# Local Triviality -> Locally Cone-Like

So, locally cone-like…what did I mean by that?  Let $X \subseteq M$ be a complex analytic subset of a complex manifold, $\mathfrak{S}$ a Whitney stratification of $X$, $p \in X$.  Suppose we’ve given $M$ a Riemannian metric, r, and denote by $B_\delta(p) = \{ q \in M | r(q,p) \leq \delta \}$ the “ball off radius $\delta$ about $p$.” Choosing local coordinates about $p$ in $M$, we might as well assume that we’re dealing with the ordinary Euclidean distance.  ANYWAY, for sufficiently small $\delta > 0$, the “boundary” $\partial B_\delta(p)$ transversely intersects all strata of $\mathfrak{S}$ (this isn’t too hard to show…suppose not, use the local finiteness criterion for $\mathfrak{S}$, and apply the Curve Selection Lemma to each stratum to achieve a contradiction).  Then, there is a homeomorphism (preserving the strata), which I’ll call a $\mathfrak{S}$-homeomorphism, of germs:

$(B_\delta(p) \cap X, p) \overset{\thicksim}{\to} (Cone(\partial B_\delta(p) \cap X), p)$

This can be rephrased a bit more efficiently.  Let $r: X \to \mathbb{R}$ be “distance squared from $p$.”  Then, for $\delta > 0$ sufficiently small, the map $r: X \to [0,\delta]$ is a proper, stratified submersion.  Think about it.   The “stratified submersion” part tells you that the level sets of $r$ are transverse to strata.  Properness allows us to invoke something called “Thom’s first isotopy lemma,” which tells us the cone bit.