Top Qs
Timeline
Chat
Perspective
Euclidean topology
Topological structure of Euclidean space From Wikipedia, the free encyclopedia
Remove ads
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.
![]() | This article duplicates the scope of other articles, specifically Real_coordinate_space#Topological_properties. (June 2025) |
Remove ads
Definition
The Euclidean norm on is the non-negative function defined by
Like all norms, it induces a canonical metric defined by The metric induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points and is
In any metric space, the open balls form a base for a topology on that space.[1] The Euclidean topology on is the topology generated by these balls. In other words, the open sets of the Euclidean topology on are given by (arbitrary) unions of the open balls defined as for all real and all where is the Euclidean metric.
Remove ads
Properties
When endowed with this topology, the real line is a T5 space. Given two subsets say and of with where denotes the closure of there exist open sets and with and such that [2]
Remove ads
See also
- Hilbert space – Type of vector space in math
- List of Banach spaces
- List of topologies – List of concrete topologies and topological spaces
References
Wikiwand - on
Seamless Wikipedia browsing. On steroids.
Remove ads