Editing Topology (section)

===Continuous functions and homeomorphisms===
{{anchor|mug-and-doughnut}}<!-- used in [[Mathematical joke]] -->
{{multiple image
 | width = 200
 | image1 = Mug and Torus morph.gif
 | image2 = Spot the cow.gif
 | footer = A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and of a cow into a sphere
}}
[[File:Topology joke.jpg|thumb|right|240px|A continuous transformation can turn a coffee mug into a donut.<br/>Ceramic model by Keenan Crane and [[Henry Segerman]].]]

{{Main|Continuous function|homeomorphism}}

A [[function (mathematics)|function]] or map from one topological space to another is called ''continuous'' if the inverse  [[image (mathematics)|image]]  of any open set is open. If the function maps the [[real number]]s to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in [[calculus]]. If a continuous function is [[injective function|one-to-one]] and [[surjective function|onto]], and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut.