Call a pair of topological spaces $(X,A)$ a good pair if $A\subseteq X$ is a closed subspace and there exists an open neighborhood $A\subseteq U \subseteq X$ such that $A$ is a deformation retract of $U$.
I have seen two definitions of the notion of a well-pointed topological space:
- It is a pointed topological space $(X,\ast)$ which is a good pair.
- It is a pointed topological space $(X,\ast)$ such that the inclusion $\{\ast\}\hookrightarrow X$ is a closed Hurewicz cofibration.
Are these two definitions equivalent?
From a homotopy theory standpoint the second definition is the right one (being well-pointed in the Strom-Hurewicz model structure on $\mathsf{Top}$).
I would say the notions are not equivalent, if you allow nasty topological spaces. I think the following is a counterexample.
Consider the Sierpinski space $S=\{\circ,\bullet\}$, where $\circ$ denotes the open point. $(S,\bullet)$ is a good pair, since the inclusion of the closed point $\{\bullet\} \rightarrow \{\circ,\bullet\}$ is a deformation retraction, since $S$ is pathconnected.
This inclusion is not a closed Hurewicz fibration however, since this is equivalent to being an NDR-pair (in the sense of Def. 3 in this question), which is contradicted by the fact that there are no non-constant maps $u:S\rightarrow[0,1]$. (I think the conditions on $u$ force $u=\text{const}_0$, so the conditions on $H$ force $H=\text{const}_\bullet$, which is a contradiction)