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Let
be a topological space, and let
.
Let
be a cardinal number. We say that
is of order less than equal to
in
, written
ord(
)
, provided that for each
such
that
, there exists
such that
and
.
We say that
is of order
in
, written
ord(
)
, provided that ord(
)
and
ord(
)
for any cardinal number
.
A concept of an order of a point
in a continuum
(in
the sense of Menger-Urysohn), written
, is defined as follows.
Let
stand for a cardinal number. We write:
provided that for every
there is an
open neighborhood
of
such that
and
;
provided that
and
for each cardinal number
the condition
does not hold;
provided that the point
has arbitrarily small open
neighborhoods
with finite boundaries
and
is not bounded by
any
.
Thus, for any continuum
we have
(convention:
); see [Kuratowski 1968, §51, I, p. 274].
Let a dendroid
and a point
be given. Then
is said to be a
point of order at least
in the classical sense
provided that
is
the center of an
-od contained in
. We say that
is a
point of order
in the classical sense provided that
is the minimum cardinality for which the above condition is
satisfied (see [Charatonik 1962, p. 229]).
Next: order preserving mapping
Up: Definitions
Previous: orbit
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30