Editing General topology (section)

==A topology on a set==
{{Main|Topological space}}
Let ''X'' be a set and let ''τ'' be a [[Family of sets|family]] of [[subset]]s of ''X''. Then ''τ'' is called a ''topology on X'' if:<ref>Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</ref><ref>Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.</ref>

# Both the [[empty set]] and ''X'' are elements of ''τ''
# Any [[union (set theory)|union]] of elements of ''τ'' is an element of ''τ''
# Any [[intersection (set theory)|intersection]] of finitely many elements of ''τ'' is an element of ''τ''

If ''τ'' is a topology on ''X'', then the pair (''X'', ''τ'') is called a ''topological space''. The notation ''X<sub>τ</sub>'' may be used to denote a set ''X'' endowed with the particular topology ''τ''.

The members of ''τ'' are called ''[[open set]]s'' in ''X''. A subset of ''X'' is said to be [[closed set|closed]] if its [[Complement (set theory)|complement]] is in ''τ'' (i.e., its complement is open). A subset of ''X'' may be open, closed, both ([[clopen set]]), or neither. The empty set and ''X'' itself are always both closed and open.

===Basis for a topology===
{{Main|Basis (topology)}}
A '''base''' (or '''basis''') ''B'' for a [[topological space]] ''X'' with [[topological space|topology]] ''T'' is a collection of [[open set]]s in ''T'' such that every open set in ''T'' can be written as a union of elements of ''B''.<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |author-link2=Howard Ensign Simmons Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=https://archive.org/details/topologicalmetho00merr/page/16 |access-date=27 July 2012 |pages=[https://archive.org/details/topologicalmetho00merr/page/16 16] |quote='''Definition.''' A collection ''B'' of subsets of a topological space ''(X,T)'' is called a ''basis'' for ''T'' if every open set can be expressed as a union of members of ''B''. |url-access=registration }}</ref><ref>{{cite book |last=Armstrong |first=M. A. |title=Basic Topology |year=1983 |publisher=Springer |isbn=0-387-90839-0  |url=https://www.springer.com/mathematics/geometry/book/978-0-387-90839-7 |access-date=13 June 2013 |page=30 |quote=Suppose we have a topology on a set ''X'', and a collection <math>\beta</math> of open sets such that every open set is a union of members of <math>\beta</math>. Then <math>\beta</math> is called a ''base'' for the topology...}}</ref>  We say that the base ''generates'' the topology ''T''. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.

===Subspace and quotient===
Every subset of a topological space can be given the [[subspace topology]] in which the open sets are the intersections of the open sets of the larger space with the subset.  For any [[indexed family]] of topological spaces, the product can be given the [[product topology]], which is generated by the inverse images of open sets of the factors under the [[projection (mathematics)|projection]] mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

A [[Quotient space (topology)|quotient space]] is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a [[surjection|surjective]] [[function (mathematics)|function]], then the [[quotient topology]] on ''Y'' is the collection of subsets of ''Y'' that have open [[inverse image]]s under ''f''. In other words, the quotient topology is the finest topology on ''Y'' for which ''f'' is continuous.  A common example of a quotient topology is when an [[equivalence relation]] is defined on the topological space ''X''.  The map ''f'' is then the natural projection onto the set of [[equivalence class]]es.

===Examples of topological spaces===
A given set may have many different topologies.  If a set is given a different topology, it is viewed as a different topological space.

====Discrete and trivial topologies====
Any set can be given the [[discrete space|discrete topology]], in which every subset is open.  The only convergent sequences or nets in this topology are those that are eventually constant.  Also, any set can be given the [[trivial topology]] (also called the indiscrete topology), in which only the empty set and the whole space are open.  Every sequence and net in this topology converges to every point of the space.  This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be [[Hausdorff space]]s where limit points are unique.

====Cofinite and cocountable topologies====
Any set can be given the [[cofinite topology]] in which the open sets are the empty set and the sets whose complement is finite.  This is the smallest [[T1 space|T<sub>1</sub>]] topology on any infinite set.

Any set can be given the [[cocountable topology]], in which a set is defined as open if it is either empty or its complement is countable.  When the set is uncountable, this topology serves as a counterexample in many situations.

====Topologies on the real and complex numbers====
There are many ways to define a topology on '''R''', the set of [[real number]]s.  The standard topology on '''R''' is generated by the [[Interval (mathematics)#Definitions|open intervals]].  The set of all open intervals forms a [[base (topology)|base]] or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the [[Euclidean space]]s '''R'''<sup>''n''</sup> can be given a topology.  In the usual topology on '''R'''<sup>''n''</sup> the basic open sets are the open [[Ball (mathematics)|ball]]s.  Similarly, '''C''', the set of [[complex number]]s, and '''C'''<sup>''n''</sup> have a standard topology in which the basic open sets are open balls.

The real line can also be given the [[lower limit topology]].  Here, the basic open sets are the half open intervals <nowiki>[</nowiki>''a'', ''b'').  This topology on '''R''' is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology.  This example shows that a set may have many distinct topologies defined on it.

====The metric topology====
Every [[metric space]] can be given a metric topology, in which the basic open sets are open balls defined by the metric.  This is the standard topology on any [[normed vector space]]. On a finite-dimensional [[vector space]] this topology is the same for all norms.

====Further examples====
* There exist numerous topologies on any given [[finite set]]. Such spaces are called [[finite topological space]]s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.
* Every [[manifold]] has a [[natural topology]], since it is locally Euclidean.  Similarly, every [[simplex]] and every [[simplicial complex]] inherits a natural topology from '''R'''<sup>n</sup>.
* The [[Zariski topology]] is defined algebraically on the [[spectrum of a ring]] or an [[algebraic variety]].  On '''R'''<sup>''n''</sup> or '''C'''<sup>''n''</sup>, the closed sets of the Zariski topology are the [[solution set]]s of systems of [[polynomial]] equations.
* A [[linear graph]] has a natural topology that generalises many of the geometric aspects of [[graph theory|graph]]s with [[Vertex (graph theory)|vertices]] and [[Graph (discrete mathematics)#Graph|edges]].
* Many sets of [[linear operator]]s in [[functional analysis]] are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
* Any [[local field]] has a topology native to it, and this can be extended to vector spaces over that field.
* The [[Sierpiński space]] is the simplest non-discrete topological space.  It has important relations to the [[theory of computation]] and semantics.
* If Γ is an [[ordinal number]], then the set Γ = [0,&nbsp;Γ) may be endowed with the [[order topology]] generated by the intervals (''a'',&nbsp;''b''), [0,&nbsp;''b'') and (''a'',&nbsp;Γ) where ''a'' and ''b'' are elements of Γ.