To construct a set with a limit point.

Question

I learned how to construct a Cantor Set, and I am asked to do the following.

"Construct a bounded set with exactly 3 limit points."

Since the Cantor set contains infinitely many points, I don't think something like it will not work. But this is the only thing that I have learned from the book that tells me anything about constructing a set that has a limit point.

I am also considering the interval $[0,1]$ and constructing a set so that the limit points are $\{0,1/2,1\}$.

If possible, I would like to see more than one simple examples because I am new to analysis and I have no teacher. It's very tough.

Answer 1

Construct a set with exactly one limit point and then add two distinct copies of it; let me be more clear: you'll surely agree with the fact that $\{0\} \cup \{{\frac{1}{n} : n \in \mathbb{N} ^{*}}\}$ has only one limit point. Thus if we repeat twice a translation we'll get what we're looking for: $(\{0\} \cup \{{\frac{1}{n} : n \in \mathbb{N} ^{*}}\}) \sqcup (\{2\} \cup \{{\frac{1}{n}+2 : n \in \mathbb{N} ^{*}}\}) \sqcup (\{4\} \cup \{{\frac{1}{n} +4: n \in \mathbb{N} ^{*}}\})$.

Answer 2

Hint: What are the limit points of $\left\{\frac{1}{n}\middle|\ n\in \mathbb{N}\right\}$?

Answer 3

Take a sequence $\{ a_n \}_{n \in \mathbb{N}}$ converging to $0$, another sequence $\{ b_n \}_{n \in \mathbb{N}}$ converging to $\frac{1}{2}$, and another sequence $\{ c_n \}_{n \in \mathbb{N}}$ converging to $1$. Then just take the set $$ S := \{ a_n,b_n,c_n; n \in \mathbb{N} \} $$ to be their union. By definition $0,\frac{1}{2}$ and $1$ will be limit points. For example, you can let $a_n := \frac{1}{n}$, $b_n := \frac{1}{2} - \frac{1}{2^n}$, $c_n := 1 - \frac{1}{n!}$. This choice also conforms to your wish that $S \subseteq [0,1]$.

Answer 4

"Construct a bounded set with exactly 3 limit points."

consider this set: $$\{0.1,\space 0.2,\space0.3,\space\space0.11,\space0.22,\space0.33,\space\space0.111,\space0.222,\space0.333,\space\space0.1111,\space0.2222,\space0.3333,...\}$$

Answer 5

Let $$A=\left\{\frac ab\;\bigg|\; a,b\in\mathbb N, a=1\text{ or }a=b-1\text{ or }b=2a\pm1\right\}$$

Answer 6

Let $$X=\left\{z\in\mathbb{C}\::\:|z|=1-\frac{1}{n},n\in\mathbb{N}_{\geq 1}, \arg z\in\{0,\frac{2}{3}\pi,\frac{4}{3}\pi\}\right\}.$$

Answer 7

{1/n ; n belongs to N} U {(n+1)/n ; n belongs to N} U {(2n+1)/n ; n belongs to N}

this is an infinite set with 3 limit points {0,1,2}