# Fort space

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{{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 space](/source/topological_space)s named after [M. K. Fort, Jr.](/source/M._K._Fort%2C_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](/source/discrete_topology) and is open and dense in ''X''. The space ''X'' is [homeomorphic](/source/homeomorphic) to the [one-point compactification](/source/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](/source/Lindel%C3%B6f_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.](/source/M._K._Fort%2C_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](/source/Counterexamples_in_Topology) | orig-date=1978 | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | edition=[Dover](/source/Dover_Publications) reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446  | year=1995}}

Category:Topological spaces

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