Cofibration and homotopy extension property

A continuous map $i:A\to X$ has the homotopy extension property (HEP), or is an unbased cofibration, if for every target $Z$, continuous $f:X\to Z$, and continuous $h:A\times I\to Z$ satisfying $h(a,0)=f(i(a))$, there is a continuous $H:X\times I\to Z$ with $H(x,0)=f(x)$ and $H(i(a),t)=h(a,t)$. No uniqueness is required. Homotopy and relative homotopy have the meaning in Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints.

For based spaces and a based map i, based HEP imposes the same condition for based f and h with $h(a_0,t)=z_0$, and requires $H(x_0,t)=z_0$. A based space is well-pointed when $\{x_0\}\hookrightarrow X$ is an unbased cofibration. Based HEP and unbased HEP are distinct quantified properties; neither is silently substituted for the other. The categorical constructions on this page use CGWH spaces and k-products; products with I have their ordinary topology. The unbased test above can also be used for arbitrary topological spaces.