In mathematics, and especially topology, a Pytkeev space is a topological space that satisfies qualities more subtle than a convergence of a sequence. They are named after E. G. Pytkeev, who proved in 1983 that sequential spaces have this property.
Definitions Let X be a topological space. For a subset S of X let S denote the closure of S. Then a point x is called a Pytkeev point if for every set A with x ∈ A \ {x}, there is a countable π {\displaystyle \pi } -net of infinite subsets of A. A Pytkeev space is a space in which every point is a Pytkeev point.
Examples Every sequential space is also a Pytkeev space. This is because, if x ∈ A \ {x} then there exists a sequence {ak} that converges to x. So take the countable π-net of infinite subsets of A to be {Ak} = {ak, ak+1, ak+2, …}. If X is a Pytkeev space, then it is also a Weakly Fréchet–Urysohn space.
References
Further reading Fedeli, Alessandro; Le Donne, Attilio (2002). "Pytkeev spaces and sequential extensions". Topology and Its Applications. 117 (3): 345–348. doi:10.1016/S0166-8641(01)00026-8. MR 1874095. Sakai, Masami (April 2003). "The Pytkeev property and the Reznichenko property in function spaces". Note di Matematica. 22 (2): 43–52. MR 2112730. Pansera, Bruno A. (2008). "Relative properties and function spaces". Far East Journal of Mathematical Sciences. 30 (2): 359–372. MR 2477776.