Topological space

Hierarchy of mathematical spaces. Inner product spaces are a subset of normed vector spaces, which are a subset of metric spaces, which in turn are a subset of topological spaces.

In mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.

A topological space is the most general type of a mathematical space in which limits, continuity, and connectedness can be defined. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

Although very general, topological spaces are fundamental and are used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called general topology (or point-set topology).