Editing Topology (section)

===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}}.