Given a topological space $X$ the cone on $X$ (sometimes denoted by $CX$ is the quotient space $X\cross [0,1]/X\cross\set{0}.$ Note that there is a natural inclusion $X\hookrightarrow CX$ which sends $x$ to $(x,1).$ If $(X,x_0)$ is a based topological space, there is a similar reduced cone construction, given by $X\cross [0,1] / (X\cross\set{0})\cup(\set{x_0}\cross[0,1]).$ With this definition, the natural inclusion $x\mapsto (x,1)$ becomes a based map, where we take $(x_0,0)$ to be the basepoint of the reduced cone.