Significant Continuous Functions to Know for Elementary Algebraic Topology

Significant continuous functions play a crucial role in understanding the structure of topological spaces. These functions, like identity and homeomorphisms, help us explore relationships between spaces, continuity, and the properties that define their topological nature.

1. Identity function

   • Maps every point in a space to itself, maintaining the structure of the space.
   • Denoted as ( id_X: X \to X ), where ( X ) is a topological space.
   • Fundamental in topology as it serves as a reference for other functions.

2. Constant function

   • Maps every point in a space to a single point in another space.
   • Denoted as ( f: X \to Y ) where ( f(x) = y_0 ) for all ( x \in X ).
   • Important for understanding the concept of continuity and limits in topology.

3. Inclusion map

   • Embeds a subspace into a larger space, denoted as ( i: A \to X ).
   • Maintains the topology of the subspace while relating it to the larger space.
   • Useful for studying properties of subspaces and their relationship to the whole space.

4. Projection map

   • Maps a product space to one of its factors, denoted as ( \pi: X \times Y \to X ).
   • Simplifies the analysis of multi-dimensional spaces by focusing on individual components.
   • Essential in understanding Cartesian products and their topological properties.

5. Homeomorphism

   • A bijective continuous function with a continuous inverse, denoted as ( f: X \to Y ).
   • Indicates that two spaces are topologically equivalent, preserving their properties.
   • Fundamental in classifying spaces and understanding their topological structure.

6. Homotopy

   • A continuous deformation of one function into another, denoted as ( H: X \times [0,1] \to Y ).
   • Captures the idea of two functions being "the same" in a topological sense.
   • Key in studying the properties of spaces that are invariant under continuous transformations.

7. Retraction

   • A continuous map ( r: X \to A ) such that ( r|_A = id_A ), where ( A \subseteq X ).
   • Indicates that the space can be "collapsed" onto a subspace while preserving the subspace's structure.
   • Important in algebraic topology for understanding fixed points and deformation.

8. Deformation retraction

   • A special type of retraction where the space is continuously shrunk to a subspace, denoted as ( H: X \times [0,1] \to X ).
   • Both the retraction and the inclusion map are continuous, showing a strong relationship between the spaces.
   • Useful for proving that certain spaces are homotopically equivalent.

9. Quotient map

   • A surjective map ( q: X \to Y ) that identifies points in ( X ) based on an equivalence relation.
   • Creates a new space ( Y ) that reflects the structure of ( X ) under the equivalence relation.
   • Important for constructing new topological spaces and understanding their properties.

10. Covering map

    • A surjective map ( p: E \to B ) such that every point in ( B ) has a neighborhood evenly covered by ( p ).
    • Allows for the study of local properties of spaces through their coverings.
    • Essential in the study of fundamental groups and covering spaces in algebraic topology.