An ordered pair \((X, \tau)\), where \(X\) is a set and \(\tau\) is a collection of subsets of \(X\) satisfying: The empty set (\(\emptyset\)) and \(X\) belong to \(\tau\) Any arbitrary (in)finite union of members of \(\tau\) still belongs to \(\tau\) The intersection of any finite number of members of \(\tau\) still belongs to \(\tau\)
A function is surjective or “onto” if For every \(y \in Y\), there exists \(x \in X\) 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 \(A \times B\). …alternatively… Let \(A\) and \(B\) be two non-empty sets, then every subset of \(A \times B\) defines a relation from \(A\) to \(B\) and ever relation from \(A\) to \(B\) is a subset of \(A \times B\). Let \(R \subseteq A \times B\) and \((a, b) \in R\). ...
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 \(\mathbb{N} = \{0, 1, 2, 3…\}\) 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++ \(\ne\) 0 for every natural number n. IV # Different natural numbers must have different successors; i. ...
Let \((X, d)\) be a metric space. Let \(a \in X\) and \(\delta > 0\). The subset of \(X\) consisting of those points \(x \in X\) such that \(d(a, x) < \delta\) is the called the open ball of radius \(\delta\) and denoted: \begin{equation} B(a;\delta) \end{equation}
The set of points surrounding a point \(p\) in a set \(V\), such that \(p \in \mathbb{R}^{n}\), within a radius \(\epsilon > 0\) 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 \(a \neq b\)) \(d(a, b) = d(b, a)\) \(d(a, c) \leq d(a, b) + d(b, c)\) (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.