Induced subhypergraph ($H[W]=(W,\{e\in E(H):e\subseteq W\})$)

For a subset $W$ of the vertex set of a hypergraph $H$, the induced subhypergraph retains exactly the edges wholly contained in $W$. This is the restriction used by a hereditary hypergraph property; it differs from intersecting every edge with $W$, which can change uniformity.