When $k_{i}=1$ there is no need to glue in an $A_{0}$ ALF space ($\mathbb{R}^{4}$ endowed with the Taub–NUT metric), since $\bm{\underline{\omega}}^{\textup{gh}}_{\epsilon}$ already extends smoothly over $\pm p_{i}$. It is however useful (we will use this in setting up the analysis for the deformation problem) to think of a rescaled Taub–NUT space localised around $\pm p_{i}$.