In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is
X
and the topologies are
σ
and
τ
then the bitopological space is referred to as
(
X
,
σ
,
τ
)
.
A map
f
:
X
→
X
′
from a bitopological space
(
X
,
τ
1
,
τ
2
)
to another bitopological space
(
X
′
,
τ
1
′
,
τ
2
′
)
is called continuous or sometimes pairwise continuous if
f
is continuous both as a map from
(
X
,
τ
1
)
to
(
X
′
,
τ
1
′
)
and as map from
(
X
,
τ
2
)
to
(
X
′
,
τ
2
′
)
.
Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.
A bitopological space
(
X
,
τ
1
,
τ
2
)
is pairwise compact if each cover
{
U
i
∣
i
∈
I
}
of
X
with
U
i
∈
τ
1
∪
τ
2
, contains a finite subcover. In this case,
{
U
i
∣
i
∈
I
}
must contain at least one member from
τ
1
and at least one member from
τ
2
A bitopological space
(
X
,
τ
1
,
τ
2
)
is pairwise Hausdorff if for any two distinct points
x
,
y
∈
X
there exist disjoint
U
1
∈
τ
1
and
U
2
∈
τ
2
with
x
∈
U
1
and
y
∈
U
2
.
A bitopological space
(
X
,
τ
1
,
τ
2
)
is pairwise zero-dimensional if opens in
(
X
,
τ
1
)
which are closed in
(
X
,
τ
2
)
form a basis for
(
X
,
τ
1
)
, and opens in
(
X
,
τ
2
)
which are closed in
(
X
,
τ
1
)
form a basis for
(
X
,
τ
2
)
.
A bitopological space
(
X
,
σ
,
τ
)
is called binormal if for every
F
σ
σ
-closed and
F
τ
τ
-closed sets there are
G
σ
σ
-open and
G
τ
τ
-open sets such that
F
σ
⊆
G
τ
F
τ
⊆
G
σ
, and
G
σ
∩
G
τ
=
∅
.
