Exercise · ZW-BASIS-OF-A-TOPOLOGY-005 · ★★☆ Intermediate
Which of these families of subsets of \(X=\{1,2,3\}\) is NOT a basis for any topology on \(X\)?
Choose one option.
Show answer
- A. \(\{\{1,2\},\{2,3\}\}\)
Explanation
To be a basis, the intersection of two members must be a union of members. Here \(\{1,2\}\cap\{2,3\}=\{2\}\) is not. The other three cover \(X\) and satisfy that condition.
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-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?