Exercise · ZW-TOPOLOGY-008 · ★★☆ Intermediate
How many open sets does the smallest topology on \(\{1,2,3\}\) containing \(\{1,2\}\) and \(\{2,3\}\) have?
Choose one option.
Show answer
- B. \(5\)
Explanation
We must add \(\varnothing\), \(X\) and the intersection \(\{2\}\): the open sets are \(\varnothing,\{2\},\{1,2\},\{2,3\},X\). Unions give nothing new.
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\?
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- 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-009 Which of the following families of subsets of ℝ is a topology?