T_1-Space
A T_1-space is a topological space fulfilling the T1-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V. In the terminology of Alexandroff and Hopf (1972), T_1-spaces are known as Fréchet spaces (though this is confusing and nonstandard).
The standard example of a T_1-space is the set of integers with the topology of open sets being those with finite complements. It is closed under finite intersection and arbitrary union so is a topology. Any integer's complement is an open set, so given two integers and using their complement as open sets, it follows that the T_1 definition is satisfied. Some T_1-spaces are not T2-spaces.
See also
Banach Space, Hausdorff Axioms, Hilbert Space, Separation Axioms, T0-Space, T2-Space, T3-Space, T4-Space, Topological Vector SpacePortions of this entry contributed by Todd Rowland
Portions of this entry contributed by Margherita Barile
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References
Alexandroff, P. and Hopf, H. Topologie, Vol. 1. New York: Chelsea, 1972.Referenced on Wolfram|Alpha
T_1-SpaceCite this as:
Barile, Margherita; Rowland, Todd; and Weisstein, Eric W. "T_1-Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/T1-Space.html