Finite sets are limited countable sets in either countable or uncountable space.
The uncountable equivalent of finite or small sets are compact sets.
Finite sets are bounded and closed. Finite sets contain their sup. and inf.
An open cover of $\mathbb{E}$ in $\mathbb{X}$ (metric) is a collection of sets $S_\alpha$ whose union covers $\mathbb{E}$. The cover of a set is a collection of sets, not a set.
A subcover of $S_\alpha$ is a subcollection $S_{\alpha_i}$ that still covers $\mathbb{E}$.
Example:
$\mathbb{E}=[1/2,1)$ has a cover $V_n$ ($n=3$ $\to \infty)$
$V_n = (1/n , 1- 1/n)$
But (0,2) is also a cover for $\mathbb{E}$.
The existence of sub covers implies that you don't need all the elements of the cover to cover the set.
A set is compact if every open cover contains a finite subcover. A set $\mathbb{K}$ is not compact in $\mathbb{X}$ if there exists $\textit{some}$ open cover with no finite subcover.
Example:
$[1/2, \ \ 1)$ is not Compact. $V_n$ is a cover for E, but $V_n$ does not have a finite sub cover.
Example:
$[0,1]$ We would have to check every open cover to know if any of them have no finite sub cover.