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order

Let (X,T)$ be a topological space, and let A\subset X$. Let \beta$ be a cardinal number. We say that A$ is of order less than equal to \beta$ in X$, written ord(A,X$)\le \beta$, provided that for each U \in T$ such that A\subset U$, there exists V\in T$ such that A
\subset V \subset U$ and \vert Bd(V)\vert\le \beta$. We say that A$ is of order \beta$ in X$, written ord(A,X$)= \beta$, provided that ord(A,X$)\le \beta$ and ord(A,X$) \not \le \alpha$ for any cardinal number \alpha
< \beta$. A concept of an order of a point p$ in a continuum X$ (in the sense of Menger-Urysohn), written \mathrm{ord}\,(p, X)$, is defined as follows. Let \mathfrak{n}$ stand for a cardinal number. We write:

\mathrm{ord}\,(p,X) \le \mathfrak{n}$ provided that for every \varepsilon > 0$ there is an open neighborhood U$ of p$ such that diam \le \varepsilon $ and \mathrm{card}\,\mathrm{bd}\,U \le
\mathfrak{n}$;

\mathrm{ord}\,(p,X) = \mathfrak{n}$ provided that \mathrm{ord}\,(p,X) \le \mathfrak{n}$ and for each cardinal number \mathfrak{m} < \mathfrak{n}$ the condition \mathrm{ord}\,
(p,X) \le \mathfrak{m}$ does not hold;

\mathrm{ord}\,(p,X) = \omega$ provided that the point p$ has arbitrarily small open neighborhoods U$ with finite boundaries \mathrm{bd}\,U$ and \mathrm{card}\,\mathrm{bd}\,U$ is not bounded by any n \in \mathbb{N}$.

Thus, for any continuum X$ we have

\displaystyle \mathrm{ord}\,(p,X) \in \{1, 2, \dots , n, \dots , \omega, \aleph_0, 2^{\aleph_0}\}
$

(convention: \omega < \aleph_0$); see [Kuratowski 1968, §51, I, p. 274]. Let a dendroid X$ and a point p \in X$ be given. Then p$ is said to be a point of order at least \mathfrak{m}$ in the classical sense provided that p$ is the center of an \mathfrak{m}$-od contained in X$. We say that p$ is a point of order \mathfrak{m}$ in the classical sense provided that \mathfrak{m}$ is the minimum cardinality for which the above condition is satisfied (see [Charatonik 1962, p. 229]).
next up previous contents index
Next: order preserving mapping Up: Definitions Previous: orbit
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30