Created on 2024-06-20Asked by Scarlett Torres (Solvelet student)
Determine if the set X={a,b,c} with the collection of subsets τ={∅,{a},{a,b},X} forms a topology.
Solution
To determine if the set X={a,b,c} with the collection of subsets τ={∅,{a},{a,b},X} forms a topology: 1. **Check the conditions for a topology:** - The empty set and the set X are in τ. - The union of any collection of sets in τ is also in τ. - The intersection of any finite collection of sets in τ is also in τ. 2. **Verify the conditions:** - ∅,X∈τ. - Unions: {a}∪{a,b}={a,b},{a,b}∪X=X,{a}∪{a,b}∪X=X. - Intersections: {a}∩{a,b}={a},{a,b}∩X={a,b},{a}∩{a,b}∩X={a}. 3. **Result:** The set X={a,b,c} with the collection τ={∅,{a},{a,b},X} forms a topology. Solved on Solvelet with Basic AI Model
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DefinitionIt is a set together with a collection of open sets (set up in such a way that the collection satifies properties that ensure that it is a topology: closed under finite intersection and union) Ex: (R, standard topology of open intervals) is a topological space.
