# Manifold over a pseudogroup

## Contents

## Definition

### Given data

A pseudogroup acting on a topological space .

### Definition part

A topological space is termed a -manifold if there is an open cover of such that are isomorphic to open sets in , and such that the transition functions are elements within .

The underlying space here is termed the *model space* for the -manifold structure.

## Examples

### Topological manifold

`Further information: topological manifold`

A topological manifold is a manifold over the pseudogroup of all homeomorphisms between open sets, with the model space being .

### Differential manifold

`Further information: differential manifold`

A differential manifold is a manifold over the pseudogroup of all diffeomorphisms between open sets, with the model space being .

### Real-analytic manifold

`Further information: real-analytic manifold`

A real-analytic manifold is a manifold over the pseudogroup of all real-analytic maps between open sets in , with the model space being .

### Measured manifold

`Further information: measured manifold`

A measured manifold is a manifold over the pseudogroup of all volume-preserving diffeomorphisms between open sets, with the model space being .