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order

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