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In a euclidean space $\mathbb{R}^k$, is the set consisting of a single point open, closed, neither, or both?

I would say that a set $E$ consisting of a single point $p$ doesn't have any limit points, so $E$ contains all of its limit points and is therefore closed. But it might be open, too, since a ball of radius zero around $p$ is a subset of $E$. When using balls to define interior points, do balls have to have radius greater than zero?

aparkerlue
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2 Answers2

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One point sets are closed in $\mathbb{R}^n$. The only closed and open sets are $\emptyset,\mathbb{R}^n$.

Batman
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    why is it not vacously open? – Charlie Parker Jun 11 '18 at 00:10
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    @CharlieParker - You can clearly see its closed, as $\cap_{n \geq 1} [x-1/n, x+1/n] = {x}$ is a countable intersection of closed sets. As for it not also being open, note that its complement is not closed -- $x$ is a limit point. Alternatively, if you know about connectedness, it is also not hard to prove that a topological space is connected if and only if the only sets which are both closed and open are the empty set and whole set: https://proofwiki.org/wiki/Connected_iff_no_Proper_Clopen_Sets. – Batman Jun 11 '18 at 01:49
  • I confused closed and open for a second. I think I see why its clearly closed (vacuously?). Since a single point doesn't have a limit point of course, then its closed. Sorry! – Charlie Parker Jun 14 '18 at 01:01
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    In any T1 space, one point sets are closed. Metric implies T1. – Batman Jun 15 '18 at 03:01
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Balls must have a radius greater than zero or else the definition would not be very useful, since that would mean any point is an interior point. Any neighborhood of radius greater than zero would include more than one point, so it cannot be a subset of E. Hence E cannot not be open.

JRB
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