Wikipedia / Initial topology
Wikipedia / 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 is the coarsest topology that makes those functions continuous.
The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
The dual notion is the final topology is the finest topology that makes those functions continuous.
Definition
and an indexed family of topological spaces is the coarsest topology is continuous.
Definition in terms of open sets
then the This topology always exists and it is equal to the topology generated by
, and the ). Explicitly, the initial topology is the collection of open sets generated is an open set under finite intersections and arbitrary unions.
are often called contains exactly one element are cylinder sets.
Examples
Several topological constructions can be regarded as special cases of the initial topology.
The subspace topology is the initial topology on the subspace with respect to the inclusion map.
The product topology is the initial topology with respect to the family of projection maps.
The inverse limit (also called projective limit) of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space.
Given a family is the supremum in the lattice of topologies is the topology generated by the union
A topological space is completely regular if and only if it has the initial topology with respect to its family of (bounded) real-valued continuous functions.
to the Sierpiński space.
Properties
Characteristic property
can be characterized by the following characteristic property:
Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.
A filter converges to if and only if the prefilter converges to
Evaluation
By the universal property of the product topology
This map is known as the .
is said to separate points is injective.
will be a topological embedding
Hausdorffness
is a Hausdorff space if and only if these maps separate points
Transitivity of the initial topology
Several important corollaries of this fact are now given.
is equal to the initial topology induced by the inclusion map
The product topology by the evaluation map separate points then the evaluation map is a homeomorphism
Separating points from closed sets
This section gives a sufficient (but not necessary) condition.
separates points from closed sets if for all closed sets denotes the closure operator.
Theorem form a base for the topology
The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.
is a T0 space must also separate points. In this case, the evaluation map will be an embedding.
Initial uniform structure
is a family of uniform structures then the This uniform always exists and it is equal to the filter generated by the filter subbase
Then the uniformly continuous.). is continuous. are uniformly continuous.
Hausdorffness is Hausdorff separate points (or equivalently, if and only if the evaluation map is injective)
Uniform continuity is uniformly continuous
Cauchy filter: A filter is a Cauchy filter
Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "transitivity of the initial topology" given above, then the resulting statement will also be true.
Categorical description
In the language of category theory be the functor from a discrete category to the category of topological spaces be the usual forgetful functor can then be thought of as a conethe category of cones
. The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism that is, a terminal object such that the following diagram commutes: which is right adjoint