# Notes - Metric Spaces MT23, Homeomorphisms

### Flashcards

What does it mean for $f : X \to Y$ to be a homeomorphism?

- $f$ is a continuous function
- $f$ is a bijection
- $f^{-1}$ is also continuous

Can you give an example of a continuous and bijective function that is not a homeomorphism?

Let $X = [0, 1) \cup [2, 3]$ and $Y = [0, 2]$. Then

\[f(x) = \begin{cases} x &\text{if } x \in [0, 1] \\\\ x-1 &\text{if } x \in [2,3]\end{cases}\]is an example.

What is special about homeomorphisms and open sets?

Homeomorphisms preserve open sets, i.e.

\[\mathcal U \text{ is open} \iff f(\mathcal U) \text{ is open}\]