What is an infinite subset of a compact set?

Question

I am attempting to work on the following proof:

If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.

I know that that this proof has been answered here already, but I am more interested in understanding the statement itself. I am just struggling to comprehend what $E$ is in this question... an infinite subset of a compact set. For some reason, I just cannot visualize this. If anyone out there is able to help me see through the fog, or even provide a specific example for me to think about, I would greatly appreciate it.

score 2 · Accepted Answer · answered Dec 07 '18 at 05:08

2

There are many possibilities. Take $(0,1)\subseteq [0,1]$ or the circle embedded as the equator of the sphere: $S^1\subseteq S^2$.

If you are familiar with it, take the middle thirds cantor set $C\subseteq [0,1]$.

answered Dec 07 '18 at 05:08

Alekos Robotis

27,579

Thank you for the examples. Would it also be correct to say that $[0,1] \subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well. – Dec 08 '18 at 05:38
1

Yes that's also correct. – Alekos Robotis Dec 08 '18 at 05:39

score 0 · Answer 2 · answered Dec 07 '18 at 05:22

There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $I$ for example, you can pick the sequence $1,\frac{1}{2},...,1/n...$ as an infinite subset, or any open(or closed) interval contained in $I$.

Although not mentioned in the description, a proof of the theorem can be found here: Infinite subset of a compact set

What is an infinite subset of a compact set?

2 Answers2