In topology and related areas of mathematics, an induced topology on a topological space is a topology which makes the inducing function continuous from/to this topological space.
Let
X
0
,
X
1
be sets,
f
:
X
0
→
X
1
.
If
τ
0
is a topology on
X
0
, then a topology coinduced on
X
1
by
f
is
{
U
1
⊆
X
1
|
f
−
1
(
U
1
)
∈
τ
0
}
.
If
τ
1
is a topology on
X
1
, then a topology induced on
X
0
by
f
is
{
f
−
1
(
U
1
)
|
U
1
∈
τ
1
}
.
The easy way to remember the definitions above is to notice that finding an inverse image is used in both. This is because inverse image preserves union and intersection. Finding a direct image does not preserve intersection in general. Here is an example where this becomes a hurdle. Consider a set
X
0
=
{
−
2
,
−
1
,
1
,
2
}
with a topology
{
{
−
2
,
−
1
}
,
{
1
,
2
}
}
, a set
X
1
=
{
−
1
,
0
,
1
}
and a function
f
:
X
0
→
X
1
such that
f
(
−
2
)
=
−
1
,
f
(
−
1
)
=
0
,
f
(
1
)
=
0
,
f
(
2
)
=
1
. A set of subsets
τ
1
=
{
f
(
U
0
)
|
U
0
∈
τ
0
}
is not a topology, because
{
{
−
1
,
0
}
,
{
0
,
1
}
}
⊆
τ
1
but
{
−
1
,
0
}
∩
{
0
,
1
}
∉
τ
1
.
There are equivalent definitions below.
A topology
τ
1
induced on
X
1
by
f
is the finest topology such that
f
is continuous
(
X
0
,
τ
0
)
→
(
X
1
,
τ
1
)
. This is a particular case of the final topology on
X
1
.
A topology
τ
0
induced on
X
0
by
f
is the coarsest topology such that
f
is continuous
(
X
0
,
τ
0
)
→
(
X
1
,
τ
1
)
. This is a particular case of the initial topology on
X
0
.
The quotient topology is the topology coinduced by the quotient map.
If
f
is an inclusion map, then
f
induces on
X
0
a subspace topology.
