TITLE: Proving another space is contractible using the homotopy extension property
QUESTION [1 upvotes]: Question 4 in the first exercise section in Spanier's, Algebraic Topology is stated as follows: 
"Prove that a space $Y$  is contractible if and only if, given a pair ($X$,$A$) having the homotopy extension property with respect to $Y$, any map $A\rightarrow Y$ can be extended over $X$."
I figured out how to go from the first statement to the second but not the other way. Any hints on how to start the other implication?

REPLY [1 votes]: Hint: Try taking $(X,A)=(Y\times [0,1], Y\times \{0,1\})$.
{"set_name": "stack_exchange", "score": 1, "question_id": 1590209}