Simple question of topological cofibration I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$.  I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$.  Also, the intersection of $U \times V$ with $X$ is $U \times V'$.
In fact, $V$ is a neighborhood of the origin in $[0,1)^k$.  Also, $V' \subset V$ is given by a set of subsets $\mathcal I$ of $\{1, \ldots, k\}$.  It is the union
$$
\bigcup_{I \in \mathcal I} \{ x_i = 0 \text{ for all } i \in I \}.
$$
That is, $V'$ is made up of some half-lines, quarter-planes, octants, etc. in $[0,1)^k$.
Question:  Is this enough to show that $X \to Y$ is a cofibration of topological spaces?
 A: I'm not sure whether this will be helpful to mpdude, but I thought I'd just point out that given a smooth manifold M and a smooth submanifold N, you can see very easily that the inclusion $N\to M$ is a cofibration, without any cell structure.  
Simply observe that a tubular neighborhood gives you a mapping cylinder neighborhood of N inside M (if you choose a metric, then the disk bundle is the mapping cylinder of the projection from the sphere bundle down to N).  The homotopy extension property is then easy to verify, because in general for a map $f: X\to Y$, the pair $(M_f, X\cup Y)$ has the homotopy extension property ($M_f$ is the mapping cylinder) so given a map on M and a homotopy on the submanifold M, you can extend your homotopy trivially outside the disk bundle of the tubular neighborhood, and use the HEP to extend it on the disk bundle itself (leaving the homotopy constant on the sphere bundle).
I suppose this procedure would also work for infinite-dimensional manifolds (with appropriate hypotheses).
Closely related is the characterization of (closed) cofibrations in terms of neighborhood deformation retracts, which appears for example in Arne Strom's Note on Cofibrations II.
A: The inclusion of a CW subcomplex $K$ into a CW complex $L$ is a cofibration. Briefly, we can extend a homotopy from the $n-1$-skeleton to the $n$-skeleton by projecting $e\times I$ to $e\times\{0\}\cup\partial E\times I$ for any $n$-cell $e$ not in $L$.
