Sketch. (i) \(\Leftrightarrow\) (ii) because the pairs
\begin{equation*}
(D^n \times I,\; D^n \times \{0\}) \quad\text{and}\quad \bigl(D^n \times I,\; (D^n \times \{0\}) \cup (S^{n-1} \times I)\bigr)
\end{equation*}
are homeomorphic: in both cases the subspace is an \(n\)-disk in the boundary sphere of the \((n+1)\)-disk \(D^n \times I\text{.}\) A homeomorphism of \(D^n \times I\) carrying one subspace onto the other turns each kind of lifting problem into the other.
(ii)
\(\Rightarrow\) (iii) by induction over the relative skeleta
\(X^{(k)} = A \cup X_k\text{.}\) A lift on
\((X \times \{0\}) \cup (X^{(k-1)} \times I)\) extends over each
\(e \times I\text{,}\) where
\(e\) is a relative
\(k\)-cell, by the HLP for
\((D^k, S^{k-1})\) applied to its characteristic map. This is the same argument as for the HEP in
Lemma 7.20,
Lemma 7.21,
Lemma 7.22.
(iii)
\(\Rightarrow\) (iv) by taking
\(A = \varnothing\text{,}\) and (iv)
\(\Rightarrow\) (i) since
\(D^n\) is a CW complex.
