History
General topology grew out of a number of areas, most importantly the following: *the detailed study of subsets of theA topology on a set
Let ''X'' be a set and let ''τ'' be a family of subsets of ''X''. Then ''τ'' is called a ''topology on X'' if: # Both theBasis for a topology
A base (or basis) ''B'' for aSubspace 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 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 is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a surjectiveExamples 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 theCofinite and cocountable topologies
Any set can be given theTopologies on the real and complex numbers
There are many ways to define a topology on R, the set of real numbers. The standard topology on R is generated by the open intervals. The set of all open intervals forms a 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 spaces R''n'' can be given a topology. In the usual topology on R''n'' the basic open sets are the openThe metric topology
EveryFurther examples
* There exist numerous topologies on any given finite set. Such spaces are called_Continuous_functions
Continuity_is_expressed_in_terms_of_ , Γ)_may_be_endowed_with_the_order_topology_generated_by_the_intervals_(''a'', ''b''),_[0, ''b'')_and_(''a'', Γ)_where_''a''_and_''b''_are_elements_of_Γ._Continuous_functions
Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhood_Continuous_functions
Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhoodContinuous functions
Continuity is expressed in terms of neighborhood (topology)">neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...Alternative definitions
Several Characterizations of the category of topological spaces, equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.Neighborhood definition
Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms ofSequences and nets
In several contexts, the topology of a space is conveniently specified in terms ofClosure operator definition
Instead of specifying the open subsets of a topological space, the topology can also be determined by aProperties
If ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'': ''X'' → ''Z''. If ''f'': ''X'' → ''Y'' is continuous and * ''X'' isHomeomorphisms
Symmetric to the concept of a continuous map is an open map, for which ''images'' of open sets are open. In fact, if an open map ''f'' has anDefining topologies via continuous functions
Given a function : where ''X'' is a topological space and ''S'' is a set (without a specified topology), the final topology on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''−1(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by ''f''. Dually, for a function ''f'' from a set ''S'' to a topological space, the initial topology on ''S'' has as open subsets ''A'' of ''S'' those subsets for which ''f''(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the subspace topology of ''S'', viewed as a subset of ''X''. A topology on a set ''S'' is uniquely determined by the class of all continuous functions into all topological spaces ''X''. Dually, a similar idea can be applied to mapsCompact sets
Formally, aConnected sets
AConnected components
The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connected components of the space. The components of any topological space ''X'' form a partition of ''X'': they are disjoint, nonempty, and their union is the whole space. Every component is aDisconnected spaces
A space in which all components are one-point sets is called totally disconnected. Related to this property, a space ''X'' is called totally separated if, for any two distinct elements ''x'' and ''y'' of ''X'', there exist disjoint open neighborhoods ''U'' of ''x'' and ''V'' of ''y'' such that ''X'' is the union of ''U'' and ''V''. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.Path-connected sets
A ''Products of spaces
Given ''X'' such that : is the Cartesian product of the topological spaces ''Xi'', indexed by , and the canonical projections ''pi'' : ''X'' → ''Xi'', the product topology on ''X'' is defined as theSeparation axioms
Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous. Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. In all of the following definitions, ''X'' is again aCountability axioms
An axiom of countability is a property of certain mathematical objects (usually in aMetric spaces
A metric space is an ordered pair where is a set and is a metric on , i.e., aBaire category theorem
TheMain areas of research
Continuum theory
A continuum (pl ''continua'') is a nonemptyDynamical systems
Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include fluid dynamics,Pointless topology
Pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann.Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5 The ideas of pointless topology are closely related to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.Dimension theory
Dimension theory is a branch of general topology dealing with dimensional invariants ofTopological algebras
A topological algebra ''A'' over a topological field K is a topological vector space together with a continuous multiplication : : that makes it anMetrizability theory
In topology and related areas of mathematics, a metrizable space is aSet-theoretic topology
Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent ofSee also
*References
Further reading
Some standard books on general topology include: * Bourbaki, Topologie Générale (General Topology), . * John L. Kelley (1955External links
* {{Areas of mathematics , collapsed