bundle with base $Y$ and typical fiber $Z$. (Again, we assume that $Y, Z$ are both compact and without boundary.) We fix once and for all a boundary defining function $x$, meaning a function $x \in C^{\infty}(M)$ with $\{ x = 0 \} = {\partial}M$ and $dx$ non-vanishing on ${\partial}M$, and thereby fix (after possibly scaling $x$ by a constant) a tubular neighborhood of the boundary, $$\label{eq:tubular}
  \mathcal{U} \simeq {\partial}M \times [0, 1)_x.$$ Note that near each point $q$ on the boundary, there is a neighborhood $V$ in $\partial M$ of $y = \pi(q)$ ($\pi$ is the projection onto the base) such that $\pi^{-1}(V) \simeq V \times Z$, where $\simeq$ denotes diffeomorphism. Below, we will say that we work ‘locally over the base’ when we fix our attention on a neighborhood of $q$ in $M$ of the form $\pi^{-1}(V) \times [0, 1)_x$. If one chooses local coordinates $y$ on the base and $z$ on $\pi^{-1}(q) \simeq Z$, then $$\label{eq:local-coords-near-boundary}
(x, y, z) \mbox{ form a coordinate chart on $M$ in a neighborhood of $q$.}$$ Let $$\label{eq:f-and-b}
  f:= \dim Z, \quad b:= \dim Y.$$

In the space $X$ defined by collapsing the fibers $Z$ over the boundary (see ), the image 