Exercise · ZW-TOPOLOGY-007 · ★★☆ Intermediate
If \(\mathcal{T}_1\) and \(\mathcal{T}_2\) are topologies on \(X\), which of these families is always a topology?
Choose one option.
Show answer
- D. \(\mathcal{T}_1\cap\mathcal{T}_2\)
Explanation
\(\varnothing\) and \(X\) lie in both, and unions and intersections of common sets remain common. The union fails (previous exercise) and both differences lose \(\varnothing\) and \(X\).
Review the concept: Topology
A topology on a set \(X\) is a family \(\mathcal{T}\) of subsets of \(X\), whose members \(U\) are called open sets, containing \(\varnothing\) and \(X\) and closed under arbitrary unions and finite intersections.
\[\varnothing,X\in\mathcal{T};\quad U_i\in\mathcal{T}\Rightarrow\bigcup_i U_i\in\mathcal{T};\quad U_1\cap U_2\in\mathcal{T}\]
More practice problems: topology
- ZW-TOPOLOGY-001 In X=\a,b,c\, which of the following families is a topology?
- ZW-TOPOLOGY-002 How many distinct topologies exist on the set \1,2\?
- ZW-TOPOLOGY-003 Each interval (-1/n,1/n) is open in ℝ. What is the intersection of all of them, n≥1?
- ZW-TOPOLOGY-004 On ℕ the open sets are ∅ and the sets with finite complement. Which set is open?
- ZW-TOPOLOGY-005 The family of all subsets of X=\1,2,3,4\ is a topology. How many open sets does it have?
- ZW-TOPOLOGY-006 In X=\a,b,c\ let T_1=\∅,X,\a\\ and T_2=\∅,X,\b\\. The union T_1cupT_2 is not a topology. Which set…
- ZW-TOPOLOGY-008 How many open sets does the smallest topology on \1,2,3\ containing \1,2\ and \2,3\ have?
- ZW-TOPOLOGY-009 Which of the following families of subsets of ℝ is a topology?