C*-algebras: Proofs on $C_0(X)$

Question

I'm looking to prove the following but am stuck, please can you help me?

$C_0(X)$ is isomorphic as a C*-algebra to $C_0(Y)$ if and only if X is homeomorphic to Y, where X and Y are locally compact Hausdorff spaces.

$C_0(X)$ is an essential ideal in C(Y) if and only if X is dense in Y.

Thanks very much.