DIAGONAL-FLEXIBLE SPACES AND ROTOIDS
A.V. Arhangel’skii A.V.
Ohio University, Athens, OH, U.S.A.
e-mail:
This article deals with several non-standard generalizations of the classical concept
of a topological group. An important common feature of these generalizations is the
fact that all of them are given in geometric terms. They are based on the concept
of a diagonal-flexible space. These spaces were introduced and studied in [6] under
the slightly different name of a diagonal resolvable space. We provide a brief survey
of results obtained so far in this direction, and also obtain some new results on the
structure of rotoids which are very close to diagonal-flexible spaces.
One of the main new results below is Theorem 3.1: Every compact rotoid of
countable tightness is metrizable. Using this fact, we establish that if
X
is a
compact hereditarily normal strong rotoid, then
X
is metrizable (Theorem 3.5).
The two theorems just mentioned suggest that compact rotoids strongly resemble, by
their topological properties, compact topological groups and, more generally, dyadic
compacta. Several open problems on rotoids are mentioned, in particular, the next
one: is every compact rotoid a dyadic compactum?
Keywords
:
diagonal-flexible, rectifiable, dyadic compactum, rotoid, topological group,
pseudocharacter,
π
-character,
π
-base, homogeneous, tightness,
G
δ
-diagonal, twister ,
retract.
1. Introduction.
General Topology is a framework inside which some
of the fundamental ideas of philosophy, such as the ideas of convergence
and continuity, can be precisely described, analyzed and applied.
In this article several generalizations of the classical concept of a
topological group are discussed. A
topological group
is a group with a
topology such that the multiplication is jointly continuous and the inverse
operation is continuous. For an introduction to topological groups, see [9]
or [20]. The famous Russian mathematician L.S. Pontryagin was one of the
founders of Topological Algebra, the domain of mathematics in which the
concept of a topological group plays a central role.
The two-headed concept of a topological group is not easy to visualize.
About 30 years ago a transparent geometric idea was introduced and used
to characterize a class of geometric objects that are very close to topological
groups, see about this [11, 13, 15, 23].
If a topological space
Y
contains two homeomorphic subspaces
A
and
B
, then it is natural to ask whether there exists a homeomorphism
h
of the
whole space
Y
onto itself which maps
A
onto
B
.
For a space
X
, the diagonal
Δ
X
is the subspace
{
(
x, x
) :
x
2
X
}
of
the square
X
×
X
. A space
X
will be called
diagonal-flexible
if, for every
z
2
X
, there exists a homeomorphism
h
of
X
×
X
onto itself such that
h
(Δ
X
) =
X
×{
z
}
. Note that diagonal-flexible spaces were called
diagonal
resolvable spaces
in [6] where they had been introduced.
A space
X
is
rectifiable
, if there exist
e
2
X
and a homeomorphism
g
of
X
×
X
onto itself such that
g
(
x, x
) = (
x, e
)
and
g
(
x, y
) = (
x, z
)
, for
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2013. № 2
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