$\{2\}$ is both open and closed in $\{1,2,3\}$, equipped with the Euclidean metric

Question

I am trying to prove that $\{2\}$ is both open and closed in $\{1,2,3\}$, equipped with the Euclidean metric. Is the following correct?

Openness: $B_{0.5}(2)\subset\{2\}$. Hence, $\{2\}$ is open in $\{1,2,3\}$.

Closedness: $\{2\}$ trivially contains all its limit points, since it has no limit points. Hence, $\{2\}$ is closed in $\{1,2,3\}$.

Thank you.

Yes. Alternatively, you could argue that on a finite set, any metric induces the discrete topology. — Anne Bauval, Feb 22 '23 at 07:55
If I was your grader, I might ask you to prove that ${2}$ has no limit points. Though it'd be easy to do based on what you've already written. — AJY, Feb 22 '23 at 07:58

score 4 · Accepted Answer · edited Feb 22 '23 at 08:28

It's correct, and the answer is OK so long as you have already proven that $\{2\}$ has no limit points. This is simple to prove, of course, but it still should be done.

Alternatively, you could also say that because $\{1\}$ is also open (by the same argument as $\{2\}$ and $\{3\}$ is also open, then the complement of $\{2\}$, which is $\{1,3\}=\{1\}\cup\{3\}$, is open, which means $\{2\}$ is closed.

$\{2\}$ is both open and closed in $\{1,2,3\}$, equipped with the Euclidean metric

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