### Context

My question is about the "proof of claim" on page 84 of Goresky and MacPherson's "Intersection Homology II". For ease of reading, here's the claim:

Claim:Suppose $X$ is a topological pseudomanifold$^*$ and $\mathbf A^\bullet$ is a topologically constructible$^*$ complex of sheaves on $X$. For any $x\in X$ there is a neighborhood basis $U_1\supset U_2\supset U_3\supset\ldots$ such that for each $i$ and $m$, the restriction map $$H^i(U_m; \mathbf{A}^\bullet) \to H^i(U_{m+1}; \mathbf{A}^\bullet)$$ is an isomorphism.

$^*$ See [$\S$1.1 and $\S$1.4, loc. cit.] for the definitions of topological pseudomanifold and topologically constructible.

### My Question

The only part of the proof I'm having issues with is the very beginning, where the authors construct the neighborhood basis itself. They start by choosing a distinguished neighborhood $N\cong \Bbb R^i\times \operatorname{cone}^\circ(L)$ of the point $x$. They then consider the join $Y=S^{i-1}*L$ and assign it a stratification. I'm OK so far. However, they then say, "Choose a stratum preserving homeomorphism $\psi\colon \operatorname{cone}^\circ(Y)\to N$ with $\psi(\text{vertex})=x$."

Question:(1) Why are $\operatorname{cone}^\circ(Y)$ and $N$ homeomorphic, and (2) why does there exist a stratum preserving homeomorphism (with $\psi(\text{vertex})=x$) between them?