In any domain of mathematics, a space has a natural topology if there is a

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that the topology in question arises ''naturally'' or ''canonically'' (see mathematical jargon) in the given context. Note that in some cases multiple topologies seem "natural". For example, if ''Y'' is a subset of a totally ordered set ''X'', then the induced order topology, i.e. the

order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...

of the totally ordered ''Y'', where this order is inherited from ''X'', is coarser than the

subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...

of the order topology of ''X''. "Natural topology" does quite often have a more specific meaning, at least given some prior contextual information: the natural topology is a topology which makes a natural map or collection of maps continuous. This is still imprecise, even once one has specified what the natural maps are, because there may be many topologies with the required property. However, there is often a finest or coarsest topology which makes the given maps continuous, in which case these are obvious candidates for ''the'' natural topology. The simplest cases (which nevertheless cover ''many'' examples) are the

initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...

and the

final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...

(Willard (1970)). The initial topology is the coarsest topology on a space ''X'' which makes a given collection of maps from ''X'' to topological spaces ''X''_''i'' continuous. The final topology is the finest topology on a space ''X'' which makes a given collection of maps from topological spaces ''X''_''i'' to ''X'' continuous. Two of the simplest examples are the natural topologies of subspaces and quotient spaces. * The natural topology on a subset of a topological space is the

. This is the coarsest topology which makes the

inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...

continuous. * The natural topology on a

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of a topological space is the

quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...

. This is the finest topology which makes the

quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...

continuous. Another example is that any metric space has a natural topology induced by its metric.

References

* (Recent edition published by Dover (2004) {{ISBN, 0-486-43479-6.) Mathematical structures Topology

See also

References