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Royal Road to Topology Flashcards

Royal Road to Topology の定義・命題・定理・記法・例・手筋カードを章/セクション順に整理。 📐 は命題/定理/補題/系カード。書籍ハブ: 20_Literature/royal_road_to_topology/_home

I.1 Relations and Maps

Section titled “I.1 Relations and Maps”

I.2 Equivalence Relations

Section titled “I.2 Equivalence Relations”

I.3 Order

Section titled “I.3 Order”

I.4 Polarities

Section titled “I.4 Polarities”

I.5 Finite and Infinite Cardinals

Section titled “I.5 Finite and Infinite Cardinals”

II.1 Convergence of Sequences

Section titled “II.1 Convergence of Sequences”

II.2 The Concept of Filter

Section titled “II.2 The Concept of Filter”

II.3 Order on Filters

Section titled “II.3 Order on Filters”

II.4 Decomposition into Free and Principal Filters

Section titled “II.4 Decomposition into Free and Principal Filters”

II.5 Supplement

Section titled “II.5 Supplement”

III.1 The Concept of Limit

Section titled “III.1 The Concept of Limit”

III.2 A Pointwise Perspective

Section titled “III.2 A Pointwise Perspective”

III.3 Order on Convergences

Section titled “III.3 Order on Convergences”

III.4 Prime Convergences

Section titled “III.4 Prime Convergences”

III.5 Finite Convergences

Section titled “III.5 Finite Convergences”

III.6 Metrizable Convergences

Section titled “III.6 Metrizable Convergences”

III.7 Supplement

Section titled “III.7 Supplement”

IV.1 Images and Preimages

Section titled “IV.1 Images and Preimages”

IV.2 Continuous Maps

Section titled “IV.2 Continuous Maps”

IV.3 Continuity and Order

Section titled “IV.3 Continuity and Order”

IV.4 Initial Convergences

Section titled “IV.4 Initial Convergences”

IV.5 Selections and Products

Section titled “IV.5 Selections and Products”

IV.6 Product Convergences

Section titled “IV.6 Product Convergences”

IV.7 Diagonal Product Maps

Section titled “IV.7 Diagonal Product Maps”

IV.8 Final Convergences

Section titled “IV.8 Final Convergences”

IV.9 Lower Semicontinuity

Section titled “IV.9 Lower Semicontinuity”

IV.10 Supplement

Section titled “IV.10 Supplement”

V.1 Isotone and Antitone Families（辞書）

Section titled “V.1 Isotone and Antitone Families（辞書）”

V.3 The Notion of Grill（辞書）

Section titled “V.3 The Notion of Grill（辞書）”

VI.1 Basic Facts about Pretopologies（辞書）

Section titled “VI.1 Basic Facts about Pretopologies（辞書）”

VI.2 Adherence（辞書）

Section titled “VI.2 Adherence（辞書）”

VI.3 Inherence（辞書）

Section titled “VI.3 Inherence（辞書）”

VI.6 Pretopologizer（辞書）

Section titled “VI.6 Pretopologizer（辞書）”

VI.7 Initial Density of the Bourdaud Pretopology（辞書）

Section titled “VI.7 Initial Density of the Bourdaud Pretopology（辞書）”

VII.1 Closed Sets, Closure

Section titled “VII.1 Closed Sets, Closure”

VII.2 Open Sets, Interior, Neighborhoods

Section titled “VII.2 Open Sets, Interior, Neighborhoods”

VII.3 Topologies

Section titled “VII.3 Topologies”

VII.4 Structure of the Class of Topologies

Section titled “VII.4 Structure of the Class of Topologies”

VII.5 Initial Density of the Sierpiński Topology

Section titled “VII.5 Initial Density of the Sierpiński Topology”

VII.6 *Remarks on Convergence of Nets

Section titled “VII.6 *Remarks on Convergence of Nets”

VII.7 Supplement

Section titled “VII.7 Supplement”

VIII.1 Adherences

Section titled “VIII.1 Adherences”

VIII.2 Covers and Inherences

Section titled “VIII.2 Covers and Inherences”

VIII.3 Compact Convergences

Section titled “VIII.3 Compact Convergences”

VIII.4 Supplement

Section titled “VIII.4 Supplement”

IX.1 Topological Separation

Section titled “IX.1 Topological Separation”

IX.2 Bases

Section titled “IX.2 Bases”

IX.3 Compact, Countably Compact, and Lindelöf Topologies

Section titled “IX.3 Compact, Countably Compact, and Lindelöf Topologies”

IX.4 Open, Closed, Topologically Quotient, and Perfect Maps

Section titled “IX.4 Open, Closed, Topologically Quotient, and Perfect Maps”

