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Let $\mathcal{K} = \mathcal{K}(\mathbb{N}^{\mathbb{N}})$ be the set of all non-empty compact subsets of the Baire space $\mathbb{N}^\mathbb{N}$ equipped with the Vietoris topology. Let $G$ be a Borel subset of $\mathbb{N}^\mathbb{N}$. Is the set $\mathcal{K}_{G} = \{K \in \mathcal{K}: K \cap G\text{ is not empty}\}$ a Borel subset of $\mathcal{K}$? analytic?

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The set $\mathcal{K}_{G}$ is in general not Borel, see Kechris Classical Descriptive Set Theory, the very beginning of Section 27.B, p209.

It is analytic. This could be shown using Exercise 4.29 in Kechris.

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