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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.

Let X be a set of ordered pairs of non-negative integers (m, n). A subset U of X is open if and only if:

  • it does not contain (0, 0), or
  • it contains (0, 0), and all but a finite number of points of all but a finite number of columns, where a column is a set {(m, n)} with fixed m.

    • In other words, an open set is only "allowed" to contain (0, 0) if only a finite number of its columns contain significant gaps. By a significant gap in a column we mean the omission of an infinite number of points.

    It is

  • Hausdorff
  • regular
  • normal

    • It is not:

  • second-countable
  • first-countable
  • metrizable
  • compact

    • References

    Arens–Fort space Wikipedia
    (Text) CC BY-SA