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F-9 Topological Characterizations of Separable Metrizable Zero-Dimensional Spaces

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F-9 Topological Characterizations of Separable Metrizable Zero-Dimensional Spaces hart v.2003/06/18 Prn:20/06/2003; 9:30 F:hartf10.tex; VTEX/J p. 1 334 Section F: Special properties f-9 Topological Characterizations of Separable Metrizable Zero-Dimensional Spaces A space X is called zero-dimensional if it is nonempty theorems give us homogeneity for free. (This phenomenon is and has a base consisting of clopen sets, i.e., if for every not uncommon in topology.) point x ∈ X and for every neighbourhood U of x there The following example of a zero-dimensional compact exists a clopen subset C ⊆ X such that x ∈ C ⊆ U.It space is of particular interest. From I =[0, 1] remove the 1 2 is clear that a nonempty subspace of a zero-dimensional interval ( /3, /3), i.e., the ‘middle-third’ interval. From the space is again zero-dimensional and that products of zero- remaining two intervals, again remove their ‘middle-thirds’, dimensional spaces are zero-dimensional. and continue in this way infinitely often. What remains of I We say that a space X is totally disconnected if for all at the end of this process is called the Cantor middle-third distinct points x,y in X there exists a clopen set C in set, C. A space homeomorphic to C is called a Cantor set. ∈ ∈ X such that x C but y/C. It is clear that every zero- It is easy to see that C is the subspace of I consisting of dimensional space is totally disconnected. The question nat- all points that have a triadic expansion in which the digit 1 urally arises whether every totally disconnected space is does not occur, i.e., the set zero-dimensional. If this were true then checking whether a given space is zero-dimensional would be simpler. However, ∞ xi the answer to this question is in the negative, as was shown x = : xi ∈{0, 2} for every i . 3i by Sierpinski´ (cf. [KU, Chapter V, §46.VI, Footnote 2]). i=1 In part 1 of this note, we are interested in theorems that state nontrivial and useful topological characterizations of The Cantor set was introduced by Cantor. It is clearly closed zero-dimensional separable metrizable spaces. So it will be in I, hence is compact. It also has no isolated points and is convenient in part 1 to let ‘space’ denote ‘separable metriz- zero-dimensional because it does not contain any nontrivial able space’. In part 2 we briefly mention a few results for interval. Interestingly, the mentioned properties topologic- general Tychonoff spaces that are in the same spirit. ally characterize C: up to homeomorphism, C is the only zero-dimensional compact space without isolated points. This is due to Brouwer [5, Theorem 1.5.5]. Observe that 1. Separable metrizable spaces the characterization theorem implies that C is homeomor- phic to the Cantor cube {0, 1}∞. Hence one can look upon It is easy to see that a zero-dimensional space can be em- Brouwer’s Theorem also as a topological characterization of bedded in the real line R, and that a nonempty subspace X ∞ {0, 1} . of R is zero-dimensional if and only if it does not contain The Cantor set C topologically surfaces ‘almost every- any nondegenerate interval. For example, the space of ra- where’. By a result of Souslin, every uncountable analytic tional numbers Q,thespace of irrational numbers P,the space contains a topological copy of C [4, Exercise 14.13]. product Q × P, etc., are all zero-dimensional. This result was partly generalized by van Douwen: if a topo- A zero-dimensional space X is strongly homogeneous X provided that all of its nonempty clopen sets are homeomor- logically complete space contains an uncountable family phic. It is easy to see that every strongly homogeneous space of pairwise disjoint homeomorphs of some compact space × is homogeneous,[5, 1.9.3]. It is tempting to conjecture that K then X contains a copy of the product C K [5, Corol- all homogeneous zero-dimensional spaces are strongly ho- lary 1.5.15]. Observe that the uncountably many pairwise mogeneous. This is not true however, as was shown by van disjoint homeomorphs of K may be irregularly embedded. Douwen. Van Douwen’s result shows that they can be replaced by If P is a topological property then a space is called a Cantor set of ‘regularly’ embedded copies of K.Since nowhere P provided that no nonempty open subset of it has Souslin’s theorem works for analytic spaces, the question P. The characterizations theorems that we will mention be- naturally arises whether van Douwen’s theorem also holds low are all of the following form: up to homeomorphism, for analytic spaces. This question was considered by Becker, there is only one zero-dimensional space which