SPM Review number: 200501
Title:
Eine Verschärfung der Eigenschaft C
Author:
F. Rothberger
Journal:
Fundamenta Mathematicae 30 (1938), 50 - 55
Reviewer:
Marion Scheepers
Rothberger starts the paper with a problem of Sierpiński:
Problem 1 (Sierpiński). Is property C preserved by continuous maps or by homeomorphisms?1
The property C has later been called strong measure zero, and was introduced in
the 1919 paper [3] by E. Borel. In connection with Sierpiński’s problem, Rothberger
introduces a property, denoted C0 . In the current notation of selection principles,
Rothberger’s property C0 is defined as follows: Let F be the collection of finite open
covers of X, and let O denote the collection of open covers of X.
Definition 1 (p. 50) A σ-totally bounded metric space X is said to have property
C0 if S1 (F, O) holds.
The relation with Borel strong measure zero is as follows:
Theorem 1 (p. 51) If X has property S1 (F, O), then X has Borel strong measure
zero.
To continue, Rothberger introduces the first example of a relative selection
principle: Let M be a σ-compact metric space and let X be a subset of M . Let
BM denote the finite open covers of M by basic open sets, and let OM,X denote
the covers of X by sets open in M .
Lemma (p. 51) If X is a subset of R the following are equivalent:
(1) X has property S1 (BR , OR,X ).
(2) X has Borel strong measure zero.
In the next two results Rothberger connects S1 (F, O) with Sierpiński’s problem:
Theorem 2 (p. 52) S1 (F, O) is preserved by continuous images.
Theorem 3 (p. 52) In the following, the statements (1) and (2) are equivalent:
(1) S1 (F, O) and Borel strong measure zero are equivalent (“gleichwertig”).
(2) Borel strong measure zero is invariant under continuous images.
Next Rothberger shows that the Continuum Hypothesis implies that there is an
uncountable set of real numbers with property S1 (F, O). In particular, he recalls
the following notion from the paper [5] by Kuratowski and Sierpiński: A set has
property (ν) if each nowhere dense subset of it is countable. And he also recalls
that a set has property L if it is “everywhere of second category”, meaning every
uncountable subset is of second category. Then he proves:
Theorem 4 (p. 53) Each set with property (ν), and a fortiori property L, has
property S1 (F, O).
Rothberger notes that since each Baire function on a set with property (ν) is,
after omitting countably many points, continuous, Theorems 2 and 4 imply that
each Baire function maps any set with property (ν) onto a set with property
S1 (F, O). Then he mentions an open problem:
1Problémes 67 in Fundamenta Mathematicae 25 (1935), p. 579.
1
2
Problem 2. (p. 54) Is S1 (F, O) invariant under transformations by Baire
functions?
Next Rothberger considers preservation of S1 (F, O) by set theoretic operations:
Theorem 5 (p. 54) The union of countably many sets with property S1 (F, O) is a
set with property S1 (F, O).
Rothberger points out that this theorem follows directly from the definition, and
alternately it also follows from invariance under continuous transformations
(Theorem 2). And then he mentions another problem:
Problem 3. (p. 54) Is S1 (F, O) hereditary? That is, does it follow from A ⊂ B,
and B has property S1 (F, O), that also A has property S1 (F, O)?
And then Rothberger introduces (near the bottom of p. 54) his property C00 ,
which in current notation is S1 (O, O)2. Rothberger states that that S1 (O, O) is
either stronger than S1 (F, O), or equivalent to it. Though he did not explicitly
mention that this is an open problem, it is clear that this was his intention with
this remark. Thus:
Problem 4. Is S1 (F, O) equivalent to S1 (O, O)?
Then Rothberger mentions that Theorems 2, 4 and 5 also hold when S1 (F, O) is
replaced by S1 (O, O), and this can be summarized as follows:
Theorem 6 (p. 54) The property S1 (O, O) is invariant under continuous images,
is preserved by countable unions, and satisfies the following inclusion:
L ⊂ (ν) ⊂ S1 (O, O) ⊂ S1 (F, O) ⊂ SMZ.
