# Abstract For a subset of a cardinal greater than ω1, fatness is strictly stronger than stationarity and strictly weaker than being closed unbounded. For many regular cardinals, being fat is a sufficient condition for having a closed unbounded subset in some generic extension. In this work we characterize fatness for subsets of Pκ(λ). We prove that for many regular cardinals κ and λ, a fat subset of Pκ(λ) obtains a closed unbounded subset in a cardinal-preserving generic extension. Additionally, we work out the conflict produced by two different definitions of fat subset of a cardinal, and introduce a novel (not model-theoretic) proof technique for adding a closed unbounded subset to a fat subset of a cardinal. # Adding Closed Unbounded Subsets of λ ## 1. Fatness **Definition 1.1 (Abraham, Shelah).** A subset $S$ of a regular cardinal $\lambda$ is *fat stationary* or simply *fat* iff for every club $C \subseteq \lambda, S \cap C$ has closed subsets of arbitrarily large order types below $\lambda$. **Definition 1.2 (Sy Friedman).** A subset $S \subseteq \omega_2$ is *fat stationary* iff $S \cap \operatorname{Cof}\left(\omega_1\right)$ is stationary in $\omega_2$ and $\forall \alpha \in S \cap \operatorname{Cof}\left(\omega_1\right)(S \cap \alpha$ contains a club subset of $\alpha)$. **Definition 1.3.** Let $\lambda$ be a cardinal, $S \subseteq \lambda$, and $\kappa \in \operatorname{Reg} \cap \lambda$. Then $\mathscr{F}_\kappa=\mathscr{F}(\lambda, S, \kappa)$ is the set of points $\alpha$ in $S$ of cofinality $\kappa$, where $S \cap \alpha$ contains a club subset: $ \mathscr{F}_\kappa=\{\alpha \in S \cap \operatorname{Cof}(\kappa): S \cap \alpha \text { contains a club subset of } \alpha\} . $ The notion of $\mathscr{F}_\kappa$ paves a way to generalize Friedman's definition. **Theorem 1.6.** If $\lambda=\kappa^{+}$ for a regular uncountable cardinal $\kappa$ and $S \subseteq \lambda$, then $S$ is fat in $\lambda$ iff $\mathscr{F}_\kappa=\mathscr{F}(\lambda, S, \kappa)$ is stationary in $\lambda$. **Theorem 1.7.** If $\lambda>\omega$ is a regular limit cardinal and $S \subseteq \lambda$, then $S$ is fat in $\lambda$ iff $\mathscr{F}_\kappa$ is stationary in $\lambda$ for cofinally many $\kappa<\lambda$.