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Characterization of Cantor Spaces

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Characterization of Cantor Spaces Characterization of Cantor Spaces Matthew Shaw (ref. Pugh Analysis Textbook) November 2019 1 Introduction The standard Cantor Middle Thirds Set is compact, perfect, nonempty, and to- tally disconnected (and since the construction takes place in R with its standard metric, the Cantor Set is also metrizable). Totally Disconnected: Every connected component is a singleton set, and equiv- alently, every point has arbitrarily small clopen neighborhoods. Perfect: A space is called perfect if it has no isolated points. There are two main theorems of this presentation: The Cantor Surjection The- orem, that every nonempty compact metric space is the image of a continuous function with the Cantor set as its domain, and a complete characterization of Cantor spaces, in particular that every compact, nonempty, perfect, and to- tally disconnected metric space is homeomorphic to the standard Cantor middle thirds set. Notation: The set of words of countably infinite length in a two character al- phabet is denoted by Ω, the standard Cantor Middle-Thirds set is denoted by C, and the set of words of length n (where n 2 N) in a two character alphabet is denoted !(n). Note that the size of !(n) is always 2n. ! with no associated number will represent a word of countably infinite length (i.e. an element of Ω) and the notation !jn for n 2 N means the word ! truncated to only its first n characters (this is an element of !(n)). t denotes disjoint union of sets. If α and β are words of finite length, then the notation αβ means the word made up of the characters of α followed by the characters of β, which is also termed concatenation. 2 Construction and Size We can use an address system to locate and name points in the Cantor set. At each stage, use 0 to denote the left interval and 2 to denote the right interval. 0 1 for example, at the first stage of construction, we use C = [0; 3 ] to denote the 2 2 left interval and C = [ 3 ; 1] to denote the right interval. Inside each interval we use the addressing the same way, so at stage 2 we inherit the first character of the name from whichever interval we use in stage 1, for example the furthest left 00 1 interval at stage 2 is denoted C = [0; 9 ], the first 0 in the name coming from 1 it being a subset of the left interval in stage 1 and the second 0 coming from it being the left interval of that set in the next stage of construction. Then at !1!2:::;!n some stage n of the construction, Cn is the disjoint union of the sets C , with each !i 2 f0; 2g. We can extend from finite cases to infinite address strings !1;!2;::: , which corresponds to the intersection of nested intervals C!1 ⊇ C!1!2 ⊇ C!1!2!3 ⊇ ::: This is an intersection of nested compact sets, so it is nonempty. Since the diameter of the sets tends to zero, the intersection contains exactly one point p(!), where ! = !1;!2;::: denotes the infinite word in the alphabet f0; 2g. In particular, \ fp(!)g = C!jn n2N Note that if two points p and p0 have different addresses ! and !0, then they are different points in the Cantor set because at some digit their addresses disagree, and since the subintervals at that stage are disjoint, they are contained in different disjoint sets and so are different points. Noting also that each point in the Cantor set has an address, we have a bijective correspondence between Ω, the set of words of countably infinite length in the alphabet f0; 2g, and the Cantor set, so the sets have the same cardinality. Clearly jΩj = jP (N)j, so the Cantor set is uncountable in size (the construction of this bijection is the same as the construction of the Cantor set via ternary expansion without 1). 