Coherent topology

In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.

Definition

Let be a topological space and let be a family of subsets of each with its induced subspace topology. Then is said to be coherent with if the topology of is recovered as the one coming from the final topology coinduced by the inclusion maps
By definition, this is the finest topology on for which the inclusion maps are continuous.
is coherent with if either of the following two equivalent conditions holds:

A subset is open in if and only if is open in for each
A subset is closed in if and only if is closed in for each

Examples

A topological space is coherent with every open cover of More generally, is coherent with any family of subsets whose interiors cover As examples of this, a weakly locally compact space is coherent with the family of its compact subspaces. And a locally connected space is coherent with the family of its connected subsets.
A topological space is coherent with every locally finite closed cover of
A discrete space is coherent with every family of subspaces.
A topological space is coherent with a partition of if and only is homeomorphic to the disjoint union of the elements of the partition.
Finitely generated spaces are those determined by the family of all their finite subspaces.
Countably generated spaces are those determined by the family of all their countable subspaces.
Compactly generated spaces are those determined by the family of all their compact subspaces.
A CW complex is coherent with its family of -skeletons

Topological union

Let be a family of topological spaces such that the induced topologies agree on each intersection
Assume further that is closed in for each Then the topological union is the set-theoretic union
endowed with the final topology coinduced by the inclusion maps. The inclusion maps will then be topological embeddings and will be coherent with the subspaces
Conversely, if is a topological space and is coherent with a family of subspaces that cover then is homeomorphic to the topological union of the family
One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings.
One can also describe the topological union by means of the disjoint union. Specifically, if is a topological union of the family then is homeomorphic to the quotient of the disjoint union of the family by the equivalence relation
for all ; that is,
If the spaces are all disjoint then the topological union is just the disjoint union.
Assume now that the set A is directed, in a way compatible with inclusion: whenever
. Then there is a unique map from to which is in fact a homeomorphism. Here is the direct (inductive) limit
of in the category Top.

Properties

Let be coherent with a family of subspaces A function from to a topological space is continuous if and only if the restrictions
are continuous for each This universal property characterizes coherent topologies in the sense that a space is coherent with if and only if this property holds for all spaces and all functions
Let be determined by a cover Then

If is a refinement of a cover then is determined by In particular, if is a subcover of is determined by
If is a refinement of and each is determined by the family of all contained in then is determined by
Let be an open or closed subspace of or more generally a locally closed subset of Then is determined by
Let be a quotient map. Then is determined by

Let be a surjective map and suppose is determined by For each let be the restriction of to Then

If is continuous and each is a quotient map, then is a quotient map.
is a closed map if and only if each is closed.

Given a topological space and a family of subspaces there is a unique topology on that is coherent with The topology is finer than the original topology and strictly finer if was not coherent with But the topologies and induce the same subspace topology on each of the in the family And the topology is always coherent with
As an example of this last construction, if is the collection of all compact subspaces of a topological space the resulting topology defines the k-ification of The spaces and have the same compact sets, with the same induced subspace topologies on them. And the k-ification is compactly generated.