite set.