Editing Metric space (section)

===Continuous maps===
{{Main|Continuous function (topology)}}
On the other end of the spectrum, one can forget entirely about the metric structure and study [[continuous function (topology)|continuous maps]], which only preserve topological structure.  There are several equivalent definitions of continuity for metric spaces.  The most important are:
* '''Topological definition.''' A function <math>f\,\colon M_1\to M_2</math> is continuous if for every open set {{mvar|U}} in {{math|''M''<sub>2</sub>}}, the [[preimage]] <math>f^{-1}(U)</math> is open.
* '''[[Sequential continuity]].''' A function <math>f\,\colon M_1\to M_2</math> is continuous if whenever a sequence {{math|(''x<sub>n</sub>'')}} converges to a point {{mvar|x}} in {{math|''M''<sub>1</sub>}}, the sequence <math>f(x_1),f(x_2),\ldots</math> converges to the point {{math|''f''(''x'')}} in {{math|''M''<sub>2</sub>}}.
: (These first two definitions are ''not'' equivalent for all topological spaces.)
* '''ε–δ definition.''' A function <math>f\,\colon M_1\to M_2</math> is continuous if for every point {{mvar|x}} in {{math|''M''<sub>1</sub>}} and every {{math|ε > 0}} there exists {{math|δ > 0}} such that for all {{mvar|y}} in {{math|''M''<sub>1</sub>}} we have <math display="block">d_1(x,y) < \delta \implies d_2(f(x),f(y)) < \varepsilon.</math>
A ''[[homeomorphism]]'' is a continuous bijection whose inverse is also continuous; if there is a homeomorphism between {{math|''M''<sub>1</sub>}} and {{math|''M''<sub>2</sub>}}, they are said to be ''homeomorphic''.  Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties.  For example, <math>\R</math> is unbounded and complete, while {{open-open|0, 1}} is bounded but not complete.