Domain representability of certain function spaces

Harold Bennett; David Lutzer

doi:10.1016/j.topol.2009.03.013

Domain representability of certain function spaces

Harold Bennett, David Lutzer

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let C_p (X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that C_p (X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then C_p (X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space C_p (X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement "every countable subset of X is closed".

Original language	English
Pages (from-to)	1937-1942
Number of pages	6
Journal	Topology and its Applications
Volume	156
Issue number	11
DOIs	https://doi.org/10.1016/j.topol.2009.03.013
State	Published - Jun 15 2009

Keywords

Domain representable space
Function space
Pointwise convergence topology
Pseudo-radial space
Pseudocomplete
Strongly Choquet complete
Transfinite sequence
Weakly Choquet complete

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10.1016/j.topol.2009.03.013

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title = "Domain representability of certain function spaces",

abstract = "Let Cp (X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp (X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp (X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp (X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement {"}every countable subset of X is closed{"}.",

keywords = "Domain representable space, Function space, Pointwise convergence topology, Pseudo-radial space, Pseudocomplete, Strongly Choquet complete, Transfinite sequence, Weakly Choquet complete",

author = "Harold Bennett and David Lutzer",

note = "Funding Information: This work was supported by the Swedish Cultural Foundation in Finland. The authors wish to thank Mr T. Karppinen for his help with the measurements.",

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T1 - Domain representability of certain function spaces

AU - Bennett, Harold

AU - Lutzer, David

N1 - Funding Information: This work was supported by the Swedish Cultural Foundation in Finland. The authors wish to thank Mr T. Karppinen for his help with the measurements.

PY - 2009/6/15

Y1 - 2009/6/15

N2 - Let Cp (X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp (X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp (X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp (X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement "every countable subset of X is closed".

AB - Let Cp (X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp (X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp (X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp (X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement "every countable subset of X is closed".

KW - Domain representable space

KW - Function space

KW - Pointwise convergence topology

KW - Pseudo-radial space

KW - Pseudocomplete

KW - Strongly Choquet complete

KW - Transfinite sequence

KW - Weakly Choquet complete

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U2 - 10.1016/j.topol.2009.03.013

DO - 10.1016/j.topol.2009.03.013

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SN - 0166-8641

VL - 156

SP - 1937

EP - 1942

JO - Topology and its Applications

JF - Topology and its Applications

IS - 11

ER -

Domain representability of certain function spaces

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