# Does any CW complex paired with a contractible space satisfy the homotopy extension property?

Does any CW complex paired with a contractible space satisfy the homotopy extension property? I am asking this question because I want to answer the question given here and I was told that it is just proposition 0.17 in hatcher, but the statement said that the pair must satisfy HEP

• A pair $(X,A)$ where $A$ is a sub-CW-complex of $X$ (more generally, if $(X,A)$ is a relative CW-complex) always has the HEP Dec 12 '19 at 7:25

## 1 Answer

No, take a circle relative the complement of one point. You can embed the circle into $$\mathbb{R}$$ at time 0, and then homotope the complement into an open line segment. This cannot be extended to a homotopy of the circle since you cannot even extend the function at time 1 to the whole circle.

One requirement of the HEP is to be closed.