Pytkeev space

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.{{citation

| last = Pytkeev | first = E. G.

| journal = Trudy Matematicheskogo Instituta imeni V. A. Steklova

| mr = 733840

| pages = 209–213

| title = Maximally decomposable spaces

| volume = 154

| year = 1983}}.

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 {{nowrap|1=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.{{cite journal|last1=Malykhin|first1=V. I.|last2=Tironi|first2=G|year=2000|title=Weakly Fréchet–Urysohn and Pytkeev spaces|journal=Topology and Its Applications|volume=104|issue=2|pages=181–190|doi=10.1016/s0166-8641(99)00027-9|doi-access=free}}

Examples

Every sequential space is also a Pytkeev space. This is because, if {{nowrap|1=x ∈ A \ {x}}} then there exists a sequence {a_k} that converges to x. So take the countable π-net of infinite subsets of A to be {{nowrap|1={A_k} = {a_k, a_k+1, a_k+2, …}}}.
If X is a Pytkeev space, then it is also a Weakly Fréchet–Urysohn space.

Pytkeev space

Definitions

Examples

References

Further reading