Most functions have several numerical inputs and produce more than one numerical output. But even generally continuity requires that we can constrain the difference in outputs by suitably constraining the difference in inputs. ‘The plane and other spaces’ asks more general questions such as ‘is the distance a car has travelled a continuous function of its speed?’ This is a subtle question as neither the input nor output are numbers, but rather functions of time, with input the speed function s(t) and output the distance function d(t). In answering the question, it considers continuity between metric spaces, equivalent metrics, open sets, convergence, and compactness and connectedness, the last two being topological invariants that can be used to differentiate between spaces.
‘Unknot or knot to be?’ explains that a knot is a smooth, simple, closed curve in 3D space. Being simple and closed means the curve does not cross itself except that its end returns to its start. All knots are topologically the same as a circle; what makes a circle knotted—or not—is how that circle has been placed into 3D space. The central problem of knot theory is a classification theorem: when is there an ambient isotopy between two knots or how do we show that no such isotopy exists? Key elements of knot theory are discussed, including the three Reidemeister moves, prime knots, adding knots, and the Alexander and Jones polynomials.
Topology is now a major area of modern mathematics, but an appreciation of topology came late in the history of mathematics. The word topology—meaning ‘the study of place’—was not coined until 1836. ‘What is topology?’ aims to provide a sense of topology’s ideas and its technical vocabulary. It discusses the concepts of letters being topologically the same or homeomorphic and then moves on to Euler’s formula, which shows that there are only five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Early problems in topology included defining what dimension means and point-set topology, which sought to address what it means to be a set or to be a space.
‘Making surfaces’ considers the shape of surfaces and discusses the work of some of the early topologists, Möbius, Klein, and Riemann. It introduces the torus shape and shows how its Euler number can be calculated along with that of a sphere. It discusses closed surfaces—ones without a boundary—and how they can be divided up into vertices, edges, and faces. It then introduces one-sided surfaces such as the Möbius strip and Klein bottle, which are examples of non-orientable surfaces. The Euler number goes a long way to separating out different surfaces, with the only missing ingredient in the classification the notion of orientability.
From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.
Many topologists might choose to describe their subject as the study of continuity. There are continuous and discontinuous functions in our everyday routines. ‘Thinking continuously’ aims to provide a more rigorous sense of what continuity entails for real-valued functions of a real variable. It focuses on functions having a single numerical input and a single numerical output. The properties of continuous functions are considered and the boundedness theorem and intermediate value theorem are also explained.
In this note we correct one argument in the proof of Theorem 1.1 of our paper Martino and Priddy (1995) [1], given as Theorem 1 below. Let G, G′ be finite groups and p be a prime number. The goal is to give necessary and sufficient algebraic conditions on thep-subgroups of G and G′ which determine if their p-completions BGp∧ and BGp′∧ are stably homotopy equivalent.
We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with the Whitney C^r topology is homeomorphic to the countable box-power of the separable Hilbert space.
We introduce a fuzzy ultrametric on the set of probability measures with compact support defined on a fuzzy metric space. The construction is a counterpart, in the realm of fuzzy ultrametric spaces, of the construction due to Vink and Rutten of an ultrametric on the set of probability measures with compact supports on an ultrametric space.It is proved that the set of probability measures with finite supports is dense in the natural topology generated by the defined fuzzy ultrametric. (C) 2009 Elsevier Ltd. All rights reserved.
A subset M of a topological vector space X is said to be dense-lineable in X if there exists an infinite dimensional linear manifold in M∪{0} and dense in X. We give sufficient conditions for a lineable set to be dense-lineable, and we apply them to prove the dense-lineability of several subsets of C[a,b]. We also develop some techniques to show that the set of differentiable nowhere monotone functions is dense-lineable in C[a,b]. Other results related to density and dense-lineability of sets in Banach spaces are also presented.
For a complex Banach algebra E, let HL(E) be the space of the mappings from E into E that are analytic in the sense of Lorch. We will show that HL(E) is a closed subalgebra of Hb(E,E) and we will give a description of its spectrum. As an application we will show that E is semi-simple if and only if HL(E) is semi-simple. We will also give descriptions of the spectra of other algebras of analytic mappings in the sense of Lorch. In particular we will study the spectrum of the Banach algebra HL∞(intBE) of the bounded mappings from intBE into E that are analytic in the sense of Lorch.
In this paper we prove that the Hirzebruch surface F2,(2,2) embedded in CP17 supports the conjecture on the structure and properties of fundamental groups of complement of branch curves of generic projections, as laid out in [M. Teicher, New Invariants for surfaces, Contemp. Math. 231 (1999) 271–281]. We use the regeneration from [M. Friedman, M. Teicher, The regeneration of a 5-point, Pure and Applied Mathematics Quarterly 4 (2) (2008) 383–425. Fedor Bogomolov special issue, part I], the van Kampen theorem and properties of B̃n-groups [M. Teicher, On the quotient of the braid group by commutators of transversal half-twists and its group actions, Topology Appl. 78 (1997) 153–186], where B̃n is a quotient of the braid group Bn, for n=16.
We consider the problem of whether a given interpolating sequence for a uniform algebra yields linear interpolation. A positive answer is obtained when we deal with dual uniform algebras. Further we prove that if the Carleson generalized condition is sufficient for a sequence to be interpolating on the algebra of bounded analytic functions on the unit ball of c0, then it is sufficient for any dual uniform algebra.
Let X be a metric space. We study the free Banach space B(X) over X, that is a predual space of the Banach space of all Lipschitz functions on X which preserve a marked point θ∈X. Some applications to the extension theory of Lipschitz and two-Lipschitz functions are obtained.
We introduce homotopical techniques in the frameworks of two-graded absolute valued algebras and absolute valued triple systems, which will simplify the study of these structures. To this end, we previously refine and concrete the known descriptions of two-graded absolute valued algebras and absolute valued triple systems, as well as characterize the fact that an absolute valued triple system is the odd part of an absolute valued two-graded algebra.
