TY - JOUR
T1 - The strong Pytkeev property in topological spaces
AU - Banakh, Taras
AU - Leiderman, Arkady
N1 - Funding Information:
The author has been partially financed by NCN grant DEC-2012/07/D/ST1/02087.
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2017/8/15
Y1 - 2017/8/15
N2 - A topological space X has the strong Pytkeev property at a point x∈X if there exists a countable family N of subsets of X such that for each neighborhood Ox⊂X and subset A⊂X accumulating at x, there is a set N∈N such that N⊂Ox and N∩A is infinite. We prove that for any ℵ0-space X and any space Y with the strong Pytkeev property at a point y∈Y the function space Ck(X,Y) has the strong Pytkeev property at the constant function X→{y}⊂Y. If the space Y is rectifiable, then the function space Ck(X,Y) is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces (Xn,⁎n), n∈ω, with the strong Pytkeev property their Tychonoff product ∏n∈ωXn and their small box-product ⊡n∈ωXn both have the strong Pytkeev property at the distinguished point (⁎n)n∈ω. We prove that a sequential rectifiable space X has the strong Pytkeev property if and only if X is metrizable or contains a clopen submetrizable kω-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.
AB - A topological space X has the strong Pytkeev property at a point x∈X if there exists a countable family N of subsets of X such that for each neighborhood Ox⊂X and subset A⊂X accumulating at x, there is a set N∈N such that N⊂Ox and N∩A is infinite. We prove that for any ℵ0-space X and any space Y with the strong Pytkeev property at a point y∈Y the function space Ck(X,Y) has the strong Pytkeev property at the constant function X→{y}⊂Y. If the space Y is rectifiable, then the function space Ck(X,Y) is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces (Xn,⁎n), n∈ω, with the strong Pytkeev property their Tychonoff product ∏n∈ωXn and their small box-product ⊡n∈ωXn both have the strong Pytkeev property at the distinguished point (⁎n)n∈ω. We prove that a sequential rectifiable space X has the strong Pytkeev property if and only if X is metrizable or contains a clopen submetrizable kω-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.
KW - Function space with compact-open topology
KW - Rectifiable space
KW - The strong Pytkeev property
KW - Topological group
KW - Topological loop
KW - Topological lop
UR - http://www.scopus.com/inward/record.url?scp=85012299350&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2017.01.015
DO - 10.1016/j.topol.2017.01.015
M3 - Article
AN - SCOPUS:85012299350
SN - 0166-8641
VL - 227
SP - 10
EP - 29
JO - Topology and its Applications
JF - Topology and its Applications
ER -