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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!

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    $\begingroup$ Doesn't the usual proof work? Any compact subspace of $E$ lies in some $E_\xi$ where $\xi < \lambda$ (but not necessarily a limit ordinal). So $\pi_i(E, p^{-1}(y_0))$ is the direct limit of the $\pi_i( E_\xi, p^{-1}(y_0))$, and so $\pi_i (E, p^{-1}(y_0)) \to \pi_i(Y, b)$ is an isomorphism if each map $\pi_i(E_\xi, p^{-1}(y_0)) \to \pi_i (X_\xi, y_0)$ is an isomorphism. This is the same argument as given in Hatcher Lemma 4K.3. $\endgroup$
    – Chris Schommer-Pries
    Commented May 24, 2021 at 15:48
  • $\begingroup$ @ChrisSchommer-Pries I'll check it out! Thanks for the pointer to Hatcher. $\endgroup$
    – Jeff Strom
    Commented May 24, 2021 at 16:42

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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.

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