Editing Connected space (section)

===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>