I am a bit confused by the wording of this exercise
Let $X = \lbrace 0 \rbrace \cup \lbrace 1, 1/2, 1/3, \dots, 1/n, \dots \rbrace$ and Y be a countable discrete space. Show that $X,Y$ don't have the same homotopy type.
Isn't $X$ a countable discrete space? Also, the hint on this exercise says to use $X$'s compactness. How is $X$ compact? Doesn't the open cover of each singleton not admit a finite subcover?
Rotman does not give the topology of $X$. An educated guess is that $X$ is meant to have the subspace topology as a subset of $\mathbb{R}$. This is consistent with the compactness hint because any open subset containing $0$ contains all but finitely many other points.