This question arises from problem 8 on pg. 366 of Munkres. Let $X$ be the union of the sets $(1/n) \times I$, $0 \times I$, and $I \times 0$, where $I = [0,1]$, with the topology it inherits from $\mathbb R^2$. I am trying to prove that the inclusion map from the singleton set $x_0 = (0,1)$ to $X$ is a homotopy equivalence, but that $x_0$ is not a deformation retract of $X$.
Intuitively, my idea is as follows: in order to obtain a homotopy from the constant map $x \mapsto x_0$ to the identity map on $X$, we must contract the entire space $X$ down to the horizontal axis, then slide it back up the vertical axis to $x_0$. But, we cannot do so while keeping $x_0$ fixed because (in order to preserve continuity) the horizontal axis must slide down along with the lines $(1/n) \times I$, which are arbitrarily close to it. However, I am having a hard time formalizing this notion. Any suggestions?