Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), let $A \subset X$, and endow $A$ with the [subspace topology](/page/Subspace%20Topology)
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\begin{align*}
\tau_A := \{A \cap O : O \in \tau\}.
\end{align*}
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Then the topological space $(A,\tau_A)$ is disconnected if and only if there exist subsets $U,V \subset A$ such that $U \neq \varnothing$, $V \neq \varnothing$, $U,V \in \tau_A$, $U \cap V = \varnothing$, and $U \cup V = A$.