Editing Connected space (section)

== Arc connectedness == <!-- Connected_space#Arc_connectedness redirects to this subsection -->
A space <math>X</math> is said to be <em>arc-connected</em> or <em>arcwise connected</em> if any two [[topologically distinguishable]] points can be joined by an [[Path (topology)|arc]], which is an [[Topological embedding|embedding]] <math>f : [0, 1] \to X</math>. An <em>arc-component</em> of <math>X</math> is a maximal arc-connected subset of <math>X</math>; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable.

Every [[Hausdorff space]] that is path-connected is also arc-connected; more generally this is true for a [[Weak Hausdorff space|<math>\Delta</math>-Hausdorff space]], which is a space where each image of a [[Path (topology)|path]] is closed. An example of a space which is path-connected but not arc-connected is given by the [[line with two origins]]; its two copies of <math>0</math> can be connected by a path but not by an arc.

Intuition for path-connected spaces does not readily transfer to arc-connected spaces. Let <math>X</math> be the [[line with two origins]]. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces:

* Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small cardinality.
* Arc-components may not be disjoint. For example, <math>X</math> has two overlapping arc-components.
* Arc-connected product space may not be a product of arc-connected spaces. For example, <math>X \times \mathbb{R}</math> is arc-connected, but <math>X</math> is not.
* Arc-components of a product space may not be products of arc-components of the marginal spaces. For example, <math>X \times \mathbb{R}</math> has a single arc-component, but <math>X</math> has two arc-components.
*If arc-connected subsets have a non-empty intersection, then their union may not be arc-connected. For example, the arc-components of <math>X</math> intersect, but their union is not arc-connected.