# Properties of Riemann Surface

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## Contents

## Theorem

This page gathers some results about Riemann surfaces.

### Riemann Surface is Path-Connected

A Riemann surface is path-connected.

### Radó's Theorem

A Riemann surface is second countable.

### Conformal Isomorphism of Universal Cover of Riemann Surface

The universal cover of a Riemann surface is conformally isomorphic to either:

- the Riemann sphere
- the complex plane

or

- the unit disk.

### Riemann Surface is Metrizable

A Riemann surface is metrizable.

### Riemann Surface admits Metric of Constant Curvature

A Riemann surface admits a metric of constant curvature.