Topology

2.

Point-Set Topology

2.1.

Topological Spaces

2.1.1.

Definition via Open Sets

2.1.1.1.

Axioms of a Topology

2.1.1.2.

Examples of Valid and Invalid Topologies

2.1.2.

Alternative Characterizations

2.1.2.1.

Definition via Closed Sets

2.1.2.2.

Definition via Neighborhoods

2.1.2.3.

Definition via Closure Operator

2.1.3.

Examples of Topological Spaces

2.1.3.1.

Discrete Topology

2.1.3.2.

Indiscrete Topology

2.1.3.3.

Cofinite Topology

2.1.3.4.

Cocountable Topology

2.1.3.5.

Standard Topology on Real Numbers

2.1.3.6.

Standard Topology on Euclidean Space

2.1.3.7.

Zariski Topology

2.1.3.8.

Lower Limit Topology

2.1.4.

Bases for Topologies

2.1.4.1.

Definition of a Basis

2.1.4.2.

Basis Criterion

2.1.4.3.

Generating Topology from a Basis

2.1.4.4.

Examples of Bases

2.1.4.5.

2.1.5.

Subbases for Topologies

2.1.5.1.

Definition of a Subbasis

2.1.5.2.

Generating Topology from a Subbasis

2.1.5.3.

Alexander's Subbase Theorem

2.2.

Fundamental Concepts

2.2.1.

Open and Closed Sets

2.2.1.1.

Properties of Open Sets

2.2.1.2.

Properties of Closed Sets

2.2.1.3.

Relationship between Open and Closed Sets

2.2.2.

2.2.2.1.

Definition of Neighborhoods

2.2.2.2.

Neighborhood Systems

2.2.2.3.

Local Properties

2.2.3.

Interior Points and Interior

2.2.3.1.

Definition of Interior Points

2.2.3.2.

Interior of a Set

2.2.3.3.

Properties of Interior

2.2.4.

Closure and Limit Points

2.2.4.1.

Closure of a Set

2.2.4.2.

2.2.4.3.

Accumulation Points

2.2.4.4.

Isolated Points

2.2.4.5.

Properties of Closure

2.2.5.

Boundary and Exterior

2.2.5.1.

Boundary of a Set

2.2.5.2.

Exterior of a Set

2.2.5.3.

Relationship between Interior, Boundary, and Exterior

2.2.6.

2.2.6.1.

Definition of Dense Sets

2.2.6.2.

Examples of Dense Sets

2.2.6.3.

Separable Spaces

2.2.7.

Nowhere Dense Sets

2.2.7.1.

Definition and Examples

2.2.7.2.

2.3.

Continuous Functions

2.3.1.

Definition of Continuity

2.3.1.1.

Preimage Definition

2.3.1.2.

Neighborhood Definition

2.3.1.3.

Sequential Definition in First Countable Spaces

2.3.2.

Properties of Continuous Functions

2.3.2.1.

Composition of Continuous Functions

2.3.2.2.

Restrictions of Continuous Functions

2.3.2.3.

Extensions of Continuous Functions

2.3.3.

Continuity and Topological Constructions

2.3.3.1.

Continuity in Subspaces

2.3.3.2.

Continuity in Product Spaces

2.3.3.3.

Continuity in Quotient Spaces

2.3.4.

Special Types of Continuous Functions

2.3.4.1.

2.3.4.2.

2.3.4.3.

2.4.

2.4.1.

Definition and Basic Properties

2.4.2.

Examples of Homeomorphic Spaces

2.4.3.

Topological Properties

2.4.3.1.

Topological Invariants

2.4.3.2.

Properties Preserved by Homeomorphisms

2.4.4.

Homeomorphism Groups

2.5.

Constructing New Topological Spaces

2.5.1.

Subspace Topology

2.5.1.1.

Definition and Construction

2.5.1.2.

Properties of Subspaces

2.5.1.3.

Examples of Subspaces

2.5.1.4.

Relative Topology

2.5.2.

Product Topology

2.5.2.1.

Finite Products

2.5.2.2.

Infinite Products

2.5.2.3.

Tychonoff Topology

2.5.2.4.

2.5.2.5.

Properties of Product Spaces

2.5.2.6.

Tychonoff's Theorem

2.5.3.

Quotient Topology

2.5.3.1.

Definition via Quotient Maps

2.5.3.2.

Identification Topology

2.5.3.3.

Examples of Quotient Spaces

2.5.3.3.1.

Cylinder Construction

2.5.3.3.2.

