Sidney A. Morris

The Free Abelian Topological Group and the Free Locally Convex Space on the Unit Interval

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) of the closed unit interval. In particular, the free abelian topological group A(X) of any finite-dimensional compact metrizable space X embeds into A(I). The situation turns out to be somewhat different for free locally convex spaces. Some results for the spaces of continuous functions with the pointwise topology are also obtained. Proofs are based on the classical Kolmogorovs Superposition Theorem.

Locally compact groups with closed subgroups open and p -adic

BY KARL H. HOFMANNFachbereich Mathematik, Technische Hochschule Darmstadt, Schlonssgartenstrasse 7,D-6100 Darmstadt, Germanye-mail: hofmann@mathematik.th-darmstadt.deSIDNEY A. MORRISFaculty of Informatics, The University Wollongong, of Wollongong,NSW 2522, Australia,e-mail: sid@wampyr.cc.uow.edu.auSHEILA OATES-WILLIAMSDepartment of Mathematics, The University of Queensland, Qld 4072, Australiae-mail: sw@maths.uq.oz.auAND V. N. OBRAZTSOVDepartment of Mathematics and Physics, Kostroma Teachers Training Institute,Kostroma 156601 Russia,(Received 14 March 1994; revised 2 August 1994)1. IntroductionAn open subgroup U of a topological group G is always closed, since U is thecomplement of the open set U{Ugg£U}. An arbitrary closed subgroup C of 0 isalmost never open, unless 0 belongs to a small family of exceptional groups. In fact,if 0 is a locally compact abelian group in which every non-trivial subgroup is open,then G is the additive group Ap of p-adic integers or the additive grou

Generators on the arc component of compact connected groups

BY KARL H. HOFMANNFachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7,Z)-6100 Darmstadt, GermanyAND SIDNEY A. MORRISFaculty of Informatics, University Wollongong, of Wollongong, NSW 2522, Australia(Received 6 July 1992)IntroductionIt is well-known that a compact connected abelian group G has weight w(G) lessthan or equal to the cardinality c of the continuum if and only if it is monothetic;that is, if and only if it can be topologically generated by one element. Hofmann andMorris [2] extended this by showing that a compact connected (not necessarilyabelian) group can be topologically generated by two elements if and only ifw(G) c, the compact connectedgroup G is not topologically generated by any finite set. In this case we look fortopological generating sets which are, in some sense, thin. A subset GX i os callef dsuitable if it topologically generates G, is discrete and (?{1} is close wher, d ien 1 isthe identity o G.f If X has the smallest cardinality of any suitabl G thee subsen t ofG is called a special subset and its cardinality is denoted b s(G).y In [2] it was provedthat if G is a connected locally compact group w(G) with > c, then s(G)^° = w(Cr)

Proofs of the Impossibilities

Within this chapter it finally becomes clear why the three famous geometrical constructions are impossible. You are now, at long last, in a position to see the solutions of problems which defied the world’s best mathematicians for over two thousand years. The key to the solutions lies in combining the geometrical ideas from Chapter 5 with the algebraic ideas from earlier chapters.

An Algebraic Postscript

We have completed the proof that the three famous constructions are impossible. In this chapter we introduce some purely algebraic results which extend results in the earlier chapters. These new results will enable you to construct fields that you have not seen before.

Other Impossibilities and Abstract Algebra

This book has introduced some basic ideas from the part of abstract algebra which deals with fields. Use of these ideas made it relatively easy for us to prove the impossibility of the constructions mentioned in the Introduction.

Transcendence of e and π

The main purpose of this chapter is to prove that the number π is transcendental, thereby completing the proof of the impossibility of squaring the circle (Problem III of the Introduction). We first give the proof that e is a transcendental number, which is somewhat easier. This is of considerable interest in its own right, and its proof introduces many of the ideas which will be used in the proof for π. With the aid of some more algebra — the theory of symmetric polynomials — we can then modify the proof for e to give the proof for π.

Algebraic Numbers and Their Polynomials

Straightedge and compass constructions can he used to produce line segments of various lengths relative to some preassigned unit length. Although the lengths are all real numbers, it turns out that not every real number can be obtained in this way. The lengths which can be constructed are rather special.

Straightedge and Compass Constructions

In the previous chapters we developed the algebraic machinery for proving that the three famous geometric constructions are impossible. In this chapter we introduce some geometry and start to show the connection between algebra and the geometry of constructions.