Finite cw basepoints have explicit homotopy extension

Given $f:X\to Z$ and a path $h(v,t)$ starting at f(v), define the homotopy on the finite zero-skeleton by h on v and by the constant f-value at every other vertex. Each summand is a point, so the finite disjoint-union topology makes it continuous and it extends the required vertex data.

At an attached m-cell pull back f along its characteristic map on $D^m\times \{0\}$ and pull back the already defined homotopy along its attaching map on $S^{m-1}\times I$. The two agree at time zero. Compose this pasted map with the explicit retraction $R(x,t)=(\lambda x,2+\lambda (t-2))$, $\lambda =1\mathrm{/}\max (1-t\mathrm{/}2,\mathrm{\parallel }x\mathrm{\parallel })$ supplied and checked in F2. It extends the required data over the whole disk cylinder. Quotient times I is quotient as in F2, so the extension descends with the existing skeleton homotopy.

Process the finitely many cells in nondecreasing dimension, applying step 2.1 at each attachment. This yields a continuous homotopy on X with initial value f and the prescribed path at v, proving HEP. A zero-dimensional complex is already treated by step 1.1. Only finitely many cells and the displayed extension formula are used; no choice of infinitely many extensions or arbitrary-CW weak-topology argument occurs.