Open quotient maps #
An open quotient map is an open map f : X → Y
which is both an open map and a quotient map.
Equivalently, it is a surjective continuous open map.
We use the latter characterization as a definition.
Many important quotient maps are open quotient maps, including
- the quotient map from a topological space to its quotient by the action of a group;
- the quotient map from a topological group to its quotient by a normal subgroup;
- the quotient map from a topological spaace to its separation quotient.
Contrary to general quotient maps,
the category of open quotient maps is closed under Prod.map
.
An open quotient map is a quotient map.
Alias of IsOpenQuotientMap.isQuotientMap
.
An open quotient map is a quotient map.
Given the following diagram with f
inducing, p
surjective,
q
an open quotient map, and g
injective. Suppose the image of A
in B
is stable
under the equivalence mod q
, then the coinduced topology on C
(from A
)
coincides with the induced topology (from D
).
A -f→ B
∣ ∣
p q
↓ ↓
C -g→ D
A typical application is when K ≤ H
are subgroups of G
, then the quotient topology on H/K
is also the subspace topology from G/K
.