Math
Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #
Topology
Topological Space
An ordered pair (X,τ), where X is a set and τ is a collection of subsets of X satisfying: The empty set (∅) and X belong to τ Any arbitrary (in)finite union of members of τ still belongs to τ The intersection of any finite number of members of τ still belongs to τ
Open Balls
Let (X,d) be a metric space. Let a∈X and δ>0. The subset of X consisting of those points x∈X such that d(a,x)<δ is the called the open ball of radius δ and denoted: B(a;δ)
Neighborhood
The set of points surrounding a point p in a set V, such that p∈Rn, within a radius ϵ>0 See also: Open Balls
Metric Space
A set and a function which defines the distance between two points on that set An ordered pair (X,d), where X is an arbitrary set and d is a metric (distance-defining function). The metric defines the following properties: d(a,b)=0 iff a=b d(a,b)>0 (if a≠b) d(a,b)=d(b,a) d(a,c)≤d(a,b)+d(b,c) (Note: #4 is a restatement of the Triangle Inequality) ...
Locality (Math)
Locality refers to: A property P, of a point x, which holds true near every point around x. As an example: A sphere (and, more generally, a manifold) is locally Euclidean. For every point on the sphere there is a neighborhood which is identical to Euclidean space.