Let $(Y,p)$ be a non-collapsing Ricci limit space, then any tangent cone of $Y$ at $p$ is some Euclidean cone $C(X)$. Denote $\Omega_{Y,p}$ as the set of all possible $X$ such that $C(X)$ can arise as a tangent cone of $Y$ at $p$. In Colding-Naber's paper: https://arxiv.org/pdf/1108.3244.pdf (page 2, 2nd paragraph) it says that $\Omega_{Y,p}$ is compact and path-connected in Gromov-Hausdorff topology.
I wonder how to show that it is path connected.
What came to my mind is that, for two different sequences $r_i,s_i\to\infty$, there is a path from $(r_iY,p)$ to $(s_iY,p)$ by changing the scale from $r_i$ to $s_i$. However, I can not see why such a path will converge to a continuous path between two limits.