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?



Simple question of topological cofibration
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 $L$.


