Hatcher Algebraic Topology 0.28

Question

Exercise 0.28 in Hatcher's Algebraic Topology states

Show that if $(X_1,A)$ satisfies the homotopy extension property, then so does every pair $(X_0 \sqcup_f X_1, X_0)$ obtained by attaching $X_1$ to a space $X_0$ via a map $f: A \to X_0$.

This question strikes me as trivial, so I wanted to check whether my solution doesn't gloss over any important subtleties.

The homotopy extension property guarantees that there is a retract from $X_1 \times I$ to $X_1 \times \{0\} \cup A \times I$. This induces a retract from $(X_0 \sqcup_f X_1)\times I$ to $(X_0 \sqcup_f X_1)\times\{0\} \cup (X_0 \sqcup_f A) \times I$, since $A \times I \subset X_1 \times I$ is unaffected by the retract and hence the space $X_0 \sqcup_f A$ just 'spectates'. This retract implies the pair $(X_0 \sqcup_f X_1, X_0)$ satisfies the homotopy extension property.

Is this proof satisfactory?

Answer 1

I would say that you have given a satisfactory intuition, but you have not given a satisfactory proof.

A satisfactory proof would be to start with what is given, namely the retract $$r : X_1 \times I \to X_1 \times \{0\} \cup A \times I $$ next use $r$ to write down an actual formula, expressed in terms of the given information, for a retract $$r' : (X_0 \sqcup_f X_1) \times I \to (X_0 \sqcup_f X_1) \times \{0\} \cup (X_0 \sqcup_f A) \times I $$ and finally cite appropriate theorems as needed to guarantee continuity of $r'$.

The one possible source of subtlety is in that final citation: since there are quotient topologies floating around in the domain and range of $r'$, presumably you'll be citing the universal properties of quotient maps in justifying continuity of $r'$.

And, presumably, this will still be very easy.

But I feel that one calls quotient topology issues "trivial" at their peril.