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The Cantor Discontinuum
The Middle Third Set or Ternary Set
by G. H. Meisters • January 17, 1997
Dfn #1. A subset P of a topological space X is called perfect if P = P 0 , where P 0
is the set of all limit points of P : That is, if P is closed (P 0 ⊂ P or P = P̄) & P ⊂ P 0 .
Dfn #2. A subset N of a topological space X is called nowhere dense if every
nonempty open set U ⊂ X contains a nonempty open set V such that V ∩ N = ∅.
Dfn #3. A subset D of a topological space is called a Cantor discontinuum if
D 6= 0, D is perfect, and D is nowhere dense.
1. The line R contains Cantor discontinuums. In particular, the Cantor Ternary Set
:= [0, 1]\
3k+1 3k+2
, 3m
= {x ∈ R : x =
xi /3i , xi = 0 or 2 } is one.
2. Every nonempty perfect set in R has the same cardinality as R. In particular, if D
is a Cantor discontinuum and if D ⊂ R, then card D = card R = 2ℵ0 .
3. Cantor’s Middle Third Set K ⊂ [0, 1] ⊂ R has Lebesgue measure zero.
4. There exist Cantor discontinuums in R of positive Lebesgue measure. In fact, for
every 0 ≤ < 1, there exist Cantor discontinuums D ⊂ [0, 1] of Lebesgue measure .
5. A closed subset N of R is nowhere dense iff it contains no open interval.
6. A nonempty bounded closed subset F of R is either a closed interval or is obtained
from some closed interval by removing a countable family of pairwise disjoint open
intervals whose end points belong to the set F .
7. Every nonempty bounded perfect set P ⊂ R is either a closed interval or is a closed
interval from which has been removed a countable number of mutually disjoint open
intervals which have no end points in common with each other or with the original
closed interval. Conversely, every such subset of R is perfect.
c G. H. Meisters January 1997