Here, SMZ denotes the collection of Borel strong measure zero sets.
Finally, Rothberger gives a definition for a concept introduced by Sierpiński
(according to footnote 7 of the paper):
Definition 2 (p. 55)A separable metric space X has property M0 if there is for each
sequence {ak } of real numbers, and for each system of neighborhoods Un (x) which
satisfies the properties:
x ∈ Un (x) for each x ∈ X and each n < ∞
and
lim Un (x) = {x}
n=∞
a subsequence Unk (xk ) (the nk here do not have to be distinct from each other) so
that:
δ(Unk (xk )) < ak and X = ∪k Unk (xk ).
Rothberger points out that it is easily seen that M0 is a strengthening of Menger’s
basis property. Menger’s basis property was defined in [6] by Menger as follows:
For each basis B of metrizable space (X, d), there is a sequence (Bn : n < ∞) of
elements of B such that limn→∞ diam(Bn ) = 0, and {Bn : n < ∞} covers X.
And then Rothberger shows:
Theorem 7 (p. 55)If a separable metric space X has property S1 (O, O), then it
has property M0 .
The paper is concluded with the statement that the following problems are open:
2The blanket assumption Rothberger makes in the beginning of the paper is that the spaces
for which he defines his new concepts are σ-totally bounded metrizable spaces.
3
Problem 5. Is the converse of Theorem 7 true?
Problem 6. How are M0 and S1 (F, O) related?
Remarks:
1) Problem 1: Solved by F. Rothberger - 1941. The answer is “No”. [8]. See also
[4], Theorem 6 (c).
2) Problem 2: Solved by I. Reclaw - 1987. The answer is ”No”. [7], Theore 3.
2) Problem 3: Solved by F. Rothberger - 1941. The answer is “No”. [8].
3) Problem 4: Solved by D.H. Fremlin and A.W. Miller - 1985. The answer is
“No”. [4], Theorem 6 (b).
4) Problem 5: Solved by L. Babinkostova - 2004. The answer is “Yes”. [2]. See
also [1], Theorem 4.2.5, Theorem 4.2.6
5) Problem 6: Solved by L. Babinkostova - 2004. D.H. Fremlin and A.W. Miller
showed that S1 (F, O) does not imply Menger’s property ([4], Theorem 6 (a)).
Also, Menger’s property does not imply S1 (F, O) ([4], Theorem 6 (c)).
Consequently, S1 (F, O) does not imply M0 . By Babinkostova’s theorem, and by
Rothberger’s Theorem 7, M0 implies S1 (F, O).
References
[1] L. Babinkostova, Selektivni Principi vo Topologijata, Ph.D. thesis, University of St Cyril and
Methodius, Skopje Macedonia (2001).
[2] L. Babinkostova, On a problem of Rothberger and Sierpiński, preprint.
[3] E. Borel, Sur la classification des ensembles de mesure nulle, Bulletin de la Societe
Mathematique de France 47 1919), 97 – 125.
[4] D. Fremlin and A.W. Miller, On some properties of Hurewicz, Menger and Rothberger,
Fundamenta Mathematicae 129 (1988), 17 - 33.
[5] C. Kuratowski and W. Sierpiński, Sur les ensembles qui ne contiennent aucun
sous-ensembles indénombrables non-dense Fundamenta Mathematicae 26 (1936), 137.
[6] K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte. Abt. 2 a,
Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie)
133 (1924), 421 – 444.
[7] I. Reclaw, On small sets, Fundamenta Mathematicae 133 (1989), 255 – 260.
[8] F. Rothberger, Sur les familles indénombrables de suites de nombres naturels et les
problèmes concernant la propriété C, Proceedings of the Cambridge Philosophical
Society 37 (1941), 109 - 126.