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.
each with its induced subspace topology.
is recovered as the one coming from the final topology coinduced by the inclusion maps
By definition, this is the finest topology on (the underlying set of)
for which the inclusion maps are continuous.
if either of the following two equivalent conditions holds: Given a topological space
there is a unique topology on (the underlying set of)
be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection
Then the topological union
endowed with the final topology coinduced by the inclusion maps
The inclusion maps will then be topological embeddings and
is a topological space and is coherent with a family of subspaces
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.
is a topological union of the family
is homeomorphic to the quotient of the disjoint union of the family
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:
is the direct (inductive) limit (colimit) of
be coherent with a family of subspaces
This universal property characterizes coherent topologies in the sense that a space
if and only if this property holds for all spaces
be a surjective map and suppose
is finer than the original topology
induce the same subspace topology on each of the
is the collection of all compact subspaces of a topological space
have the same compact sets, with the same induced subspace topologies on them.