We consider the category $\mathsf{CG}$ of compactly generated spaces (since the question becomes obsolete for compactly generated weakly hausdorff spaces). A Hurewicz-fibration is a map with the homotopy lifting property. A Hurewicz-cofibration is a map with the homotopy extension property. We will call a map with the LLP against trivial Hurewicz-fibrations a Strom-Hurewicz-cofibration, since they are the cofibrations in the Strom model structure.
Is there an immediate argument, why Strom-Hurewicz-cofibrations are closed maps? Or even better, why they are precisely the closed Hurewicz cofibrations?
Of cause we can invoke the recognition principle, saying that Strom-Hurewicz-cofibrations are precisely the NDR-pairs, but this result requires quite some work. I was wondering, whether there is a more direct argument (at least for the "necessarily closed" part) along the following lines:
An inclusion $f:A \rightarrow B$ is closed, if and only if the lifting problem $$\begin{array}{ccc} A & \xrightarrow{\Bbb 1_A} & \{0,1\}_S\\ {\small f}\downarrow && \downarrow\\ B & \xrightarrow{\Bbb 1_{f(A)}} & \{0,1\}_C \end{array}$$ admits a solution, where $\{0,1\}_S$ denotes the Sierpinski space with $\{1\}$ closed, $\{0,1\}_C$ is equipped with the chaotic/indiscrete topology, the horizontal maps are characteristic/indicating functions and the underlying function of the right map is the identity. It is a fun exercise to show that $\{0,1\}_S$ and $\{0,1\}_C$ are contractible and that the right map constitutes a homotopy equivalence. But unfortunately it is not a Hurewicz-fibration, since lifting properties of the form $$\begin{array}{ccc} * & \rightarrow & \{0,1\}_S\\ {\small in_0}\downarrow\,\; && \downarrow\\ [0,1] & \rightarrow & \{0,1\}_C \end{array}$$ need not admit a solution. Moreover I don't know (or rather missed to check) wether the spaces $\{0,1\}_S$ and $\{0,1\}_C$ are actually compactly generated. Anyway I think this would have been a very neat argument to see the necessity of Strom-Hurewicz-cofibrations being closed. In other words
Can we use a lifting problem against a trivial Hurewicz-cofibration to show that maps with LLP against trivial Hurewicz-fibrations are closed?
Thank you for your time and considerations!