Editing Connected space (section)

===Disconnected spaces===
A space in which all components are one-point sets is called [[Totally disconnected space|<em>{{visible anchor|totally disconnected}}</em>]]. Related to this property, a space <math>X</math> is called <em>{{visible anchor|totally separated}}</em> if, for any two distinct elements <math>x</math> and <math>y</math> of <math>X</math>, there exist disjoint [[open sets]] <math>U</math> containing <math>x</math> and <math>V</math> containing <math>y</math> such that <math>X</math> is the union of <math>U</math> and <math>V</math>. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers <math>\Q</math>, and identify them at every point except zero. The resulting space, with the [[quotient topology]], is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even [[Hausdorff space|Hausdorff]], and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.