Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$.
My try (edited using comments below): The retraction I use is the map $$r:C(X)-P\longrightarrow X\times \{1\}; \;\;[(x,t)]\mapsto (x,1)$$ Now I want to construct a homotopy $$H_t:C(X)-P\rightarrow C(X)-P$$ such that $H_0=id_{C(X)-P}$ and $H_1=i\circ r$ where $$i:X\times \{1\} \rightarrow C(X)-P;\;\;(x,1)\mapsto (x,1)$$ is the inclusion map. The homotopy is given by $$H_t: C(X)-P\longrightarrow C(X)-P;\;\; [x,s]\mapsto [x,(1-t)s+t]$$