Convergence foundations of topology / Szymon Dolecki (Mathematical Institute of Burgundy, France), Frédéric Mynard (New Jersey City University, USA).

Format:

Book

Author/Creator:

Dolecki, Szymon.

Contributor:

Mynard, Frédéric, 1973-

Rosengarten Family Fund.

Language:

English

Subjects (All):

Topology--Textbooks.

Topology.

Convergence.

Topological groups.

Genre:

Textbooks.

Physical Description:

xix, 548 pages ; 23 cm

Place of Publication:

New Jersey : World Scientific, 2016.

Contents:

Machine generated contents note: I. Introduction

1. Preliminaries and conventions

2. Premetrics and balls

3. Sequences

4. Cofiniteness

5. Quences

6. Almost inclusion

7. When premetrics and sequences do not suffice

7.1. Pointwise convergence

7.2. Riemann integrals

II. Families of sets

1. Isotone families of sets

2. Filters

2.1. Order

2.2. Free and principal filters

2.3. Sequential filters

2.4. Images, preimages, products

3. Grills

4. Duality between filters and grills

5. Triad: filters, filter-grills and ideals

6. Ultrafilters

7. Cardinality of the set of ultrafilters

8. Remarks on sequential filters

8.1. Countably based and Frechet filters

8.2. Infima and products of filters

9. Contours and extensions

III. Convergences

1. Definitions and first examples

2. Preconvergences on finite sets

2.1. Preconvergences on two-point sets

2.2. Preconvergences on three-point sets

3. Induced (pre)convergence

4. Premetrizable convergences

5. Adherence and cover

6. Lattice of convergences

7. Finitely deep modification

8. Pointwise properties of convergence spaces

9. Convergences on a complete lattice

IV. Continuity

1. Continuous maps

2. Initial and final convergences

3. Initial and final convergences for multiple maps

4. Product convergence

4.1. Finite product

4.2. Infinite product

5. Functional convergences

6. Diagonal and product maps

6.1. Diagonal map

6.2. Product map

7. Initial and final convergences for product maps

8. Quotient

9. Convergence invariants

9.1. Premetrizability, metrizability

9.2. Isolated points, paving number, finite depth

9.3. Characters and weight

9.4. Density and separability

V. Pretopologies

1. Definition and basic properties

2. Principal adherences and inherences

3. Open and closed sets, closures, interiors, neighborhoods

4. Topologies

4.1. Topological modification

4.2. Induced topology

4.3. Product topology

5. Open maps and closed maps

6. Topological defect and sequential order

6.1. Iterated adherence and topological defect

6.2. Sequentially based convergence and sequential order

VI. Diagonality and regularity

1. More on contours

2. Diagonality

2.1. Various types of diagonality

2.2. Diagonal modification

3. Self-regularity

4. Topological regularity

5. Regularity with respect to another convergence

VII. Types of separation

1. Convergence separation

2. Regularity with respect to a family of sets

3. Functionally induced convergences

4. Real-valued functions

5. Functionally closed and open sets

6. Functional regularity (aka complete regularity)

7. Normality

8. Continuous extension of maps

9. Tietze's extension theorem

VIII. Pseudotopologies

1. Adherence, inherence

2. Pseudotopologies

3. Pseudotopologizer

4. Regularity and topologicity among pseudotopologies

5. Initial density in pseudotopologies

6. Natural convergence

7. Convergences on hyperspaces

IX. Compactness

1. Compact sets

2. Regularity and topologicity in compact spaces

3. Local compactness

4. Topologicity of hyperspace convergences

5. Stone topology

6. Almost disjoint families

7. Compact families

8. Conditional compactness

8.1. Paratopologies

8.2. Countable compactness

8.3. Sequential compactness

9. Upper Kuratowski topology

10. More on covers

11. Cover-compactness

12. Pseudocompactness

X. Completeness in metric spaces

1. Complete metric spaces

2. Completely metrizable spaces

3. Metric spaces of continuous functions

4. Uniform continuity, extensions, and completion

XI. Completeness

1. Completeness with respect to a collection

2. Cocompleteness

3. Completeness number

4. Finitely complete convergences

5. Countably complete convergences

6. Preservation of completeness

7. Completeness of subspaces

8. Completeness of products

9. Conditionally complete convergences

10. Baire property

11. Strict completeness

XII. Connectedness

1. Connected spaces

2. Path connected and arc connected spaces

3. Components and quasi-components

4. Remarks on zero-dimensional spaces

XIII. Compactifications

1. Introduction

2. Compactifications of functionally regular topologies

3. Filters in lattices

4. Filters in lattices of closed and functionally closed sets

5. Maximality conditions

6. Cech-Stone compactification

XIV. Classification of spaces

1. Modifiers, projectors, and coprojectors

2. Functors, reflectors and coreflectors

3. Adherence-determined convergences

3.1. Reflective classes

3.2. Composable classes of filters

3.3. Conditional compactness

4. Convergences based in a class of filters

5. Other F0-composable classes of filters

6. Functorial inequalities and classification of spaces

7. Reflective and coreflective hulls

8. Conditional compactness and cover-compactness

XV. Classification of maps

1. Various types of quotient maps

1.1. Remarks on the quotient convergence

1.2. Topologically quotient maps

1.3. Hereditarily quotient maps

1.4. Quotient maps relative to a reflector

1.5. Biquotient maps

1.6. Almost open maps

1.7. Countably biquotient map

2. Interactions between maps and spaces

3. Compact relations

4. Product of spaces and of maps

XVI. Spaces of maps

1. Evaluation and adjoint maps

2. Adjoint maps on spaces of continuous maps

3. Fundamental convergences on spaces of continuous maps

4. Pointwise convergence

5. Natural convergence

5.1. Continuity of limits

5.2. Exponential law

5.3. Finer subspaces and natural convergence

5.4. Continuity of adjoint maps

5.5. Initial structures for adjoint maps

6. Compact subsets of function spaces (Ascoli-Arzela)

XVII. Duality

1. Natural duality

2. Modified duality

3. Concrete characterizations of bidual reflectors

4. Epitopologies

5. Functionally embedded convergences

6. Exponential hulls and exponential objects

7. Duality and product theorems

8. Non-Frechet product of two Frechet compact topologies

9. Spaces of real-valued continuous functions

9.1. Cauchy completeness

9.2. Completeness number

9.3. Character and weight

XVIII. Functional partitions and metrization

2. Perfect normality

3. Pseudometrics

4. Functional covers and partitions

5. Paracompactness

6. Fragmentations of partitions of unity

7. Metrization theorems

A. Set theory

1. Axiomatic set theory

2. Basic set theory

3. Natural numbers

4. Cardinality

5. Continuum

6. Order

7. Lattice

8. Well ordered sets

9. Ordinal numbers

10. Ordinal arithmetic

11. Ordinal-cardinal numbers.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

9789814571517

9814571512

9789814571524

9814571520

OCLC:

945169917

Publisher Number:

99996451734