Editing Connected space (section)

== Path connectedness ==<!-- This section is linked from [[Covering space]] and [[path-connected]] -->
[[File:Path-connected space.svg|thumb|This subspace of '''R'''² is path-connected, because a path can be drawn between any two points in the space.]]

A <em>{{visible anchor|path-connected space}}</em> is a stronger notion of connectedness, requiring the structure of a path. A <em>[[Path (topology)|path]]</em> from a point <math>x</math> to a point <math>y</math> in a [[topological space]] <math>X</math> is a continuous function <math>f</math> from the [[unit interval]] <math>[0,1]</math> to <math>X</math> with <math>f(0)=x</math> and <math>f(1)=y</math>. A <em>{{visible anchor|path-component}}</em> of <math>X</math> is an [[equivalence class]] of <math>X</math> under the [[equivalence relation]] which makes <math>x</math> equivalent to <math>y</math> if and only if there is a path from <math>x</math> to <math>y</math>. The space <math>X</math> is said to be <em>path-connected</em> (or <em>pathwise connected</em> or <math>\mathbf{0}</math><em>-connected</em>) if there is exactly one path-component.  For non-empty spaces, this is equivalent to the statement that there is a path joining any two points in <math>X</math>.  Again, many authors exclude the empty space.

Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended [[Long line (topology)|long line]] <math>L^*</math> and the [[topologist's sine curve]].

Subsets of the [[real line]] <math>\R</math> are connected [[if and only if]] they are path-connected; these subsets are the [[interval (mathematics)|intervals]] and rays of <math>\R</math>.
Also, open subsets of <math>\R^n</math> or <math>\C^n</math> are connected if and only if they are path-connected.
Additionally, connectedness and path-connectedness are the same for [[finite topological space]]s.<ref name="Munkres">{{cite book |last1=Munkres |first1=James Raymond |title=Topology |date=2000 |publisher=Prentice Hall |location=Upper Saddle River (N. J.) |isbn=0-13-181629-2 |pages=155–157 |edition=2nd |url=https://math.ucr.edu/~res/math205B-2018/Munkres%20-%20Topology.pdf |access-date=24 March 2025 |language=en}}</ref>