It might be ridiculous but I cannot comprehend why only closed subsets of compact sets are compact. I know there are easy counter examples like $(0,1) \subseteq [0,1]$ where $(0,1)$ is not compact but $[0,1]$ is compact. However, I am still confused.
Let $K$ is a compact set in topological space. Then for ever open cover, there exist some finite subcover.
If $\bigcup_{ \alpha \in I} O_{\alpha}$ is an open cover for $K$ there exists $\bigcup_{ \alpha \in J} O_{\alpha}$ is a finite open cover for $K$ with $J\subseteq I$
Let $A$ is any subset of $K$.
Why I cannot say $A \subseteq K \subseteq \bigcup_{ \alpha \in J} O_{\alpha}$ ? Why $\bigcup_{ \alpha \in J} O_{\alpha}$ doesn’t cover $A$?
I need help for getting rid of this confusion. Thanks in advance