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E56

[0JK] Difficulty:*.Suppose \(E⊆ X\) is disconnected, can we assume that

\begin{equation} ∃ A,B∈ 𝜏, E{\mathbf{i}}A ~ ∧~ E{\mathbf{i}}B~ ∧~ E⊆ A∪ B~ ∧~ A{\mathbf{d}}B~ .\label{eq:E_ sconnesso_ disgiunti} \end{equation}

that is, that there exist two disjoint open sets, each of which intersects \(E\), and that \(E\) is covered by their union?

Solution 1