 Hereditarily small set

For an infinite cardinal number $\kappa$, $H_{\kappa }$ consists of the sets $x$ for which $|\mathrm{tc}(\{x\})|<\kappa$, where $\mathrm{tc}$ is transitive closure. The entire transitive membership ancestry is bounded, not only the size of $x$. These collections are transitive sets. At $\kappa ={\omega }_1$ the elements are the hereditarily countable sets.