Applications of abstract algebra: Mug-Torus homeomorphism

Post Reply
User avatar
Eli
Senior Expert Member
Reactions: 183
Posts: 5214
Joined: 9 years ago
Location: Tanzania
Has thanked: 75 times
Been thanked: 88 times
Contact:

#1

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

Formally,

A function between two topological spaces and is called a homeomorphism if it has the following properties:
  • is a bijection (one-to-one and onto),
  • is continuous,
  • the inverse function is continuous ( is an open mapping).
A function with these three properties is sometimes called bicontinuous. If such a function exists, we say and are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes.

A continuous deformation between a coffee mug and a donut (torus) illustrating that they are homeomorphic ( a continuous mapping with a continuous inverse function ).
Attachments
220px-Mug_and_Torus_morph.gif
220px-Mug_and_Torus_morph.gif (195.08 KiB) Viewed 1708 times
220px-Mug_and_Torus_morph.gif
220px-Mug_and_Torus_morph.gif (195.08 KiB) Viewed 1708 times
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
User avatar
Eli
Senior Expert Member
Reactions: 183
Posts: 5214
Joined: 9 years ago
Location: Tanzania
Has thanked: 75 times
Been thanked: 88 times
Contact:

#2

What is holomorphic function?
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
Post Reply
  • Similar Topics
    Replies
    Views
    Last post

Return to “Abstract Algebra”

  • Information
  • Who is online

    Users browsing this forum: No registered users and 0 guests