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Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?

Definition: Suppose that $(X,d)$ is a metric space. A set $E \subseteq X$ is said to be disconnected if there exist two non-empty open sets $G_1$ and $G_2$ such that

$G_1 \cap E \ne \emptyset$, $G_2 \cap E \ne \emptyset$, $G_1 \cap G_2 \cap E = \emptyset$ and $E \subseteq G_1 \cup G_2$.


Or is a set of two elements $\{x,y\}$ connected in metric space $(X,d)$?

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A singleton $\{x\}$ is a connected set in any topological space whatsoever: it clearly cannot be written as the union of two disjoint non-empty sets of any kind, let alone relatively open ones.

A two-element set $\{x,y\}$ in a metric space $\langle X,d\rangle$ is not connected: if $\epsilon=\frac12d(x,y)>0$, you can set $G_1=B(x,\epsilon)$ and $G_2=B(y,\epsilon)$ in your definition to show that $\{x,y\}$ is not connected. (Here $B(x,\epsilon)$ is the open ball of radius $\epsilon$ centred at $x$.)

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