Question :Let $M$ be a (say smooth, possibly non compact) manifold with boundary. Is the inclusion $\partial M\hookrightarrow M$ a cofibration in the Serre-Quillen model structure of topological spaces ?

I'm only really interested in the following special case :

Let $C\subset\Bbb R^n$ be a closed convex subset with non empty interior. Assume, if necessary, $C$ to be a (possibly unbounded) polyhedron, in that locally near any of its points, $C$ looks like the intersection of a finite number of closed half-spaces $D_i=\lbrace x\mid\varphi(x)\geq\alpha\rbrace$, $i=1,\dots,N$, with linear forms $\varphi_i$ and real numbers $\alpha_i$. Let $F\subset\partial C$ be a closed subset of the boundary; if necessary, assume $F$ to be the union of facets of the polyhedron $C$.

It seems reasonable to expect $\Gamma=C-F$ to be homeomorphic to $B-G$, where $B$ is the closed unit ball, and $G\subset S$ is a closed subset of the unit sphere $S=\partial B$, so that $\Gamma$ is homeomorphic to the smooth manifold with boundary $B-G$ (whose boundary as a manifold ought to be $S-G$).

Real Question :let $G\subset S$ be a closed subset of the sphere, is the inclusion $S-G\hookrightarrow B-G$ a Serre cofibration ?If this fails for general closed subsets $G\subset S$, will $\partial^{\text{man}}\Gamma\hookrightarrow \Gamma$, that is $\partial C-F\hookrightarrow C-F$ for $C$ closed polyhedron, and $F$ a closed subset of $\partial C$ union of facets, be a Serre cofibration ?

I expect the inclusion of the boundary of a smooth manifold $M$ to be a Hurewicz cofibration by virtue of the existence of collarings. I was told, and have found supporting evidence online, that smooth manifolds admit smooth triangulations, so that one might expect $(M,\partial M)$ to be pair homeomorphic to $(X,A)$ for $X$ a CW complex and $A$ a subcomplex.

Are these guesses correct ? Are inclusions $\partial C-F\hookrightarrow C-F$ Serre cofibrations ? A simple Yes / No answser, and if possible, a reference would satisfy me!

(I picked what seemed like a reasonable set of tags, but feel free to change them.)

**EDIT.** Milnor-Stasheff provide just the theorem I need in chapter 20 of their book *Characteristic Classes*, along with two references, one being @ThiKu's reference to Whitehead's paper.