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Which rules must a collection of open sets satisfy to define a topology?
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topology
Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.
To define a topology on a set X, a collection of subsets T (called open sets) must satisfy the following three axioms:
- Existence of extremes: Both the set X itself and the empty set must be members of the collection T.12
- Finite intersections: The intersection of any finite number of sets in T must also be a member of T.132
- Arbitrary unions: The union of any collection of sets in T—whether finite or infinite—must be a member of T.132
When these conditions are met, the set X is referred to as a topological space.