Editing General topology (section)

===Path-connected sets===
[[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 ''[[path (topology)|path]]'' from a point ''x'' to a point ''y''  in a [[topological space]] ''X'' is  a [[continuous function (topology)|continuous function]] ''f'' from the [[unit interval]] [0,1] to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. A ''[[Path component|path-component]]'' of ''X'' is an [[equivalence class]] of ''X'' under the [[equivalence relation]], which makes ''x''  equivalent to ''y'' if there is a path from ''x'' to ''y''. The space ''X'' is said to be ''[[path-connected space|path-connected]]'' (or ''pathwise connected'' or ''0-connected'') if there is at most one path-component; that is, if there is a path joining any two points in ''X''. 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]] ''L''* and the ''[[topologist's sine curve]]''.

However, subsets of the [[real line]] '''R''' are connected [[if and only if]] they are path-connected; these subsets are the [[interval (mathematics)|intervals]] of '''R'''. Also, [[open subset]]s of '''R'''<sup>''n''</sup> or '''C'''<sup>''n''</sup> are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for [[finite topological space]]s.