Английская Википедия:Arens–Fort space

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In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr.

Definition

The Arens–Fort space is the topological space <math>(X,\tau)</math> where <math>X</math> is the set of ordered pairs of non-negative integers <math>(m, n).</math> A subset <math>U \subseteq X</math> is open, that is, belongs to <math>\tau,</math> if and only if:

  • <math>U</math> does not contain <math>(0, 0),</math> or
  • <math>U</math> contains <math>(0, 0)</math> and also all but a finite number of points of all but a finite number of columns, where a column is a set <math>\{ (m, n) ~:~ 0 \leq n \in \mathbb{Z} \}</math> with <math>0 \leq m \in \mathbb{Z}</math> fixed.

In other words, an open set is only "allowed" to contain <math>(0, 0)</math> if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points.

Properties

It is

It is not:

There is no sequence in <math>X \setminus \{ (0, 0) \}</math> that converges to <math>(0, 0).</math> However, there is a sequence <math>x_{\bull} = \left( x_i \right)_{i=1}^{\infty}</math> in <math>X \setminus \{ (0, 0) \}</math> such that <math>(0, 0)</math> is a cluster point of <math>x_{\bull}.</math>

See also

References

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Arens–Fort_space&oldid=13234914