Problem 3.17.
Is the Souslin number of any compact (strong) rotoid
countable?
Recall that a space
X
is
Mal’tsev
if there exists a continuous mapping
F
:
X
3
→
X
such that
F
(
x, x, y
) =
y
=
F
(
y, x, x
)
, for every
x, y
2
X
.
The Souslin number of every Mal’tsev compactum is countable (see [23],
[9]). Thus, Problem 3.18 below is related to Problem 3.17.
Problem 3.18.
Is every compact rotoid a Mal’tsev space?
Problem 3.19.
Is every compact strong rotoid homogeneous?
In the non-compact case, rotoids and strong rotoids are not so close
to topological groups as in the compact case. This became clear after the
following remarkable result in [10]:
Sorgenfrey line is a strong rotoid
. Thus,
a first-countable strong rotoid needn’t be metrizable, unlike a topological
group.
Observe that Sorgenfrey line is a paratopological group. The answer to
the next question, motivated by the result in [10] we just mentioned, will
be probably in the negative:
Problem 3.20.
Is every paratopological group a rotoid?
REFERENCES
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On bicompacta hereditarily satisfying Souslin’s condition.
Tightness and free sequences.
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199 (1971). P. 1227–1230.
English translation: Soviet Math. Dokl. 12 (1971). P. 1253–1257.
2.
Arhangel’skii A.V.
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subspaces.
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35:3 (1980). P. 1–23.
3.
Arhangel’skii A.V.
Topological Function Spaces. Kluwer, 1992.
4.
Arhangel’skii A.V.
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28:1 (2004). P. 1–18.
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Arhangel’skii A.V.
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Questions and Answers
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Arhangel’skii A.V.
and
Burke D.K.
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G
δ
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153 (2006). P. 1917–1929.
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Arhangel’skii A.V.
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Arhangel’skii A.V.
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Bennett H.
,
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Lutzer D.
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216 (2012). P. 147–161.
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Choban M.M.
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Choban M.M.
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18 (1992). P. 129–137.
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Choban M.M.
Some topics in topological algebra.
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54 (1993).
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Engelking R.
General Topology. PWN, Warszawa, 1977.
26
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
