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 \((X, \tau)\), 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 \(\tau\)
Open Balls
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}
Neighborhood
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
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 \(a \neq b\)) \(d(a, b) = d(b, a)\) \(d(a, c) \leq d(a, b) + d(b, c)\) (Note: #4 is a restatement of the Triangle Inequality) ...
Locality (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.