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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*