Continuous Bijections from Compact Space to a Hausdorff Space are Homeomorphisms

Last updated Nov 1, 2022

Suppose $X$ is a Compact space, $Y$ is a Hausdorff space, and $f: X \to Y$ is a continuous Bijection. Then $f$ is a Homeomorphism.

Continuous Functions from a Compact Space to a Hausdorff Space are Closed Maps and A Function is a Homeomorphism iff it is a Bijective Continuous Closed Map. $\square$

1. This is a very applicable result. For example, this shows us that any injective Continuous Function from $\mathbb{S}^{1} \to \mathbb{R}^{2}$ must be a Topological Embedding.
2. According to Nick Hanson, this is called the Homeomorphism Sledgehammer.

• Euclidean Sphere
• Euclidean Space