Topology Atlas
Document # paca-24
The fourteen subsets problem: interiors, closures and complements
Henno Brandsma
a note in Topology Explained
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It is a famous and well-known question how many different subsets can be
formed from a given subset A, when we repeatedly apply the operations of
complementation, closure and interior to A, in any order we choose. The
answer turns out to be 14 at most, and there are spaces and subsets that
realise all 14 of them. In this note I will show all of this, by turning
the problem into an algebraic one, giving a set of rules that determine
all relations (inclusions) that are generally true (for all subsets and
all spaces), and deducing the result from that.
Date: December 22, 2003
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