How many distinct topologies exist on the set \(\{1,2\}\)?

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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.

More practice problems: topology

• ZW-TOPOLOGY-001 In X=\a,b,c\, which of the following families is a topology?
• 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?
• ZW-TOPOLOGY-009 Which of the following families of subsets of ℝ is a topology?

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