Let $X$ Polish space, and $K(X):\{K \subset X : K \text{ is compact}\}$
I need a hint how to prove the family of sets $\{K \in K(X): K \cap U \neq \emptyset\}$ and $\{K \in K(X): K \subset U\}$ generates $\mathbf{B}(K(X))$.
However, it's not a $\sigma$-algebra because the complement is not in the family. We know by this solution form a subbase of Vietoris topology, but the objective is to solve without knowing this fact.