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 -net of infinite subsets of A. A Pytkeev space is a space in which every point is a Pytkeev point.
-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
- Pytkeev, E. G. (1983), "Maximally decomposable spaces", Trudy Matematicheskogo Instituta Imeni V. A. Steklova, 154: 209–213, MR 0733840.
- ^ Malykhin, V. I.; Tironi, G (2000). "Weakly Fréchet–Urysohn and Pytkeev spaces". Topology and Its Applications. 104 (2): 181–190. doi:10.1016/s0166-8641(99)00027-9.
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 (FJMS). 30 (2): 359–372. MR 2477776.
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