IX.5 Supplement

Section titled “IX.5 Supplement”

X.1 The Real Line

Section titled “X.1 The Real Line”

X.2 Real-Valued Functions

Section titled “X.2 Real-Valued Functions”

X.3 More on Metrics and Metrizable Topologies

Section titled “X.3 More on Metrics and Metrizable Topologies”

X.4 Completely Metrizable Topologies

Section titled “X.4 Completely Metrizable Topologies”

X.5 Completion

Section titled “X.5 Completion”

X.6 Functionally Closed and Open Sets

Section titled “X.6 Functionally Closed and Open Sets”

X.7 Functional Separation

Section titled “X.7 Functional Separation”

X.8 *A Non-Trivial Regular Topology with Each Continuous Function Constant

Section titled “X.8 *A Non-Trivial Regular Topology with Each Continuous Function Constant”

X.9 Functionally Initial Topologies

Section titled “X.9 Functionally Initial Topologies”

X.10 Supplement

Section titled “X.10 Supplement”

XII.1 A Short Story of Compactness

Section titled “XII.1 A Short Story of Compactness”

XII.2 Compact Sets in Topological Spaces

Section titled “XII.2 Compact Sets in Topological Spaces”

XII.3 Weaker Variants of Compactness

Section titled “XII.3 Weaker Variants of Compactness”

XII.4 Sequentially Compact Topologies

Section titled “XII.4 Sequentially Compact Topologies”

XII.5 The Cantor Set

Section titled “XII.5 The Cantor Set”

XII.6 *Cardinality Aspects

Section titled “XII.6 *Cardinality Aspects”

XII.7 The Stone Topology

Section titled “XII.7 The Stone Topology”

XII.8 *Almost Disjoint Families

Section titled “XII.8 *Almost Disjoint Families”

XII.9 *Hyperconvergences and Semicontinuities

Section titled “XII.9 *Hyperconvergences and Semicontinuities”

XII.10 Supplement

Section titled “XII.10 Supplement”

XV.1 Preuniformities

Section titled “XV.1 Preuniformities”

XV.2 Uniformities

Section titled “XV.2 Uniformities”

XV.3 Operations

Section titled “XV.3 Operations”

XV.4 Completeness

Section titled “XV.4 Completeness”

XV.5 Supplement

Section titled “XV.5 Supplement”

XVI.3 Sequentially Founded Convergences（辞書）

Section titled “XVI.3 Sequentially Founded Convergences（辞書）”

XVII.3 Reflective and Coreflective Classes（辞書）

Section titled “XVII.3 Reflective and Coreflective Classes（辞書）”

XVII.4 Initial and Final Functors（辞書）

Section titled “XVII.4 Initial and Final Functors（辞書）”

XVIII.2 Foundations（辞書）

Section titled “XVIII.2 Foundations（辞書）”

XVIII.3 Pseudotopologies（辞書）

Section titled “XVIII.3 Pseudotopologies（辞書）”

XXII.9 Measure-theoretic Convergences

Section titled “XXII.9 Measure-theoretic Convergences”

XX.6 Compactoid and Compact Families（辞書）

Section titled “XX.6 Compactoid and Compact Families（辞書）”

XX.9 Local Compactness（辞書）

Section titled “XX.9 Local Compactness（辞書）”

XX.10 Hyperconvergences Revisited（辞書）

Section titled “XX.10 Hyperconvergences Revisited（辞書）”

XXIII.1 Fundamental Filters and Completeness（辞書）

Section titled “XXIII.1 Fundamental Filters and Completeness（辞書）”

XXIII.4 Countably Complete Convergences（辞書）

Section titled “XXIII.4 Countably Complete Convergences（辞書）”

XXIV.0 Introduction

Section titled “XXIV.0 Introduction”

XXIV.1 Dual (Natural) Convergence

Section titled “XXIV.1 Dual (Natural) Convergence”

XXIV.2 Exponentiation, Transposition, and Adjoint Maps

Section titled “XXIV.2 Exponentiation, Transposition, and Adjoint Maps”

XXIV.3 Exponential Law

Section titled “XXIV.3 Exponential Law”

XXIV.4 Hyperconvergences as Dual Convergences

Section titled “XXIV.4 Hyperconvergences as Dual Convergences”

XXIV.5 Other Convergences on Function Spaces

Section titled “XXIV.5 Other Convergences on Function Spaces”

XXIV.6 A Unification of Functional Convergences and Hyperconvergences

Section titled “XXIV.6 A Unification of Functional Convergences and Hyperconvergences”

XXIV.7 Consonance

Section titled “XXIV.7 Consonance”

XXIV.8 Compactness in Function Spaces

Section titled “XXIV.8 Compactness in Function Spaces”