Prove that $\mathbb{N}$ is not homotopic to $X=\{\frac{1}{n}|n\in \mathbb{N}\}\cup \{0\}$.

Question

I have the following two subsets of Real Numbers.

$X=\{\frac{1}{n}|n\in \mathbb{N}\}\cup \{0\}$

$Y=\mathbb{N}$

I want to prove that $\mathbb{N}$ is not homotopic to $X=\{\frac{1}{n}|n\in \mathbb{N}\}\cup \{0\}$.

EDITS

We say two spaces $X$ and $Y$ are homotopic if there are continuous maps $f:X\rightarrow Y$ and $g: Y\rightarrow X$ such that $f\circ g$ is homotopic to $Id_{Y}$ and $g\circ f$ is homotopic to $Id_{X}$

I am reading from Hatcher's book.

Answer 1

Hint 1: If a function $h:\Bbb N\to \Bbb N$ is a function which is homotopic to the identity, what can you say about $h$? For instance, what could $h(1)$ possibly be?

Hint 2: What do you know about the image of continuous maps where the domain is compact?