# deformation retract should not imply strong deformation retract

Suppose $$Y$$ is a deformation retract of $$X$$. Then, there exists a homotopy $$H:X\times I\rightarrow X$$ such that $$H(x,0)=x$$, $$H(x,1)\in Y$$, and $$H(y,1)=y$$ for any $$y\in Y$$, $$x\in X$$. So $$h(x)=H(x,1)$$ is a retraction from $$X$$ to $$Y$$. Suppose $$i$$ is the inclusion from $$Y$$ to $$X$$. Then, $$h\circ i=Id_Y$$. Define $$F$$ from $$X\times I$$ to $$X$$ as $$F(x,t)=tx+(1-t)i\circ h(x)$$. So, $$i\circ h\simeq Id_X$$ and $$F(y,t)=y$$ for any $$y\in Y$$, which means that Y is a strong deformation retract.

May I ask what is the problem of this proof? If this works, deformation retract will imply strong deformation retract. So there must be something wrong with it.

You're assuming that $$X$$ is a space where $$+$$ makes sense, and where every point on the line segment from $$x$$ to $$h(x)$$ is contained in $$X$$.