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?