Let \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},X\}\) on \(X=\{1,2,3\}\). How many elements does the smallest basis of \(\mathcal{T}\) have?

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Review the concept: Basis of a 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 \(\mathcal{B}\).

More practice problems: basis of a topology

• ZW-BASIS-OF-A-TOPOLOGY-001 Let X=\1,2,3\ have the discrete topology. Which of these families is a basis?
• ZW-BASIS-OF-A-TOPOLOGY-002 In ℝ with the usual topology, which of these families is a basis?
• ZW-BASIS-OF-A-TOPOLOGY-004 The family B=\\1\,\2,3\\ is a basis for a topology on X=\1,2,3\. How many open sets does that topol…
• ZW-BASIS-OF-A-TOPOLOGY-005 Which of these families of subsets of X=\1,2,3\ is NOT a basis for any topology on X?
• ZW-BASIS-OF-A-TOPOLOGY-006 If B is a basis of the topology T, which statement is true?
• ZW-BASIS-OF-A-TOPOLOGY-007 In ℝ with the usual topology, let B=\(p,q):p<q\ rational\. Which statement is correct?
• ZW-BASIS-OF-A-TOPOLOGY-008 Let B=\\1\,\2\,\3\,\1,2\\ on X=\1,2,3\. Which topology does it generate?
• ZW-BASIS-OF-A-TOPOLOGY-009 Let B be a basis of T, UinT and x∈ U. Why is there BinB with x∈ B⊆ U?

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