gen-define-a-homeomorphism-and-give-one-short-worked-example-bd67

Define a homeomorphism and give one short worked example.

# Homeomorphism ## Definition A **homeomorphism** is a continuous bijection $f: X \to Y$ between topological spaces whose inverse $f^{-1}: Y \to X$ is also continuous. Two spaces are called **homeomorphic** if there exists a homeomorphism between them (written $X \cong Y$). Homeomorphic spaces are considered "the same" from a topological perspective. ## Worked Example **Claim:** The open interval $(0,1)$ is homeomorphic to $\mathbb{R}$. **Proof:** Define $f: (0,1) \to \mathbb{R}$ by $$f(x) = \tan\left(\pi\left(x - \frac{1}{2}\right)\right)$$ - **Bijective:** As $x$ ranges over $(0,1)$, the argument $\pi(x - \frac{1}{2})$ ranges over $(-\frac{\pi}{2}, \frac{\pi}{2})$, and $\tan$ maps this interval bijectively onto $\mathbb{R}$. - **Continuous:** $f$ is a composition of continuous functions (linear map and tangent on its domain). - **Inverse continuous:** The inverse is $f^{-1}(y) = \frac{1}{\pi}\arctan(y) + \frac{1}{2}$, which is also continuous. Therefore $f$ is a homeomorphism. ∎ This example shows that a bounded interval can be homeomorphic to an unbounded space—boundedness is not a topological property!