# 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

## 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