Locally homotopy-negligible subset

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

Definition

A subset $A$ of a topological space $X$ is said to be locally homotopy-negligible if for every open subset $U$ of $X$ containing $A$, the inclusion map from $U\setminus A$ to $U$ is a homotopy equivalence.