Questions about Hurewicz model structure

Question

How to see every object in the Hurewicz (Strom's) model structure is every object is both a fibrant object and a cofibrant object? Is there a reference for the proof. In particular, i wonder why the statement is not true in the model category but true in the Hurewicz model structure?

Answer 1

The Hurewicz model structure has cofibrations as the maps with the homotopy extension property with respect to all spaces and the fibrations the Hurewicz fibrations.

All you have to do is check that the empty set inclusion has this extension property, which it obviously does, and that we can lift homotopies over a point, provided we are given an initial lift. The lift can be taken to be the constant homotopy from the initial lift to itself.