Dyadicity index and metrizability of compact continuous images of function spaces
type: Item
instance of: scholarly article
title: Dyadicity index and metrizability of compact continuous images of function spaces
zbMATH Open document ID: 1078.54008
published in: Topology and its Applications
publication date: 2005-06-01T00:00:00Z
review text: All spaces are supposed to be Tychonoff. The authors note that in [Russ. Math. Surv. 35, 145--156 (1980; Zbl 0462.54013)], \textit{B. E. Shapirovsky} defined \textit{the dyadicity index} id\((X)\) for an arbitrary space \(X\) to be the minimal cardinal \(\kappa\) such that \(X\) cannot be continuously mapped onto \([0,1]^{\kappa^+}\). And they note that in the preceding article a number of deep results were presented that concern dyadicity index and its relationships with \(\pi\)-character and tightness. The authors show that the dyadicity index can be increased by taking the square even in the class of second countable spaces. They prove that, for any infinite cardinal \(\kappa\), a compact space \(K\) with \(\chi(x,K)\geq\kappa\) for any \(x\in K\) cannot be represented as a union of \(\leq\kappa\)-many subspaces of network weight \(<\kappa\). The authors prove that if \(K\) is an \(\omega_1\)-monolithic Lindelöf \(\Sigma\)-space then every compact continuous image of \(C_p(K)\) is metrizable. For any cardinal \(\kappa\) an example is given of a compact space \(K\) such that \(C_p(K)\) maps continuously onto the Tychonoff cube of weight \(\kappa\). They also prove that Luzin's axiom (\(2^{\omega_1}>\mathfrak c\)) is equivalent to the metrizability of all compact continuous images of \(C_p(K)\) whenever \(K\) is a separable compact space. They conclude by presenting a number of open problems concerning dyadicity index properties and the metrizability of compact continuous images of ``nice'' function spaces.
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 2173008
zbMATH Keywords: dyadicity index
zbMATH Keywords: dense subspaces of topological groups
zbMATH Keywords: dense subspaces of products
zbMATH Keywords: \(\kappa\)-monolithic space
zbMATH Keywords: factorizations theorems
zbMATH Keywords: strongly \(\kappa\)-cosmic space
zbMATH Keywords: metrizability
author: Mikhail Tkachenko
author: Vladimir Tkachuk
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
full work available at URL: j.topol.2004.09.010
OpenAlex ID: W1998187272
cites work: Q5572091
cites work: Q3049484
cites work: Q3715852
cites work: Q3736526
cites work: Q3684795
cites work: Q4150804
cites work: Q3344197
cites work: Compact spaces representable as unions of nice subspaces
cites work: Q3938195
cites work: On Lindelöf \(\Sigma\)-spaces of continuous functions in the pointwise topology
cites work: ON $ \kappa$-METRIZABLE SPACE
cites work: MAPS ONTO TIKHONOV CUBES
cites work: Q4819398
cites work: Q3860490
cites work: Q3945217
cites work: Q3935830
cites work: Q3959085
cites work: Continuous images of function spaces
cites work: Factorization theorems for topological groups and their applications
cites work: Unions of chains in dyadic compact spaces and topological groups
cites work: Q4734032
DOI: 10.1016/J.TOPOL.2004.09.010
Recommended article: Q4702001
Recommended article: Generalizing dyadicity and Esenin-Vol'pin's theorem
Recommended article: Q3357064
Recommended article: Q4500934
Recommended article: Q4643356
Recommended article: Q3126967
Recommended article: Dense, sequentially continuous maps on dyadic compacta
Recommended article: On dense subspaces of countable pseudocharacter in function spaces
Recommended article: On the properties of subclasses of weakly dyadic compact spaces
Recommended article: Chain conditions and continuous mappings on \(C_ p(X)\)
description: scientific article; zbMATH DE number 2173008