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.
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.|
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.
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.|
A homeomorphism preserves 'nearness' but allows 'rubber geometry' deformations.
Note: homeomorphism is a different concept from homomorphism.
|We can project helix onto circle and still keep continuity.|