TITLE: the cone is contractible
QUESTION [23 upvotes]: Let $X$ be a topological space. I want to show that the cone $CX$ is contractible. Here we construct a deformation retraction from $CX$ to 
the tip point of the cone 
$$H_t: CX\to CX;\; (x,t')\mapsto (x,t'(1-t))$$
is this correct?

REPLY [2 votes]: First define the auxiliary function $f:I\times I\longrightarrow I$ given by $$f(s, t)=(1-s)t+s.$$ Define $H:C(X)\times I\longrightarrow C(X)$ setting $$H([x, t], s)=[x, f(s, t)].$$ It is easy to check $H$ is homotopy from $I_{C(X)}$ to a constant.
Obs: You must take care with the function $f$ to be chosen. You must assure $f$ assumes its values on $I$. For instance, $f(s, t)=2t-s$ wouldn't work because, for example, $f(0, 1)=2$ and there is no such $[(x, 2)]$ on $C(X)$.