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

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

(convention: $\omega < \aleph_0$); see [55, §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 [8, p. 229]).