# Is this set closed? The set is in $\mathbb{R}^2$

If I have $\{ (x, 1/x) \in \mathbb{R}^n : 0 < x \leq 1 \}$, is this set closed?

I know that almost every point is a limit point (I drew the graph in the first quadrant), but should I test whether 0 has a neighbourhood that contains other points in the set? Or is it okay I can forget about it since it isn't even in the set anyways?

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Yes, since $\{ (x, 1/x) \in \mathbb{R}^n : 0 < x \leq 1 \} = ([0,1]\times\mathbb{R}) \cap \{(x,y) | x y = 1 \}$, and both of the sets on the right hand side are clearly closed (the latter set being $\phi^{-1}\{1\}$, where $\phi(x,y) = xy$).