3 Topological Properties The Cantor set is perfect, compact, nonempty, and totally disconnected. Nonempty is obvious. Compact follows from Heine-Borel as it is a closed and bounded subset of R. To see why C is perfect and totally disconnected, first note that C contains every endpoint of any interval in any Cn stage of its construction, denote this set of endpoints by E. Now let x 2 C, let " > 0, and 1 n 1 let n 2 N such that 3n < ". Note that x is in one of the 2 intervals of size 3n of Cn, call this interval I. Note that E \ I ⊆ (x − "; x + ") contains infinitely many elements, since that interval is contained in Cn and so will have middle thirds removed infinitely many more times, so contributing infinitely many more endpoints inside the interval I. Thus no x is isolated and C is perfect. Note that I is closed in Cn and since Cn n I is made up of finitely many closed intervals, it is also closed in Cn, so both of these are clopen in Cn. Then I \ C and (Cn n I) \ C are both clopen in C with C \ I ⊆ (x − "; x + "), so every open neighborhood about any x 2 C contains a clopen neighborhood of x, and so C is totally disconnected. 2 4 Cantor Surjection Theorem The statement of the theorem is as follows: Given any nonempty compact met- ric space M, there exists a continuous surjection from C onto M. The proof is broken up into multiple lemmas. Definition: Given a compact metric space M, we call a compact nonempty sub- set of M a piece of M. Lemma 1: If M is a compact, nonempty metric space and " > 0, then M can be covered by finitely many pieces, each with diameter no more than ". " Proof: Make a covering of M by open balls with radius 2 . By compactness, take a finite subcover. Then the closure of each ball in the subcover constitutes a covering of M by finitely many pieces of diameter no more than ". This cov- ering is made up of pieces rather than just subsets as closed subsets of compact spaces are compact. Using that lemma, divide the space M into finitely many pieces of diameter 1 and denote this collection of pieces by Σ1. Since this is finite in size, choose n1 n1 2 N large enough that 2 ≥ jΣ1j. Let w1 : !(n1) ! Σ1 be any onto function (since j!(n1)j ≥ jΣ1j, such a function must exist). Then we say w1 labels Σ1 and for L 2 Σ1 if w1(α) = L, we call α the label of L. Then since each piece L is compact, we can repeat the process, breaking each into finitely many pieces 1 of diameter no more than 2 and for any L 2 Σ1, we call the collection of such pieces of L Σ2(L). Then the family [ Σ2 = Σ2(L) L2Σ1 is a covering of M by finitely many smaller pieces. Choose n2 2 N large enough n2 that 2 ≥ max(jΣ2(L)j : L 2 Σ1). Then label Σ2 with words αβ 2 !(n1 + n2) such that if L = w1(α) then αβ labels all the pieces S 2 Σ2(L) as β varies with α fixed. This is an onto labeling w2 : !(n1 + n2) ! Σ2 which is coherent with w1 in the sense that matching leading digits in a labeling means the points are close, i.e. contained in the same stage 1 piece. Then proceeding in this fashion inductively, we construct a sequence of divisions (Σk)k2N of M and surjections wk : !k(n1 + n2 + ··· + nk) ! Σk for k 2 N such that the following hold: 1 The maximum diameter of the pieces L 2 Σk tends to zero as k tends to infinity (specifically we can do the construction by making pieces of 1 diameter no more than n at stage n of the construction). 2 Σk+1 refines Σk in the sense that 8S 2 Σk+1; 9L 2 Σk such that S ⊆ L (i.e. subpieces aren't broken up across multiple pieces). 3 If L 2 Σk and Σk+1(L) is the set fS 2 Σk+1 : S ⊆ Lg then [ L ⊆ S S2Σk+1(L) 3 (i.e. each piece is entirely covered by its subpieces). 4 The labelings wk are coherent, i.e. if wk(α) = L 2 Σk, then the association β ! wk+1(αβ) labels every S 2 Σk+1(L) as β varies in !(nk+1) (with α fixed). Cantor Surjection Theorem: From here the surjection we wish to construct is the natural one given that we know how to label pieces of any nonempty compact space with words of count- ably infinite length in two letters and we already have a bijective correspondence between such words and the standard Cantor set. Formally, for any p = p(!) 2 C, use the labeling provided by \ p(!) = C!jn n2N (technically this isn't correct as the right side is a set and the left side is a point, but the set on the right is exactly the singleton set containing p, so this notation shouldn't be too confusing). Then construct the labeling of M with words in a two character alphabet as outlined previously. Then for p 2 C, consider the sequence of pieces of Lk(p) 2 Σk of M, where Lk(p) = wk(!j(n1 +n2 +···+nk)), which is the collection of pieces at stage k of the labeling of M, labeled by the first (n1 + n2 + ··· + nk) characters in the name associated with p.
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