Fort space

In mathematics, there are a few topological spaces named after M. K. Fort, Jr..

Fort space

Fort space[1] is defined by taking an infinite set X, with a particular point p in X, and declaring open the subsets A of X such that:

  • A does not contain p, or
  • A contains all but a finite number of points of X.

Note that the subspace has the discrete topology and is open and dense in X. X is homeomorphic to the one-point compactification of a infinite discrete space.

Modified Fort space

Modified Fort space[2] is similar but has two particular points. So take an infinite set X with two distinct points p and q, and declare open the subsets A of X such that:

  • A contains neither p nor q, or
  • A contains all but a finite number of points of X.

The space X is compact and T1, but not Hausdorff.

Fortissimo space

Fortissimo space[3] is defined by taking an uncountable set X, with a particular point p in X, and declaring open the subsets A of X such that:

  • A does not contain p, or
  • A contains all but a countable number of points of X.

Note that the subspace has the discrete topology and is open and dense in X. The space X is not compact, but it is a Lindelöf space. It is obtained by taking an uncountable discrete space, adding one point and defining a topology such that the resulting space is Lindelöf and contains the original space as a dense subspace. Similarly to Fort space being the one-point compactification of an infinite discrete space, one can describe Fortissimo space as the one-point Lindelöfication[4] of an uncountable discrete space.

See also

Notes
  1. Steen & Seebach, Examples #23 and #24
  2. Steen & Seebach, Example #27
  3. Steen & Seebach, Example #25
  4. https://dantopology.wordpress.com/tag/one-point-lindelofication/

References
  • M. K. Fort, Jr. "Nested neighborhoods in Hausdorff spaces." American Mathematical Monthly vol.62 (1955) 372.
  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
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