I have a quick question about the difference between the two concepts in the title. The question is basically ex.6 (b) in Hatcher's book titled "Algebraic Topology". Let $X$ be the subspace of $R^2$ consisting of the horizontal segment $[0,1] \times \{0\}$ together with the vertical segments $\{r\} \times [0,1-r]$ for $r$ a rational number in $[0,1]$. Now let $Y$ be the space that is the union of an infinite number of copies of $X$ arranged in a zig zag formation. See below -
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Now my question is why can't one deformation retract $Y$ to a point in the darkened zig zag line? Surely the darkened zig zag line is homeomorphic to $\mathbb{R}$, which is deformation retractable to a point, and each of the vertical lines of each copy of $X$ deformation retracts to its segment of the zig zag line! I must be missing something here as one has to prove that $Y$ does not deformation retract to any point!