the homeomorphisms betwen two spaces looks like broom

Question

Let $Y=${$(x,x/n)\in \mathbb{R} \times \mathbb{R}: x\in [0,1],n \in \mathbb{N}$} and
$X=\cup_{n\in \mathbb{N}}{[0,1]\times (n)}$

and $(0,n) R (0,m),\forall n,m \in \mathbb{N}$.
Then does $X/R $ is homeomorphic to $Y$?

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The figure of both the spaces are looks like broom and looks like they are homeomorphic but I cant prove or disprove it.
Can I get some help?

Answer 1

Suppose I gave you a point $[(x,n)]_R$ in $X/R$. What do you think would be an appropriate point to map this element to in $Y$? Hint. Consider the cases $x=0$ and $x\neq 0$ separately.

You want this map to be bijective and continuous. You can then use the fact that $X/R$ is compact and $Y$ is Hausdorff to conclude that the map is a homeomorphism.