Exercise · ZW-BASIS-OF-A-TOPOLOGY-010 · ★★★ Advanced
Two different bases can generate the same topology . In \(X=\{1,2,3\}\), which pair illustrates this?
Choose one option.
A \(\{\{1\},\{2,3\}\}\) and \(\{\{1\},\{2\},\{3\}\}\) B \(\{\{1\},\{2\},\{3\}\}\) and \(\{\{1\},\{2\},\{3\},\{1,2\}\}\) C \(\{X\}\) and \(\{\{1\},\{2\},\{3\}\}\) D None: different bases always generate different topologies Show answer B. \(\{\{1\},\{2\},\{3\}\}\) and \(\{\{1\},\{2\},\{3\},\{1,2\}\}\)← Back to the question
Explanation The singletons and the singletons together with \(\{1,2\}\) both generate the discrete topology . With \(\{\{1\},\{2,3\}\}\) and the singletons, the first generates only 4 open sets and the second 8; with \(\{X\}\) and the singletons, the trivial versus the discrete topology.
A basis \(\mathcal{B}\) of a topology \(\mathcal{T}\) on \(X\) is a family of open sets such that every open set \(U\) of \(\mathcal{T}\) is a union of members of \(\mathcal{B}\).
\[\forall U\in\mathcal{T}:\ U=\bigcup\{B\in\mathcal{B}:B\subseteq U\}\]
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