nach oben

2015 | Buch

Kapitel lesen Erstes Kapitel lesen

Topology

verfasst von: Marco Manetti

Verlag: Springer International Publishing

Buchreihe : UNITEXT

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises.

The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; connectedness and compactness; Alexandrov compactification; quotient topologies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups; and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced. It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Geometrical Introduction to Topology

Abstract

Let us start off with an excerpt from the introduction to Chapter V in What is mathematics? by R. Courant and H. Robbins.

Marco Manetti

Chapter 2. Sets

Abstract

In this book we’ll always work within the so-called naïve set theory and completely avoid the axiomatic backgrounds. This choice has the advantage of leaving the reader free to pick his own preferred notions of ‘set’ ‘element’, for instance ones suggested by conventional wisdom, or those learnt during undergraduate lectures on algebra, analysis and geometry. The only exception will concern the axiom of choice, whose content is less evident from the point of view of elementary logic, or to the layman.

Marco Manetti

Chapter 3. Topological Structures

Abstract

According to some psychologists, before the age of two-and-a-half children simply scribble. Between two-and-a-half and four they develop an understanding of sorts for topology, because they draw different pictures for open and closed figures. From four years on they can replicate all topological notions: a point inside a figure, outside it or on the border. Only between four and seven they start distinguishing simple figures (like squares, triangles) based on their size or angles.

Marco Manetti

Chapter 4. Connectedness and Compactness

Abstract

The concept of a topological space is way too general to allow to prove interesting and wide ranging results. When a mathematician faces useful notions that are too general, there is only one thing he does: classify.

Marco Manetti

Chapter 5. Topological Quotients

Abstract

The dual notion to an immersion is that of an identification.

Marco Manetti

Chapter 6. Sequences

Abstract

A topological space is said to be second countable if the topology admits a countable basis of open sets.

Marco Manetti

Chapter 7. Manifolds, Infinite Products and Paracompactness

Abstract

A sub-basis of a topological space is a family $\mathcal {P}$ of open sets such that finite intersections in $\mathcal {P}$ form a basis of the topology.

Marco Manetti

Chapter 8. More Topics in General Topology $$\varvec{\curvearrowright }$$ ↷

Abstract

According to the unsophisticated and naïve definition, a set is determined by its elements (with no further conditions); therefore it might seem that the same would hold for the set of all sets, i.e. the set whose elements are all possible sets.

Marco Manetti

Chapter 9. Intermezzo $$\varvec{\curvearrowright }$$ ↷

Abstract

In this shorter chapter, and only here, we will allows ourself to be less rigorous in the arguments and a little more vague in defining notions. The purpose is to explain in an intuitive manner, sometimes even slightly punningly, a few foundational ideas of the branch of mathematics called ‘algebraic topology’.

Marco Manetti

Chapter 10. Homotopy

Abstract

There’s an intuitive notion of ‘equivalence of shapes’ in topology that is broader than the concept of homeomorphism. Two connected regular subsets in $\mathbb {R}^2$ — where regular is meant naively, as the opposite of complicated, strange, pathological, unreasonable etc. — have equivalent shape if they have the same number of holes.

Marco Manetti

Chapter 11. The Fundamental Group

Abstract

We keep on writing I to denote the closed unit interval [0, 1].

Marco Manetti

Chapter 12. Covering Spaces

Abstract

At this stage we are only able to show that certain spaces are simply connected: normally it’s quite difficult to prove directly the existence of loops that aren’t homotopic to constant paths. There are indirect methods for this, and this chapter is dedicated to one such.

Marco Manetti

Chapter 13. Monodromy

Abstract

From this point onwards we shall normally assume, to make the theory more agile and simplify both statements and proofs, that total spaces and bases of all coverings $E\rightarrow X$ are locally path connected. In particular, given a connected covering $E\rightarrow X$, both $E,X$ will be path connected.

Marco Manetti

Chapter 14. van Kampen’s Theorem

Abstract

The notation for this chapter will be as follows: if $G$ is a group and $S\subset G$ a subset we will write $\langle S\rangle \subset G$ or $\langle s\mid s\in S\rangle \subset G$ for the normal subgroup generated by $S$, which is the intersection of all normal subgroups in $G$ containing $S$. It’s easy to show that $\langle S\rangle $ coincides with the subgroup generated by all elements of the type $gsg^{-1}$, for $s\in S$ and $g\in G$.

Marco Manetti

Chapter 15. Selected Topics in Algebraic Topology $$\varvec{\curvearrowright }$$ ↷

Abstract

Among the most riveting concepts in category theory are natural transformation and equivalence of categories; much of the mathematics of the last 60 years wouldn’t exist without them. Rephrasing MacLane [ML71], one may say that functors were introduced for the purpose of defining natural transformations, and categories in order to define functors.

Marco Manetti

Chapter 16. Hints and Solutions

Abstract

This final chapter contains the solutions to the exercises marked with $\heartsuit $ and a number of suggestions and tips.

Marco Manetti

Backmatter

Titel: Topology
verfasst von: Marco Manetti
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-16958-3
Print ISBN: 978-3-319-16957-6
DOI: https://doi.org/10.1007/978-3-319-16958-3