Introduction to Topology
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About this ebook
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
Simone Malacrida (1977) Ha lavorato nel settore della ricerca (ottica e nanotecnologie) e, in seguito, in quello industriale-impiantistico, in particolare nel Power, nell'Oil&Gas e nelle infrastrutture. E' interessato a problematiche finanziarie ed energetiche. Ha pubblicato un primo ciclo di 21 libri principali (10 divulgativi e didattici e 11 romanzi) + 91 manuali didattici derivati. Un secondo ciclo, sempre di 21 libri, è in corso di elaborazione e sviluppo.
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Introduction to Topology - Simone Malacrida
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
––––––––
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