Editing Connected space (section)

== Local connectedness ==<!-- This section is linked from [[Covering space]] -->

{{main|Locally connected space}}

A topological space is said to be <em>[[Locally connected space|locally connected]]</em> at a point <math>x</math> if every neighbourhood of <math>x</math> contains a connected open neighbourhood. It is <em>locally connected</em> if it has a [[Base (topology)|base]] of connected sets. It can be shown that a space <math>X</math> is locally connected if and only if every component of every open set of <math>X</math> is open.

Similarly, a topological space is said to be <em>{{visible anchor|locally path-connected}}</em> if it has a base of path-connected sets.
An open subset of a locally path-connected space is connected if and only if it is path-connected.
This generalizes the earlier statement about <math>\R^n</math> and <math>\C^n</math>, each of which is locally path-connected. More generally, any [[topological manifold]] is locally path-connected.
[[File:Topologists (warsaw) sine curve.png|thumb|314x314px|The topologist's sine curve is connected, but it is not locally connected]]
Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two [[Separated sets|separated]] intervals in <math>\R</math>, such as <math>(0,1) \cup (2,3)</math>.

A classic example of a connected space that is not locally connected is the so-called [[topologist's sine curve]], defined as <math>T = \{(0,0)\} \cup \left\{ \left(x, \sin\left(\tfrac{1}{x}\right)\right) : x \in (0, 1] \right\}</math>, with the [[Euclidean topology]] [[Induced topology|induced]] by inclusion in <math>\R^2</math>.