Exercise · ZW-BASIS-OF-A-TOPOLOGY-008 · ★★☆ Intermediate
Let \(\mathcal{B}=\{\{1\},\{2\},\{3\},\{1,2\}\}\) on \(X=\{1,2,3\}\). Which topology does it generate?
Choose one option.
Show answer
- D. The discrete one
Explanation
\(\mathcal{B}\) covers \(X\) and satisfies the intersection condition (for example \(\{1,2\}\cap\{2\}=\{2\}\)), so it is a basis. The singletons already form any subset: it generates the discrete topology, with 8 open sets.
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 \(\mathcal{B}\).
\[\forall U\in\mathcal{T}:\ U=\bigcup\{B\in\mathcal{B}:B\subseteq U\}\]
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-003 Let T=\∅,\1\,\1,2\,X\ on X=\1,2,3\. How many elements does the smallest basis of T have?
- 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-009 Let B be a basis of T, UinT and x∈ U. Why is there BinB with x∈ B⊆ U?