Here is the statement that bothering me.
Let the sets $A, K$ and $X$ $s.t.$ $K \subset A \subset X$ for topological space $(X,T)$ and its relative topology (subspace), $(A, T_A)$
Then, Is this statement is true?
$K$ is a compact set on $(A, T_A)$ $\iff$ $K$ is a compact on $(X,T)$
(Surely, this implies the case $K = A$)
Intuitively, It looks like true. But Don't have any confidence that is true or not If true Why does that is hold? or not What condition we need make that hold?
Answer would be appreciated.
It should be clear that if $K$ is compact in $(X,T)$, then it is certainly compact in $(A, T_A)$ via the restriction of open sets (i.e. any open cover in $A$ of $K$ can be extended to an open cover in $X$ of $K$ which will have a finite subcover).
Consider the opposite direction: $K$ is compact in $(A,T_A)$ and let $\{U_\alpha \; | \; \alpha \in I\}$ be an open cover of $K$ in $X$. Then this restricts to an open cover $\{A\cap U_\alpha \; | \; \alpha \in I\}$ in $A$ which has a finite subcover $\{A\cap U_{\alpha_1}, \ldots, A\cap U_{\alpha_k}\}$, proving that there a finite subcover $\{U_{\alpha_1}, \ldots, U_{\alpha_k}\}$ of $K$ in $X$.