Is the following statement true: $K \subset X$ and $K \subset Y$ such that $X \cap Y \ne \varnothing$ and $X \not\subset Y, Y \not\subset X$, then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$?
In Rudin's Chapter 2, and Theorem 2.33 states that: Suppose $K \subset Y \subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$. But is this embedding of $X$ and $Y$ necessary? What if $Y$ and $X$ is not subset of one another? I think this statement should still be true since compactness is an intrinsic property that does not depend on metric spaces? But I don't have enough techniques to prove it yet. Insights are appreciated!