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.[1]
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 \( \pi \) -net of infinite subsets of A. A Pytkeev space is a space in which every point is a Pytkeev point.[2]
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, …}.[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.
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