By definition,
Perfect set $E_1$ is closed set without isolated points.
Compact set $E_2$ is bounded and closed set in Euclidean space; $\mathbb{R}^n$.
Is the following equation true? $$E_2 \subseteq E_1$$
In my understanding, examples of perfect set; $$ \mathbb{R}^n, [10, \infty), [-10, 20] \cup [30, 40], \mbox{Cantor set}, [(0,0) (20, \infty)], \cdots $$
and examples of compact set; $$ [10, 20], [-7, -2]\cup[0, 5], [(10, 20), (15, 23)], \cdots $$
In a nutshell, my question is whether every compact set is perfect.
In addition, are there any problems in my examples?
If $p\in\Bbb R^n$, the set $\{p\}$ is compact but not perfect, since $p$ is an isolated point. The same is true of any finite set. There are also infinite compact sets that are not perfect; a simple example in $\Bbb R$ is $\{0\}\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$, in which $0$ is the only non-isolated point.