Math Set-Theory Topology Real-Analysis Main

Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #

Topological Space

Math Topology

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

Math Topology

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}


Math Topology Set-Theory

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

Math Topology

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)

Math Topology Set-Theory

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.