Assume that $C$ is a compact set and $p\in X\backslash C $.

Question

$(X,d)$ is a metric space.Assume that $C$ is a compact set and $p\in X\backslash C $.
Construct two disjoint open sets $E_1$ and $E_2$ such that $p\in E_1 $ and $C\subset E_2$.

I want to denote $d=d(p,C)$ which $d(p,C)$ is the distance between $p$ and $C$. And $E_1= B(p,\frac{d}{2})$ , $E_2=C$. I once thought it was figured out. But C is not open ...

So is there any brief construction?

Answer 1

For $x\in C$, define $U_x = B(x,\frac{1}{2}d(x,p)), V_x=B(p,\frac{1}{2}d(x,p))$.

Then $U_x \ \cap \ V_x = \emptyset$, and letting $x$ vary in $C$ we get an open cover $\{U_x\}_{x \in C}$.

Can you take it from here?