Editing Topology (section)

==Concepts==

===Topologies on sets===
{{Main|Topological space}}

The term ''topology'' also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the [[real line]], the [[complex plane]], and the [[Cantor set]] can be thought of as the same set with different topologies.

Formally, let {{mvar|X}} be a set and let {{mvar|τ}} be a [[family of sets|family]] of subsets of {{mvar|X}}. Then {{mvar|τ}} is called a topology on {{mvar|X}} if:

# Both the empty set and {{mvar|X}} are elements of {{mvar|τ}}.
# Any union of elements of {{mvar|τ}} is an element of {{mvar|τ}}.
# Any intersection of finitely many elements of {{mvar|τ}} is an element of {{mvar|τ}}.

If {{mvar|τ}} is a topology on {{mvar|X}}, then the pair {{math|(''X'', ''τ'')}} is called a topological space. The notation {{math|''X''<sub>''τ''</sub>}} may be used to denote a set {{mvar|X}} endowed with the particular topology {{mvar|τ}}. By definition, every topology is a [[Pi-system|{{pi}}-system]].

The members of {{mvar|τ}} are called ''open sets'' in {{mvar|X}}. A subset of {{mvar|X}} is said to be closed if its complement is in {{mvar|τ}} (that is, its complement is open). A subset of {{mvar|X}} may be open, closed, both (a [[clopen set]]), or neither. The empty set and {{mvar|X}} itself are always both closed and open. An open subset of {{mvar|X}} which contains a point {{mvar|x}} is called an open [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}.

===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.

===Manifolds===
{{Main|Manifold}}
While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A ''manifold'' is a topological space that resembles Euclidean space near each point.  More precisely, each point of an {{mvar|n}}-dimensional manifold has a [[neighborhood (mathematics)|neighborhood]] that is [[homeomorphic]] to the Euclidean space of dimension {{mvar|n}}.  [[Line (geometry)|Lines]] and [[circle]]s, but not [[Lemniscate|figure eights]], are one-dimensional manifolds.  Two-dimensional manifolds are also called [[Surface (topology)|surfaces]], although not all [[surface (mathematics)|surfaces]] are manifolds.  Examples include the [[Plane (geometry)|plane]], the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the [[Klein bottle]] and [[real projective plane]], which cannot (that is, all their realizations are surfaces that are not manifolds).