Menger-Urysohn order

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Let $(X,\tau)$ be a topological space. We define the order of a point $p \in X$, written $\mathrm{ord}(p,X)$ in the following way:

  1. $\mathrm{ord}(p,X) \leq n$ provided for every open neighborhood $U$ of $p$ there is an open neighborhood $V$ of $p$ such that $V \subset U$ and $\mathrm{card} \partial V \leq \mathfrak{n}$
  2. $\mathrm{ord} (p,X) = \mathfrak{n}$ provided that $\mathrm{ord}(p,X) \leq \mathfrak{n}$ and for each cardinal $\mathfrak{m} < \mathfrak{n}$,
  3. $\mathrm{ord}(p,X)=\omega$ provided that for every open neighborhood $U$ of $p$ there are open neighborhoods $V$ of $p$ such that $V \subset U$ with finite boundaries $\partial V$ and the numbers $\mathrm{card} \partial V$ are not bounded by any positive integer $n$.