To explain sets with this property, James Munkres argued “topological sets are not like a door.” For 10 points each:
[10m] Identify this property exhibited by the empty set and the entire set in any topology. In the discrete topology, every set has this property.
ANSWER: clopen [accept closed-open set accept any description mentioning the set is both closed and open; prompt on closed or open alone]
[10e] With the union and intersection, the clopen subsets of a topological space form an algebraic structure named for this person. Logical operators like “AND” and “OR” are used in a branch of algebra named for this English mathematician.
ANSWER: George Boole [accept Boolean algebras; accept Boolean lattices]
[10h] A representation theorem named for this mathematician states that every Boolean algebra is isomorphic to the set of clopen subsets of some topological space. With a Czech mathematician, this man names a method for generating the largest compact space containing a given topological space.
ANSWER: Marshall Harvey Stone (The results mentioned are the Stone representation theorem of Boolean algebras and the Stone-Čech compactification.)
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