## Definition

If is

- one-one and onto (1-1 correspondence/bijection)
- continuous
- is continuous

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

## Properties and Theorems

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

## Examples

- 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].
- Consider f(x)=log x. Then f is a homeomorphism of (0,∞) onto

### Counterexamples

- (0,1) and [0,1] are not homeomorphic.
- and 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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