Mathematics Branches, Topics, and Sub-Topics

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54Cxx Maps and general topological spaces

This subtopic introduces the core ideas in maps and general topological spaces, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.

Specific topics

54C05 Continuous maps in general topology

Overview

Continuous maps in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C08 Weak and generalized continuity in general topology

Overview

Weak and generalized continuity in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C10 Special maps on topological spaces (open, closed, etc.)

Overview

Special maps on topological spaces (open, closed, etc.). This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C15 Retraction in general topology

Overview

Retraction in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C20 Extension of maps in general topology

Overview

Extension of maps in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C25 Embedding of topological spaces

Overview

Embedding of topological spaces. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C30 Real-valued functions in general topology

Overview

Real-valued functions in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C35 Function spaces in general topology

Overview

Function spaces in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C40 Algebraic properties of function spaces in general topology

Overview

Algebraic properties of function spaces in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C45 $C$- and $C^*$-embedding in general topology

Overview

$C$- and $C^*$-embedding in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C50 Special sets defined by functions in general topology

Overview

Special sets defined by functions in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract, absolute retract spaces

Overview

Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract, absolute retract spaces. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C56 Shape theory in general topology

Overview

Shape theory in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C60 Set-valued maps in general topology

Overview

Set-valued maps in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C65 Selections in general topology

Overview

Selections in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks

54C70 Entropy in general topology

Overview

Entropy in general topology. This topic analyzes continuous maps and general morphisms between topological spaces, emphasizing structural and categorical behavior.

Related Wikipedia Page

Wikipedia: Continuous function

Useful Links

Key Ideas

  • continuity and separation interactions
  • map classes and factorization
  • function-space viewpoints

Typical Uses

Used to classify spaces and transformations through map properties and invariants.

Applications

  • Dynamical systems
  • Geometric topology
  • Analysis on topological spaces

References

Recommended Textbooks