has P but is van Engelen and van Mill; it is undecidable. nowhere Q.HereP and Q can be quite complex topological The Cantor set is a universal object for the class of all properties. An interesting consequence will be that all non- zero-dimensional spaces. By a result of Alexandroff and empty clopen sets of the spaces considered share the same Urysohn, every compact space is a continuous image of C properties. This means that all these spaces are strongly ho- [5, Theorem 1.5.10]. This is also some sort of universal prop- mogeneous and hence homogeneous. So the characterization erty. hart v.2003/06/18 Prn:20/06/2003; 9:30 F:hartf10.tex; VTEX/J p. 2 f-9 Topological characterizations of separable metrizable zero-dimensional spaces 335 Cantor sets are widely studied in geometric topology. It So not all homogeneous zero-dimensional absolute Borel is easy to prove that all Cantor sets in R are topologically sets are topological groups. One could revive van Douwen’s equivalent. The same result is also true in R2 but the proof problem by asking whether every zero-dimensional homo- is more difficult. In R3 there are ‘wild’ Cantor sets how- geneous absolute Borel set admits a transitive actionbya ever, the most famous one of which is Antoine’s necklace. topological group. It was recently shown by the author that Similar sets can be constructed in Rn for n 3 and in the the answer to this question is in the affirmative. Hilbert cube Q. In contrast, in the countable infinite prod- The difficult problem to topologically characterize all ∞ uctoflinesR , all Cantor sets are tame. For details, and zero-dimensional homogeneous absolute Borel sets was references, see [2, 5]. solved by van Engelen in [3]. He used the hierarchy of small The rational numbers Q can also be characterized in topo- 0 Borel classes in 3 to characterize all homogeneous zero- logical terms. It is, up to homeomorphism, the only count- dimensional absolute Borel sets of ambiguous class 2. How- able space without isolated points. This is due to Sierpinski´ 0 ever, when extended to the classes α for α<ω1,thishier- [5, Theorem 1.9.6]. As is the case with Cantor sets, topolo- archy turned out to be too coarse to distinguish between all gical copies of Q surface ‘almost everywhere’. It was shown zero-dimensional homogeneous absolute Borel sets. Instead, for example by Hurewicz that if a space X is not a Baire he used the so-called Wadge hierarchy of Borel sets devel- space then X contains a closed copy of Q (the simple proof oped by Wadge (see [4]) and powerful results of Louveau, of this which was presented in [5, Theorem 1.9.12] is due to Martin, and Steel. He concluded that there are precisely ω1 van Douwen). homogeneous zero-dimensional absolute Borel sets and that Alexandroff and Urysohn proved that, up to homeomor- they can all be topologically characterized. The question P phism, is the only zero-dimensional topologically com- which of those spaces admits the structure of a topological plete space which is nowhere compact [5, Theorem 1.9.8]. group is not completely solved. It is known that if a first P N∞ This implies for example that is homeomorphic to , category zero-dimensional absolute Borel set is homeomor- N where is the (discrete) space of natural numbers. They phic to its own square then it is a topological group (van Q × also proved that C is up to homeomorphism the only σ - Engelen). The conjecture is that the converse holds. compact zero-dimensional space which is nowhere compact Call a space rigid if the identity is its only homeomor- and nowhere countable. A similar result was obtained by the phism. It is a natural question related to the results men- Q × P author: is the topologically unique zero-dimensional tioned here whether there exist rigid zero-dimensional ab- space which is a countable union of closed topologically solute Borel sets. This question was posed by van Douwen. complete subspaces and which in addition is nowhere to- The zero-dimensionality is essential for there are simple ex- pologically complete and nowhere σ -compact. It was sub- amples of rigid continua. Van Engelen, Miller and Steel an- sequently shown by van Engelen [3]thatQ∞ is the only swered it in the negative, [3]. So any zero-dimensional ab- zero-dimensional absolute F which is first category and σδ solute Borel set admits a nontrivial homeomorphism. As far nowhere an absolute G . δσ as we know, it is unknown whether the same result can be Observe that all the spaces considered so far are topolo- proved for zero-dimensional analytic spaces. gical groups. This is clear for Q, being a subgroup of R.Itis also clear for C and P since their characterization theorems imply that they are homeomorphic to the topological groups ∞ ∞ 2.
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