2 problems

Let $T\subseteq {^{<\omega_2}}\omega_2$ be an $\omega_2$-Kurepa tree, and let $[T]$ denote its set of cofinal branches. For $\sigma\in T$, let $N_\sigma$ be the basic neighborhood…

Let $\kappa$ be an infinite cardinal, let ${\mathfrak{K}}^\kappa$ be the generalized Cantor space, and let $\boldsymbol{\text{S}{\mathcal{N}}}_\kappa$ denote the family of $\kappa$…