Editing Topological space (section)

== Definitions ==
{{main|Axiomatic foundations of topological spaces}}

The utility of the concept of a ''topology'' is shown by the fact that there are several equivalent definitions of this [[mathematical structure]]. Thus one chooses the [[axiomatization]] suited for the application. The most commonly used is that in terms of {{em|[[open set]]s}}, but perhaps more intuitive is that in terms of {{em|[[Neighbourhood (mathematics)|neighbourhood]]s}} and so this is given first.

=== Definition via neighbourhoods{{anchor|Neighborhood definition|Neighbourhood definition}} ===

This axiomatization is due to [[Felix Hausdorff]].
Let <math>X</math> be a (possibly empty) set. The elements of <math>X</math> are usually called {{em|points}}, though they can be any mathematical object. Let <math>\mathcal{N}</math> be a [[Function (mathematics)|function]] assigning to each <math>x</math> (point) in <math>X</math> a non-empty collection <math>\mathcal{N}(x)</math> of subsets of <math>X.</math> The elements of <math>\mathcal{N}(x)</math> will be called {{em|neighbourhoods}} of <math>x</math> with respect to <math>\mathcal{N}</math> (or, simply, {{em|neighbourhoods of <math>x</math>}}). The function <math>\mathcal{N}</math> is called a [[Neighbourhood (topology)|neighbourhood topology]] if the [[axiom]]s below{{sfn|Brown|2006|loc=section 2.1}} are satisfied; and then <math>X</math> with <math>\mathcal{N}</math> is called a '''topological space'''.

# If <math>N</math> is a neighbourhood of <math>x</math> (i.e., <math>N \in \mathcal{N}(x)</math>), then <math>x \in N.</math>  In other words, each point of the set <math>X</math> belongs to every one of its neighbourhoods with respect to <math> \mathcal{N} </math>.
# If <math>N</math> is a subset of <math>X</math> and includes a neighbourhood of <math>x,</math> then <math>N</math> is a neighbourhood of <math>x.</math>  I.e., every [[superset]] of a neighbourhood of a point <math>x \in X</math> is again a neighbourhood of <math>x.</math>
# The [[Intersection (set theory)|intersection]] of two neighbourhoods of <math>x</math> is a neighbourhood of <math>x.</math>
# Any neighbourhood <math>N</math> of <math>x</math> includes a neighbourhood <math>M</math> of <math>x</math> such that <math>N</math> is a neighbourhood of each point of <math>M.</math>

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of <math>X.</math>

A standard example of such a system of neighbourhoods is for the real line <math>\R,</math> where a subset <math>N</math> of <math>\R</math> is defined to be a {{em|neighbourhood}} of a real number <math>x</math> if it includes an open interval containing <math>x.</math>

Given such a structure, a subset <math>U</math> of <math>X</math> is defined to be '''open''' if <math>U</math> is a neighbourhood of all points in <math>U.</math> The open sets then satisfy the axioms given below in the next definition of a topological space. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining <math>N</math> to be a neighbourhood of <math>x</math> if <math>N</math> includes an open set <math>U</math> such that <math>x \in U.</math>{{sfn|Brown|2006|loc=section 2.2}}

=== Definition via open sets {{anchor|topology}} ===
{{anchor|topological space}}
A ''topology'' on a [[Set (mathematics)|set]] {{mvar|X}} may be defined as a collection <math>\tau</math> of [[subset]]s of {{mvar|X}}, called '''open sets''' and satisfying the following axioms:{{sfn|Armstrong|1983|loc=definition 2.1}}

# The [[empty set]] and <math>X</math> itself belong to <math>\tau.</math>
# Any arbitrary (finite or infinite) [[Union (set theory)|union]] of members of <math>\tau</math> belongs to <math>\tau.</math>
# The intersection of any finite number of members of <math>\tau</math> belongs to <math>\tau.</math>

As this definition of a topology is the most commonly used, the set <math>\tau</math> of the open sets is commonly called a '''topology''' on <math>X.</math> 

A subset <math>C \subseteq X</math> is said to be {{em|closed}} in <math>(X, \tau)</math> if its [[Complement (set theory)|complement]] <math>X \setminus C</math> is an open set.

==== Examples of topologies ====

[[Image:Topological space examples.svg|frame|right|Let <math>\tau</math> be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set <math>\{1,2,3\}.</math> The bottom-left example is not a topology because the union of <math>\{2\}</math> and <math>\{3\}</math> [i.e. <math>\{2,3\}</math>] is missing; the bottom-right example is not a topology because the intersection of <math>\{1,2\}</math> and <math>\{2,3\}</math> [i.e. <math>\{2\}</math>], is missing.]]

# Given <math>X = \{ 1, 2, 3, 4\},</math> the [[Trivial topology|trivial]] or {{em|indiscrete}} topology on <math>X</math> is the [[Family of sets|family]] <math>\tau = \{ \{ \}, \{ 1, 2, 3, 4 \} \} = \{ \varnothing, X \}</math> consisting of only the two subsets of <math>X</math> required by the axioms forms a topology on <math>X.</math>
# Given <math>X = \{ 1, 2, 3, 4\},</math> the family <math display="block">\tau = \{ \varnothing, \{ 2 \}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}, X \}</math> of six subsets of <math>X</math> forms another topology of <math>X.</math>
# Given <math>X = \{ 1, 2, 3, 4\},</math> the [[discrete topology]] on <math>X</math> is the [[power set]] of <math>X,</math> which is the family <math>\tau = \wp(X)</math> consisting of all possible subsets of <math>X.</math> In this case the topological space <math>(X, \tau)</math> is called a ''[[discrete space]]''.
# Given <math>X = \Z,</math> the set of integers, the family <math>\tau</math> of all finite subsets of the integers plus <math>\Z</math> itself is {{em|not}} a topology, because (for example) the union of all finite sets not containing zero is not finite and therefore not a member of the family of finite sets. The union of all finite sets not containing zero is also not all of <math>\Z,</math> and so it cannot be in <math>\tau.</math>

=== Definition via closed sets ===
Using [[de Morgan's laws]], the above axioms defining open sets become axioms defining '''[[closed set]]s''':

# The empty set and <math>X</math> are closed.
# The intersection of any collection of closed sets is also closed.
# The union of any finite number of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set <math>X</math> together with a collection <math>\tau</math> of closed subsets of <math>X.</math> Thus the sets in the topology <math>\tau</math> are the closed sets, and their complements in <math>X</math> are the open sets.

=== Other definitions ===
There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.

Another way to define a topological space is by using the [[Kuratowski closure axioms]], which define the closed sets as the [[Fixed point (mathematics)|fixed points]] of an [[Operator (mathematics)|operator]] on the [[power set]] of <math>X.</math>

A [[Net (mathematics)|net]] is a generalisation of the concept of [[sequence]].  A topology is completely determined if for every net in <math>X</math> the set of its [[Topology glossary|accumulation points]] is specified.