Exercise · ZW-TOPOLOGY-009 · ★★★ Advanced
Which of the following families of subsets of \(\mathbb{R}\) is a topology?
Choose one option.
Show answer
- A. \(\varnothing\), \(\mathbb{R}\) and all rays \((a,\infty)\), \(a\in\mathbb{R}\)
Explanation
Two rays intersect in a ray, and a union of rays is a ray or \(\mathbb{R}\). In the other options an infinite union (the integers, \(\bigcup[n,n+1]\) or \(\bigcup[-n,n]\)) leaves the family.
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-007 If T_1 and T_2 are topologies on X, which of these families is always a topology?
- ZW-TOPOLOGY-008 How many open sets does the smallest topology on \1,2,3\ containing \1,2\ and \2,3\ have?