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.
- The set G⊂
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.
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)
4. Which of the following is true? (Multiple answers may be true)
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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