I have to check if the set of points $p_n = (\frac{1}{n},\frac{1}{n})$ is closed in the metric space $(\mathbb{R}^2, d_R)$, where $d_R$ is the river metric.

Intuition tells me that this is an open set, but I don't know how to show that.

Edit: the river metric is defined as follows:

if $x_1 = x_2$

$d_R((x_1,y_1),(x_2,y_2)) = |y_1 - y_2|$

else $d_R((x_1,y_1),(x_2,y_2)) = |x_1 - x_2| + |y_1| + |y_2|$

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    $\begingroup$ What is your definition of the river metric? $\endgroup$ – upanddownintegrate Jan 12 at 18:04
  • $\begingroup$ Note that $d_R(p_n,(0,0))=\frac{1}{n}+0+\left|\frac{1}{n}-0\right|\to0$ as $n\to\infty$. Therefore, $p_n\to (0,0)$ as $n\to\infty$, while $(0,0)$ is not a point of the form $\left(\frac{1}{n},\frac{1}{n}\right)$. $\endgroup$ – MoonLightSyzygy Jan 12 at 18:11
  • $\begingroup$ I've edited the question to add my definition of the river metric $\endgroup$ – schrader2 Jan 12 at 18:52
  • $\begingroup$ Is is equivalent to the standard metric? $\endgroup$ – orientablesurface Jan 12 at 18:54
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    $\begingroup$ @topologicalmagician No, for example the sequence $q_n=(1/n,1)$ is not Cauchy for $d_R$, since $d_R((1/n,1),(1/m,1))=\left|\frac{1}{n}-\frac{1}{m}\right|+2\geq2$, while it is Cauchy for $d((1/n,1),(1/m,1))=\sqrt{(1/n-1/m)^2+(1-1)^2}$. $\endgroup$ – MoonLightSyzygy Jan 12 at 19:02

The set (a sequence really) has $(0,0)$ as a limit point, so is not closed. And it has no interior points, as open balls are uncountable.

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