Scattered countable metric spaces X for which X × X ≈ X-香川大学学術情報リポジトリ

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Scattered countable metric spaces X for which X × X X Mem. Fac. Educ., Kagawa Univ. II, 59（2009）, 59 61

Scattered countable metric spaces X for which X × X

_X

by

Shinpei O

Abstract

  As an application of [2], we shall give a characterization of locally compact countable metric spaces X for which X × X X, namely X LC(ωγ_{) if leng (X) > 1. We shall also give 2}ℵ0 _many

scattered countable metric spaces X for which X × X X.

  In a classical paper ([1]), Mazurkiewicz and Sierpi ski counted the number of locally compact countable metric spaces and scattered countable metric spaces. The former was found to be ℵ1 and the

latter to be 2ℵ0. We shall show that there are sufficiently many scattered countable metric spaces X for which X × X X.

 Definition ([2]). Let α > 0, β > 0 be ordinals. Using Cantor s normal form, represent α, β uniquely as

     α = ωγ1n

1 + ωγ2n2 + … + ωγknk , β = ωγ1m1 + ωγ2m2 + … + ωγkmk,

γ1 > γ2 > … > γk , 0 ni < ω, 0 mi < ω, so that ni = 0 = mi does not occur.

 Now put          l = min{max{i | ni≠0} , max{j | mj≠0}} and define     π(α,β) =      Σli=1 ωγi(ni + mi) if l < k       (Σk1 i=1ωγi(ni + mi)) + ωγk(nk + mk−1) if l = k ,

where l = k is, of course, equivalent to nk≠0≠mk.

 For convenience, define π(α, 0) = π(0, α) = 0 for every ordinal α.  We write simply π(α, β) = α*β.

 Proposition 1. α*α = α if and only if α = 0 or α = ωγ

for some ordinal γ.

 Thus the ordinals of the form ωγ _{play a peculiar role in products of scattered spaces.}

 Proposition 2. A scattered space X satisfies leng(X × X) = leng(X) if and only if

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X = or leng(X) = ωγ _{with γ an ordinal.}

 Let α < ω 1 be a limit ordinal. A locally compact countable metric space X with

length α is uniquely determined and is denoted by LC(α). Indeed, the Alexandrov

one-point compactification X ∪ {p} of X satisfies {p} = (X ∪ {p})(α)

and is homeomorphic to MS(α + 1, 1). This fact is combined with Proposition 2 to give

 Theorem 3. A locally compact countable metric space X satisfies X × X X if and only if X , the one point space, the countable discrete space Nor LC(ωγ_{) with}

1 γ< ω1. In particular, there are just ℵ1many topological types of locally compact countable metric spaces X for which X × X X.

 Remark. There is no compact countable metric space X satisfying X × X X with the trivial exceptions of the empty set and the one point space. Indeed, the length of a compact countable metric space X is a non-limit ordinal so that leng(X × X) > leng(X)

whenever leng(X) > 1 by Proposition 1.

 Let us turn to scattered countable metric spaces.  Theorem 4. There are just 2ℵ0

many topological types of scattered countable metric spaces X for which X×X X.

 Proof. Let Φ : {1, 2, 3, } → {r, s} be a function of the narurals to the two point set {r, s}. Put X = MS(ωω_{+ 1, 1) and}

       Y = X − ∪{X(ω k₎ | Φ(k) = s}.

Let y ∈ Y(ω k₎ . Note that y is a regular point of Y  if Φ(k) = r,        y is a singular point of Y   if Φ(k) = s.

Here y ∈ Y(ω k) is called a regular point (resp a singular point) of Y if y has a clopen

neighborhood base U1 ⊇ U2 ⊇ U3 ⊇ … in Y satisfying leng(Um − Um+1) < ωk (resp

leng(Um − Um+1) = ωk) for every m . Now define

         Z = N ⊕ωY⊕ωY 2_⊕ωY 3_⊕…,

where⊕means the topological sum and, for example, Y 3 _{= Y × Y × Y and ωY} 3 _denotes

the topological sum of countably many Y 3_{’s. Clearly Z is a scattered countable metric}

space and satisfies Z × Z Z. Let Φ' : {1, 2, 3, } → {r, s} be another function, say

Φ(j) ≠ = Φ'(j). Assume for instance Φ(j) = r , Φ'(j) = s . Let Y', Z' be defined in the

same way as above for Φ'. To show Z Z' note that, for instance,

   (Y 3₎

(ωj₎ = (Y_(ωj₎× Y₍₀₎× Y₍₀₎) ∪ (Y₍₀₎× Y_(ωj₎× Y₍₀₎) ∪ (Y₍₀₎× Y₍₀₎× Y_(ωj₎) by [2, Proposition 2 and Theorem 2]. Since Y(0)× Y(0) is a discrete set of Y 2 and each

point of Y(ωj) is a regular point of Y it follows that each point of (Y 3)(ωj) is also a regular

point of Y 3_{. Consequently we see that every point of Z}

(ωj₎ is a regular point of Z. By the same reason, every point of Z' (ωj₎ is a singular point of Z'. Thus Z Z' which implies that there are 2ℵ0 _{many scattered countable metric spaces Z for which Z × Z ≈ Z. Just 2}ℵ0

follows from [1, Theorem 3].

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Scattered countable metric spaces X for which X × X X

References

 [1] S. Mazurkiewicz and W. Sierpi ski, Contribution à la topologie des ensembles

dénombrables, Fund. Math. 1 (1920), 17 27.

 [2] S. Oka, On Telgárski’s formula, RIMS Kokyuroku 1634 (2009), 70 73.