TY - JOUR
T1 - The Cp-stable closure of the class of separable metrizable spaces
AU - Banakh, Taras
AU - Gabriyelyan, Saak
N1 - Funding Information:
The first author was partially supported by NCN grant DEC-2012/07/D/ST1/02087.
Publisher Copyright:
© Instytut Matematyczny PAN, 2017.
PY - 2017/1/1
Y1 - 2017/1/1
N2 - Denote by Cp[M0] the Cp-stable closure of the class M0 of all separable metrizable spaces, i.e., Cp[M0] is the smallest class of topological spaces that contains M0 and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces Cp(X, Y). Using a recent deep result of Chernikov and Shelah, we prove that Cp[M0] coincides with the class of all Tychonoff spaces of cardinality strictly less than ℶω1. Being motivated by the theory of generalized metric spaces, we also characterize other natural Cp-type stable closures of M0.
AB - Denote by Cp[M0] the Cp-stable closure of the class M0 of all separable metrizable spaces, i.e., Cp[M0] is the smallest class of topological spaces that contains M0 and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces Cp(X, Y). Using a recent deep result of Chernikov and Shelah, we prove that Cp[M0] coincides with the class of all Tychonoff spaces of cardinality strictly less than ℶω1. Being motivated by the theory of generalized metric spaces, we also characterize other natural Cp-type stable closures of M0.
KW - Countable network
KW - Function space
KW - Ordered field
KW - Separately continuous function
KW - Topology of pointwise convergence
UR - http://www.scopus.com/inward/record.url?scp=85015449650&partnerID=8YFLogxK
U2 - 10.4064/cm6475-3-2016
DO - 10.4064/cm6475-3-2016
M3 - Article
AN - SCOPUS:85015449650
SN - 0010-1354
VL - 146
SP - 283
EP - 294
JO - Colloquium Mathematicum
JF - Colloquium Mathematicum
IS - 2
ER -