Editing Connected space (section)

==Examples==

* The closed interval <math>[0, 2)</math> in the [[Euclidean space|standard]] [[subspace topology]] is connected; although it can, for example, be written as the union of <math>[0, 1)</math> and <math>[1, 2),</math> the second set is not open in the chosen topology of <math>[0, 2).</math>
* The union of <math>[0, 1)</math> and <math>(1, 2]</math> is disconnected; both of these intervals are open in the standard topological space <math>[0, 1) \cup (1, 2].</math>
* <math>(0, 1) \cup \{ 3 \}</math> is disconnected.
* A [[convex set|convex subset]] of <math>\R^n</math> is connected; it is actually [[Simply connected set|simply connected]].
* A [[Euclidean space|Euclidean plane]] excluding the origin, <math>(0, 0),</math> is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
* A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
* <math>\R</math>, the space of [[real number]]s with the usual topology, is connected.
* The [[Lower limit topology|Sorgenfrey line]] is disconnected.<ref>{{cite book|title=General Topology|author=Stephen Willard|publisher=Dover|year=1970|page=191|isbn=0-486-43479-6}}</ref> 
* If even a single point is removed from <math>\mathbb{R}</math>, the remainder is disconnected. However, if even a countable infinity of points are removed from <math>\R^n</math>, where <math>n \geq 2,</math> the remainder is connected. If <math>n\geq 3</math>, then <math>\R^n</math> remains simply connected after removal of countably many points.
* Any [[topological vector space]], e.g. any [[Hilbert space]] or [[Banach space]], over a connected field (such as <math>\R</math> or <math>\Complex</math>), is simply connected.
* Every [[discrete topological space]] with at least two elements is disconnected, in fact such a space is [[Totally disconnected space|totally disconnected]].  The simplest example is the [[discrete two-point space]].<ref>{{cite book|title=Introduction to Topology and Modern Analysis|author=George F. Simmons|author-link=George F. Simmons|publisher=McGraw Hill Book Company|year=1968|page=144|isbn=0-89874-551-9}}</ref>
* On the other hand, a finite set might be connected. For example, the spectrum of a [[discrete valuation ring]] consists of two points and is connected. It is an example of a [[Sierpiński space]].
* The [[Cantor set]] is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
* If a space <math>X</math> is [[Homotopy|homotopy equivalent]] to a connected space, then <math>X</math> is itself connected.
* The [[topologist's sine curve]] is an example of a set that is connected but is neither path connected nor locally connected.
* The [[general linear group]] <math>\operatorname{GL}(n, \R)</math> (that is, the group of <math>n</math>-by-<math>n</math> real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, <math>\operatorname{GL}(n, \Complex)</math> is connected. More generally, the set of invertible bounded operators on a complex Hilbert space is connected.
* The spectra of commutative [[local ring]] and integral domains are connected. More generally, the following are equivalent<ref>[[Charles Weibel]], [http://www.math.rutgers.edu/~weibel/Kbook.html The K-book: An introduction to algebraic K-theory]</ref>
*# The spectrum of a commutative ring <math>R</math> is connected
*# Every [[finitely generated projective module]] over <math>R</math> has constant rank.
*# <math>R</math> has no [[idempotent]] <math>\ne 0, 1</math> (i.e., <math>R</math> is not a product of two rings in a nontrivial way).

An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an [[Annulus (mathematics)|annulus]] removed, as well as the union of two disjoint closed [[Disk (mathematics)|disks]], where all examples of this paragraph bear the [[Subspace (topology)|subspace topology]] induced by two-dimensional Euclidean space.