Kerodon

Definition 3.4.6.4. Let $X$ be a topological space and let $\operatorname{\mathcal{U}}$ be a collection of subsets of $X$. We say that a singular $n$-simplex $\sigma : | \Delta ^{n} | \rightarrow X$ is $\operatorname{\mathcal{U}}$-small if its image is contained in $U$, for some $U \in \operatorname{\mathcal{U}}$. We let $\operatorname{Sing}_{n}^{\operatorname{\mathcal{U}}}(X)$ denote the subset of $\operatorname{Sing}_{n}(X)$ consisting of the $\operatorname{\mathcal{U}}$-small simplices. Note that the subsets $\{ \operatorname{Sing}_{n}^{\operatorname{\mathcal{U}}}(X) \} _{n \geq 0}$ are stable under the face and degeneracy operators of the simplicial set $\operatorname{Sing}_{\bullet }(X)$, and therefore determine a simplicial subset which we will denote by $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \subseteq \operatorname{Sing}_{\bullet }(X)$.