In mathematics, there are a few
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s named after
M. K. Fort, Jr.
Fort space
Fort space 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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
and is open and dense in ''X''.
''X'' is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Ale ...
of an infinite discrete space.
Modified Fort space
Modified Fort space 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
1, but not Hausdorff.
Fortissimo space
Fortissimo space 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 In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of '' compactness'', which requires the existence of a ''finite'' sub ...
. 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''
of an uncountable discrete space.
See also
*
*
*
Notes
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
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) hav ...
, origyear=1978 , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, location=Berlin, New York , edition=
Dover
Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidstone ...
reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995
Topological spaces