The most important part of this subject is the relationship between different topologies. The mappings or morphisms between them.

For example, the continuous 'rubber geometry' deformations as between a toroid and a mug. |

There are two related mapping types continuous mappings and homeomorphisms, we start with continuous mappings.

One condition for a continuous mapping is that the intersections and unions of neighbourhoods play together properly. | |

This can be shown in terms of logic like this: | |

Or we can show it in terms of a simplex: | |

We can see, for instance, that if there is a meet (A/\B) in the domain but not the codomain this implies a sort of tear. Where to elements of A/\B map to? |

### Open Set Criterion for Continuity

A map ƒ: M_{1} -> M_{2} is continuous if and only if the inverse image of every open set is open.

#### Definition 1

Here is one definition of continuity, based on open sets:

Let X and Y be topological spaces. A function f : X->Y is continuous if f^{-1}(V) is open for every open set V in Y. |

#### Definition 2

Another definition of continuity, based on neighbourhoods, is equivalent to the above definition.

Let X and Y be topological spaces. A function f : X->Y is continuous if for every xX and every open set U containing f(x), there exists a neighbourhood V of x such that f(V)U. |

More detail about continuity on page here.

### What does a continuous mapping look like?

A continuous mapping may collapse multiple points to single points. | |

Here we convert that to neigbourhoods. | |

So this looks like a fibre bundle as discussed on the page here.

### Counter Examples

### Examples - Möbius Band to Circle

Here we have a morphism which collapses the width of the Möbius band. | |

when we reverse this morphism we keep the global structure and loose some local structure. We also loose the twist in the Möbius band. |

### Examples - Helix to Circle

Here we have a morphism which collapses a helix to a circle. | |

### Homeomorphisms

A homeomorphism preserves 'nearness' but allows 'rubber geometry' deformations. Note: homeomorphism is a different concept from homomorphism. |

### Projections

We can project helix onto circle and still keep continuity. |