Quasifibrations and transfinite filtrations This question takes place in the category $\mathrm{CGWH}$
of compactly generated weak Hausdorff spaces.
Let $\lambda$ be a limit ordinal, and suppose we have
a diagram $\Phi: \lambda \to \mathrm{CGWH}$, as indicated
$$
X_0 \hookrightarrow
X_1 \hookrightarrow
\cdots
\hookrightarrow
X_\xi \hookrightarrow
X_{\xi+1} \hookrightarrow
\cdots
.
$$
We'll assume the inclusion maps are as nice as could be reasonably hoped for: they are all obtained by pushouts from closed cofibrations.  I'm even prepared to go so far as to say each inclusion is a relative CW complex.  Let's also assume  that if $\xi< \lambda$ is a limit ordinal, then $X_\xi = \mathrm{colim}\, \Phi|_\xi$. Write $Y = \mathrm{colim}\, \Phi$.
Now suppose we have a map $p: E\to Y$, and we hope to prove that it is a quasifibration.  If $\lambda = \omega$, then the diagram $\Phi$ can be taken to be $\mathbb{N}$-indexed, and there is a "classical" theorem with various technical conditions, whose heuristic import is that if all of the pullback maps $p_n : E_n \to X_n$ are quasifibrations, then so is $p$ (see, for example Theorem 2.7 in Peter May's paper "Weak equivalences and quasifibrations", available at https://www.math.uchicago.edu/~may/PAPERS/67.pdf).
Can this be extended to the more general ordinal-indexed case, possibly at the expense of imposing some additional conditions?
EDIT:  If it is as easy and technical as Chris Schommer-Pries suggests, then it would be really nice to have a reference to point to!
 A: It appears that the notion of a closed inclusion into a nice enough (meaning heriditarily normal) space is defined by a left lifting property with respect to a certain map of finite topological spaces (in the category of all topological spaces). Could this observation be helpful for your argument, i.e. take " the inclusion maps are as nice as could be reasonably hoped for" to mean this lifting property ? If I understand correctly, this observation in other words can be stated as saying that cofibrations between nice enough spaces are characterised by a left lifting property with respect to a certain map of finite topological spaces.
Actually, I am interested to know the right statement characterising cofibrations in this way if there is one.
This characterisation is discussed in the following question.
Closed embedding into a normal Hausdorff space and left lifting property
Namely, a map into a heriditarily normal Hausdorff space is a closed embedding iff it satisfies the left lifting property with respect to a certain map of finite topological spaces.
