Editing Connected space (section)

==Formal definition==
A [[topological space]] <math>X</math> is said to be <em>{{visible anchor|disconnected}}</em> if it is the union of two disjoint non-empty open sets. Otherwise, <math>X</math> is said to be <em>connected</em>.  A [[subset]] of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the [[empty set]] (with its unique topology) as a connected space, but this article does not follow that practice.

For a topological space <math>X</math> the following conditions are equivalent:

#<math>X</math> is connected, that is, it cannot be divided into two disjoint non-empty open sets.
#The only subsets of <math>X</math> which are both open and closed ([[clopen set]]s) are <math>X</math> and the empty set.
#The only subsets of <math>X</math> with empty [[Boundary (topology)|boundary]] are <math>X</math> and the empty set.
#<math>X</math> cannot be written as the union of two non-empty [[separated sets]] (sets for which each is disjoint from the other's closure). 
#All [[Continuous function#Continuous functions between topological spaces|continuous]] functions from <math>X</math> to <math>\{ 0, 1 \}</math> are constant, where <math>\{ 0, 1 \}</math> is the two-point space endowed with the [[discrete topology]].

Historically this modern formulation of the notion of connectedness (in terms of no partition of <math>X</math> into two separated sets) first appeared (independently) with N.J. Lennes, [[Frigyes Riesz]], and [[Felix Hausdorff]] at the beginning of the 20th century. See {{harv|Wilder|1978}} for details.

===Connected components===

Given some point <math>x</math> in a topological space <math>X,</math> the union of any collection of connected subsets such that each contains <math>x</math> will once again be a connected subset. 
The <em>connected component of a point</em> <math>x</math> in <math>X</math> is the union of all connected subsets of <math>X</math> that contain <math>x;</math> it is the unique largest (with respect to <math>\subseteq</math>) connected subset of <math>X</math> that contains <math>x.</math> 
The [[Maximal element|maximal]] connected subsets (ordered by [[Subset|inclusion]] <math>\subseteq</math>) of a non-empty topological space are called the <em>connected components</em> of the space.
The components of any topological space <math>X</math> form a [[Partition of a set|partition]] of&nbsp;<math>X</math>: they are [[Disjoint sets|disjoint]], non-empty and their union is the whole space.
Every component is a [[closed subset]] of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the [[rational number]]s are the one-point sets ([[Singleton (mathematics)|singletons]]), which are not open. Proof: Any two distinct rational numbers <math>q_1<q_2</math> are in different components. Take an irrational number <math>q_1 < r < q_2,</math> and then set <math>A = \{q \in \Q : q < r\}</math> and <math>B = \{q \in \Q : q > r\}.</math> Then <math>(A,B)</math> is a separation of <math>\Q,</math> and <math>q_1 \in A, q_2 \in B</math>. Thus each component is a one-point set.

Let <math>\Gamma_x</math> be the connected component of <math>x</math> in a topological space <math>X,</math> and <math>\Gamma_x'</math> be the intersection of all [[clopen]] sets containing <math>x</math> (called [[Locally connected space|quasi-component]] of <math>x</math>). Then <math>\Gamma_x \subset \Gamma'_x</math> where the equality holds if <math>X</math> is compact Hausdorff or locally connected.<ref>{{cite book |title=The Stacks Project |publisher=Columbia University |url=https://stacks.math.columbia.edu/tag/0059 |access-date=17 March 2025 |archive-url=https://web.archive.org/web/20250317125700/https://stacks.math.columbia.edu/tag/0059 |archive-date=17 March 2025 |language=English |chapter=5.12 Quasi-compact spaces and maps}}</ref>

===Disconnected spaces===
A space in which all components are one-point sets is called [[Totally disconnected space|<em>{{visible anchor|totally disconnected}}</em>]]. Related to this property, a space <math>X</math> is called <em>{{visible anchor|totally separated}}</em> if, for any two distinct elements <math>x</math> and <math>y</math> of <math>X</math>, there exist disjoint [[open sets]] <math>U</math> containing <math>x</math> and <math>V</math> containing <math>y</math> such that <math>X</math> is the union of <math>U</math> and <math>V</math>. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers <math>\Q</math>, and identify them at every point except zero. The resulting space, with the [[quotient topology]], is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even [[Hausdorff space|Hausdorff]], and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.