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 one of these algebraic structures named for an English mathematician. Operations on these structures include “AND” and “OR.”
ANSWER: 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 H(arvey) Stone [accept Stone representation theorem; accept Stone-Čech compactification]
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