Suppose I have some Metric $(E,D)$ and some point $P_0 \in E$. Suppose this point has the property $d(P_0 , P) > \epsilon$ for all points $P \in E , P \neq P_0$ and for some $\epsilon > 0 $. Now consider the set $ S = \{ P \in E , d(P_0 , P) < \epsilon \} $. This set will be the the set will be the open set containing the single point $P_0$. However if we look at the set $ S = \{ P \in E , d(P_0 , P) \leq \epsilon \} $, this will be the same set containing only $P_0$ and by definition a closed set. Does this mean we can classify this set as both a closed and an open set? I am slightly confused on how a set can be both open and closed and was not able to find too many examples outside the universal set and the empty set, and would greatly appreciate any help.
2026-04-12 11:33:06.1775993586
Understanding clopen sets
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Yes, $\{P\}$ is indeed a clopen set (all finite sets are closed in any metric space, and it is an "open ball", hence open (also in all metric spaces).
It's perfectly fine for a set to be open and closed ($E$ and $\emptyset$ also are) it's just the names that confuse you. If we'd called open sets "young sets" instead (OK, silly) and closed sets "beautiful sets", would your confusion be the same?
In $[0,1] \cup \{2\}$ we have an example, and in a discrete metric (so $d(x,y)=1$ except when $x=y$) all subsets are clopen.