Show that if a space $X$ deformation retracts to a point $x ∈ X$, then for each neighborhood $U$ of $x$ in $X$ $\exists$ a neighborhood $V ⊂ U$ of $x$ such that the inclusion map $V \rightarrow U$ is nullhomotopic.
Could anyone lay out the main arguments and connections needed for this? I'm not quite sure what to do.
This question has been asked before, see here.
However, I'm not understanding it, if it is even correct. It also refers to a tube lemma, which I would like to avoid for the time being.