How to Prove that these Spaces are not Homotopically Equivalent

Question

Let $X=\{0\}\cup\{1/n:n\in\mathbb{N}\}$ and $Y$ be any countable discrete space. Show that $X$ and $Y$ are not homotopically equivalent.

The hint says: Every continuous map from $X$ to $Y$ takes all but finitely many points of $X$ to a common point of $Y$. I have proved this, but I cannot proceed any further using this.

Thanks for any help.

Answer 1

Hint. Suppose there is an homotopy $H \colon Y \times [0,1] \to Y$ such that $H(\cdot, 0) = \mathrm{id}_Y$. For $y\in Y$, $H(y,\cdot)$ is a path in $Y$ with starting point $y$ : what are then the allowed values for the ending point (that is $H(y,1)$) ?