Maths - Morphisms Between Topologies

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: M1 → M2 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. continuity diagram

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 x∈X and every open set U containing f(x), there exists a neighbourhood V of x such that f(V)containsU. continuity diagram

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. diagram helix map
 

 

Homeomorphisms

A homeomorphism preserves 'nearness' but allows 'rubber geometry' deformations.

Note: homeomorphism is a different concept from homomorphism.

homeomorphism

Projections

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

 

 


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