{{Multiple issues|{{improve categories|date=September 2025}} {{more sources|date=September 2025}}}}{{Short description|Examples of topological spaces}} In mathematics, there are a few topological spaces named after M. K. Fort, Jr.
== Fort space == '''Fort space'''<ref>Steen & Seebach, Examples #23 and #24</ref> 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''.
The subspace <math>X\setminus\{p\}</math> has the discrete topology and is open and dense in ''X''. The space ''X'' is homeomorphic to the one-point compactification of an infinite discrete space.
== Modified Fort space == '''Modified Fort space'''<ref>Steen & Seebach, Example #27</ref> 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 T<sub>1</sub>, but not Hausdorff.
== Fortissimo space == '''Fortissimo space'''<ref>Steen & Seebach, Example #25</ref> 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''.
The subspace <math>X\setminus\{p\}</math> 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''<ref>{{Cite web|url=https://dantopology.wordpress.com/tag/one-point-lindelofication/|title = One-point Lindelofication| date=May 2014 }}</ref> of an uncountable discrete space.
== See also ==
* {{annotated link|Arens–Fort space}} * {{annotated link|Cofinite topology}} * {{annotated link|List of topologies}}
== Notes == {{reflist}}
==References== *M. K. Fort, Jr. "Nested neighborhoods in Hausdorff spaces." ''American Mathematical Monthly'' vol.62 (1955) 372. *{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | orig-date=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
Category:Topological spaces