Fort space

Fort spaceSteen & Seebach, Examples #23 and #24 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 $X\backslash setminus\backslash \{p\backslash \}$ 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 spaceSteen & Seebach, Example #27 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 spaceSteen & Seebach, Example #25 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 $X\backslash setminus\backslash \{p\backslash \}$ 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{{Cite web|url=https://dantopology.wordpress.com/tag/one-point-lindelofication/|title = One-point Lindelofication| date=May 2014 }} of an uncountable discrete space.

• {{annotated link|Arens–Fort space}}
• {{annotated link|Cofinite topology}}
• {{annotated link|List of topologies}}

{{reflist}}

• 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