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Sidney Morris

La Trobe University, Department of Mathematics and Statistics, Adjunct

Federation University Australia, School of Science, Information Technology and Engineering, Emeritus

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I am a retired Professor of Mathematics and of Informatics. I have published over 160 papers in refereed international journals and 4 undergraduate, graduate and research books , the largest and most recent being on Compact Groups ofabout 1000 pages. I also have edited 2 books and published an online book called Topology Without Tears , translated into 8 languages. My main interests are in topological groups and topology.

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Snezana Filipovic

Susana A Eisenchlas

Griffith University

SEAMEO RELC

Michael Ndemanu

Patricia Watson

Mahidol University

Michael de Percy

Oxford Brookes University

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Papers by Sidney Morris

International Education and Innovative Approaches to University Teaching

The Australian Universities' review, 1995

In their article for the 1995 volume of Australia Univer- sities ’ Review on Postgraduate Studies and Postgraduate

work for radically redefining international education in

International Education and Innovative Approaches to University Teaching

Australian Universities Review, 1995

Australian healthcare: a smart card for a clever country

International Journal of Bio-medical Computing, 1995

In this paper we give an overview of smart card technology how a smart card could be used as a he... more In this paper we give an overview of smart card technology how a smart card could be used as a healthcare card and the benefits that would most likely result from doing so. The smart card memory can be zoned into different security levels. The top security zone may contain an individual's full medical history while the lowest security zone

Hilbert 13: Are there any genuine continuous multivariate real-valued functions?

Bulletin of the American Mathematical Society

This article begins with a provocative question: Are there any genuine continuous multivariate re... more This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert's 13th problem conjectured that there exists a continuous function f : I 3 → R, where I = [0, 1], which cannot be expressed in terms of composition and addition of continuous functions from R 2 → R, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert's conjecture is false. This article discusses the solution.

The Structure of Compact Groups

Observations on the Separable Quotient Problem for Banach Spaces

Axioms

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional ... more The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.

The Tubby Torus as a Quotient Group

Axioms

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then... more Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

Varieties of topological groups, Lie groups and SIN-groups

Colloquium Mathematicum

In this paper we answer three open problems on varieties of topological groups by invoking Lie gr... more In this paper we answer three open problems on varieties of topological groups by invoking Lie group theory. We also reprove in the present context that locally compact groups with arbitrarily small invariant identity neighborhoods can be approximated by Lie groups. Recall that a class V of topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products is called a variety of topological groups. The following problems on varieties were formulated in Colloquium Mathematicum [10]:

Embedding of the Free Abelian Topological Group Into

Mathematika

We consider the following question: for which metrizable separable spaces $X$ does the free abeli... more We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$ . While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$ .

On Lie groups in varieties of topological groups

Colloquium Mathematicum

Separability of Topological Groups: A Survey with Open Problems

Axioms

Separability is one of the basic topological properties. Most classical topological groups and Ba... more Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

Locally compact groups, residual Lie groups, and varieties generated by Lie groups

Topology and its Applications

Varieties of topological groups a survey

Colloquium Mathematicum

Embeddings in contractible or compact objects

Colloquium Mathematicum

The variety of topological groups generated by the free topological group on [0,1]

Colloquium Mathematicum

Free products of Lie groups

Colloquium Mathematicum

Maximally almost periodic groups and varieties of topological groups

Fundamenta Mathematicae

Just-non-singly generated varieties of locally convex spaces

Colloquium Mathematicum

Locally compact groups and β-varieties of topological groups

Fundamenta Mathematicae

International Education and Innovative Approaches to University Teaching

The Australian Universities' review, 1995

In their article for the 1995 volume of Australia Univer- sities ’ Review on Postgraduate Studies and Postgraduate

work for radically redefining international education in

International Education and Innovative Approaches to University Teaching

Australian Universities Review, 1995

Australian healthcare: a smart card for a clever country

International Journal of Bio-medical Computing, 1995

In this paper we give an overview of smart card technology how a smart card could be used as a he... more In this paper we give an overview of smart card technology how a smart card could be used as a healthcare card and the benefits that would most likely result from doing so. The smart card memory can be zoned into different security levels. The top security zone may contain an individual's full medical history while the lowest security zone

Hilbert 13: Are there any genuine continuous multivariate real-valued functions?

Bulletin of the American Mathematical Society

This article begins with a provocative question: Are there any genuine continuous multivariate re... more This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert's 13th problem conjectured that there exists a continuous function f : I 3 → R, where I = [0, 1], which cannot be expressed in terms of composition and addition of continuous functions from R 2 → R, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert's conjecture is false. This article discusses the solution.

The Structure of Compact Groups

Observations on the Separable Quotient Problem for Banach Spaces

Axioms

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional ... more The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.

The Tubby Torus as a Quotient Group

Axioms

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then... more Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

Varieties of topological groups, Lie groups and SIN-groups

Colloquium Mathematicum

In this paper we answer three open problems on varieties of topological groups by invoking Lie gr... more In this paper we answer three open problems on varieties of topological groups by invoking Lie group theory. We also reprove in the present context that locally compact groups with arbitrarily small invariant identity neighborhoods can be approximated by Lie groups. Recall that a class V of topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products is called a variety of topological groups. The following problems on varieties were formulated in Colloquium Mathematicum [10]:

Embedding of the Free Abelian Topological Group Into

Mathematika

We consider the following question: for which metrizable separable spaces $X$ does the free abeli... more We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$ . While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$ .

On Lie groups in varieties of topological groups

Colloquium Mathematicum

Separability of Topological Groups: A Survey with Open Problems

Axioms

Separability is one of the basic topological properties. Most classical topological groups and Ba... more Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

Locally compact groups, residual Lie groups, and varieties generated by Lie groups

Topology and its Applications

Varieties of topological groups a survey

Colloquium Mathematicum

Embeddings in contractible or compact objects

Colloquium Mathematicum

The variety of topological groups generated by the free topological group on [0,1]

Colloquium Mathematicum

Free products of Lie groups

Colloquium Mathematicum

Maximally almost periodic groups and varieties of topological groups

Fundamenta Mathematicae

Just-non-singly generated varieties of locally convex spaces

Colloquium Mathematicum

Locally compact groups and β-varieties of topological groups

Fundamenta Mathematicae

Topological Groups Advances, Surveys, and Open Questions

MDPI, 2019

A collection of papers on topological groups