Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

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 K := [0, 1]\ S k≥0,m≥1 3k+1 3k+2 , 3m 3m = {x ∈ R : x = P∞ i=1 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 1