Topology/Analysis: Homeomorphism

Definition

If f:M_1\to M_2 is

  1. one-one and onto (1-1 correspondence/bijection)
  2. continuous
  3. f^{-1} is continuous

Then we call f a homeomorphism between two metric spaces M_1,M_2. The metric space M_1,M_2 are said to be homeomorphic.

Properties and Theorems

  • If f is a homeomorphism between metric spaces M_1,M_2,
    • The set G⊂M_1 is open if and only if the image f(G)⊂M_2 is open.
    • The set F⊂M_1 is closed if and only if the image f(F) is closed.

Examples

  1. Consider metric spaces <[0,1],ρ>,<[0,2],ρ> and the function f:[0,1]→[0,2] given by f(x)=2x is a homeomorphism from [0,1] onto [0,2].
  2. Consider f(x)=log⁡ x. Then f is a homeomorphism of (0,∞) onto \mathbb{R}

Counterexamples

  1. (0,1) and [0,1] are not homeomorphic.
  2. \mathbb{R} and \mathbb{R}_d are not homeomorphic

(more examples and counterexamples)

Simple Quizzes

1. Suppose is a homeomorphism. Which of the following is true?
f([0,1]) is open in
If F is closed in , then f(F) is closed in since f is a homeomorphism.
f([0,1]) is closed in
If F is closed in , then f(F) is closed in since f is a homeomorphism.
f([0,1]) is closed and open in
If F is closed in , then f(F) is closed in since f is a homeomorphism. Similarly, if G is open in , then is open in since f is a homeomorphism.
f([0,1]) is open but not closed in
If F is closed in , then f(F) is closed in since f is a homeomorphism.
2. Suppose , , where is real numbers with discrete metric. Then
f is continuous
If F is closed in , then f(F) is closed in since f is a homeomorphism.
is continuous
If F is closed in , then f(F) is closed in since f is a homeomorphism.
f is a homeomorphism
If F is closed in , then f(F) is closed in since f is a homeomorphism. Similarly, if G is open in , then is open in since f is a homeomorphism.
None of the above
If F is closed in , then f(F) is closed in since f is a homeomorphism.
3. Which of the following is\are homeomorphisms? (Multiple choices may be correct)
is not a funciton
4. Which of the following is true? (Multiple answers may be true)
and are homeomorphic
Check out the f(x) = log x example above.
(0,2) and [0,2] are homeomorphic
(0,1) and are homeomorphic
Read Math.SE Article
None of the above
Two options are correct

Memory Trigger

Two spaces are homeomorphic if they share similar structure – maybe this image will help you remember the concept better!

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