Topology

## 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 #

## 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.