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
A function is surjective or “onto” if For every y∈Y, there exists x∈X such that f(x)=y.
Concepts # Null Hypothesis
Definition # A relation R from the elements of set A to the elements of set B is a subset of A×B. …alternatively… Let A and B be two non-empty sets, then every subset of A×B defines a relation from A to B and ever relation from A to B is a subset of A×B. Let R⊆A×B and (a,b)∈R. ...
The theoretical foundation which underlies calculus Peano Axioms Surjective Function Bijective Function Injective Function Cartesian Product
A natural number is any element of the set N={0,1,2,3…} I # 0 is a natural number. II # If n is a natural number, then n++ is also a natural number. III # 0 is not the successor of any natural number; i.e., we have n++ ≠ 0 for every natural number n. IV # Different natural numbers must have different successors; i. ...
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)
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
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 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.