Möbius Strip Construction

2.5.3.3.3.

Torus Construction

2.5.3.3.4.

Klein Bottle Construction

2.5.3.3.5.

Projective Spaces

2.5.3.4.

Properties of Quotient Spaces

2.5.4.

Disjoint Union Topology

2.5.5.

Wedge Sum Topology

2.6.

2.6.1.

Connected Spaces

2.6.1.1.

Definition via Separation

2.6.1.2.

Alternative Characterizations

2.6.1.3.

Examples and Non-Examples

2.6.2.

Connected Subsets

2.6.2.1.

Connected Subsets of Real Numbers

2.6.2.2.

Intervals and Connectedness

2.6.3.

Connected Components

2.6.3.1.

Definition and Properties

2.6.3.2.

Totally Disconnected Spaces

2.6.4.

Path-Connectedness

2.6.4.1.

Definition of Path-Connected Spaces

2.6.4.2.

Relationship to Connectedness

2.6.4.3.

Path Components

2.6.5.

Local Connectedness

2.6.5.1.

Locally Connected Spaces

2.6.5.2.

Locally Path-Connected Spaces

2.6.6.

Applications of Connectedness

2.6.6.1.

Intermediate Value Theorem

2.6.6.2.

Fixed Point Theorems

2.7.

2.7.1.

Definition via Open Covers

2.7.1.1.

Finite Subcover Property

2.7.1.2.

Examples of Compact Spaces

2.7.2.

Equivalent Characterizations

2.7.2.1.

Sequential Compactness

2.7.2.2.

Limit Point Compactness

2.7.2.3.

Finite Intersection Property

2.7.3.

Properties of Compact Spaces

2.7.3.1.

Closed Subsets of Compact Spaces

2.7.3.2.

Continuous Images of Compact Spaces

2.7.3.3.

Compact Subsets of Hausdorff Spaces

2.7.4.

Compactness in Euclidean Space

2.7.4.1.

Heine-Borel Theorem

2.7.4.2.

Bolzano-Weierstrass Theorem

2.7.5.

Local Compactness

2.7.5.1.

Definition and Examples

2.7.5.2.

One-Point Compactification

2.7.5.3.

Stone-Čech Compactification

2.7.6.

Paracompactness

2.7.6.1.

Definition and Properties

2.7.6.2.

Relationship to Normality

2.8.

Separation Axioms

2.8.1.

2.8.1.1.

Kolmogorov Spaces

2.8.1.2.

Examples and Properties

2.8.2.

2.8.2.1.

Fréchet Spaces

2.8.2.2.

Characterizations

2.8.3.

2.8.3.1.

Hausdorff Spaces

2.8.3.2.

Properties and Examples

2.8.3.3.

Uniqueness of Limits

2.8.4.

2.8.4.1.

2.8.4.2.

Regularity Conditions

2.8.5.

2.8.5.1.

2.8.5.2.

Normality Conditions

2.8.5.3.

Urysohn's Lemma

2.8.5.4.

Tietze Extension Theorem

2.8.6.

Completely Regular Spaces

2.8.6.1.

Tychonoff Spaces

2.8.6.2.

Embedding in Cubes

2.8.7.

Relationships between Separation Axioms

2.9.

2.9.1.

Definition and Examples

2.9.1.1.

2.9.1.2.

Euclidean Metric

2.9.1.3.

Discrete Metric

2.9.1.4.

2.9.1.5.

2.9.1.6.

2.9.2.

Metric Topology

2.9.2.1.

Open Balls and Neighborhoods

2.9.2.2.

Topology Induced by Metric

2.9.2.3.

Metrizable Spaces

2.9.3.

Convergence in Metric Spaces

2.9.3.1.

Convergent Sequences

2.9.3.2.

Cauchy Sequences

2.9.3.3.

Relationship to Topological Convergence

2.9.4.

2.9.4.1.

Complete Metric Spaces

2.9.4.2.

Examples of Complete Spaces

2.9.4.3.

Completion of Metric Spaces

2.9.4.4.

Baire Category Theorem

2.9.5.

Compactness in Metric Spaces

2.9.5.1.

Equivalence of Compactness Notions

2.9.5.2.

Total Boundedness

2.9.5.3.

Arzelà-Ascoli Theorem

2.9.6.

2.9.6.1.

Urysohn Metrization Theorem

2.9.6.2.

Nagata-Smirnov Metrization Theorem

2.9.6.3.

Bing Metrization Theorem

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3. Algebraic Topology