Editing Topological space (section)

=== 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}}