Introduction to Topology E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

The following topics are presented in this book:
introduction to topology
topological structures such as spaces, groups and varieties
topological properties
topological successions

Leseprobe

Table of Contents

“Introduction to Topology”

INTRODUCTION

BASIC CONCEPTS

TOPOLOGICAL STRUCTURES

TOPOLOGICAL CHARACTERISTICS

TOPOLOGICAL SUCCESSION

“Introduction to Topology”

SIMONE MALACRIDA

The following topics are presented in this book:

introduction to topology

topological structures such as spaces, groups and varieties

topological properties

topological successions

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – BASIC CONCEPTS

Graphs and topological geometry

Continuity

Cardinality

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II - TOPOLOGICAL STRUCTURES

Topological spaces

Inside, closure and surroundings

Metric spaces

Subspaces, embeddings and topological products

Hausdorff spaces

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III - TOPOLOGICAL CHARACTERISTICS

Density and uniformity

Connection

Coverings

Compactness

Wallace and Baire theorems

Topological groups

Topological varieties _ _

Morphisms

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IV - TOPOLOGICAL SUCCESSION

Successions

Completeness and compactness of metric spaces

INTRODUCTION

This book deals with a mathematical topic of primary importance, given by topology.

As is known, the conceptual leap between elementary and advanced mathematics was evident only after the introduction of mathematical analysis.

The fact that this discipline was local, and not punctual, led to the study and development of topology, understood as the study of places and spaces not only in a geometric sense, but in a much broader sense.

Hence, topology assumes a decisive role in the understanding of mathematical analysis and every other discipline connected to it, such as functional and complex analysis, differential and tensor geometry.

Topology has its roots in mathematical logic, in the theory of sets and in that of functions, changing some basic aspects such as the concepts of cardinality, countability and the relationships that can be established.

On this, a series of successive results are built such as topological, metric and regulated spaces, groups, varieties with properties such as completeness, compactness and connection.

Ultimately, topology studies the "living space" in which mathematical analysis moves, defining the majority of the hypotheses of the latter's theorems.

I

BASIC CONCEPTS

Graphs and topological geometry

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A graph G is an ordered pair of sets V and E, where V is the set of nodes and E the set of edges such that the elements of E are pairs of elements of V.

Two nodes joined by an arc are called endpoints of the arc and the arc is identified by the pair of numbers of its endpoints.

An arc having two coincident extremes is called a loop, while multiple arcs joining the same extremes generate a multi-arc.

A graph without loops and multi-edges is called simple, otherwise it is called multi-graph.

The graph obtained by eliminating all loops and replacing each multi-edge with a single edge having the same endpoints is called a skeleton of a graph.

The number of edges existing on a node is called the degree of the node.

The minimum and maximum degree of a graph are, respectively, the degree of the node with the minimum number of subsistent edges and the degree of the node with the maximum number of subsistent edges.

A graph is said to be regular when the maximum and minimum degree coincide and, in this case, the graph is said to be regular of order equal to the degree.

A graph is said to be planar if, in the plane, the edges intersect only at the nodes.

A planar graph is said to be maximal if, with the addition of any new node, it is no longer planar.