Incoming links: Neighbourhood (mathematics)
We map each point and a small enough neighbourhood of it to , so we can talk about the manifold points in terms of coordinates.
Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.
A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.
Given a function :we want to find the points of the domain of where the value of is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of .
- from some space. For beginners the real numbers but more generally topological spaces should work in general
- to the real numbers
In the case of Functionals, this problem is treated under the theory of the calculus of variations.
Topology is the plumbing of calculus.
The key concept of topology is a neighbourhood.
Just by havin the notion of neighbourhood, concepts such as limit and continuity can be defined without the need to specify a precise numerical value to the distance between two points with a metric.
As an example. consider the orthogonal group, which is also naturally a topological space. That group does not usually have a notion of distance defined for it by default. However, we can still talk about certain properties of it, e.g. that the orthogonal group is compact, and that the orthogonal group has two connected components.