
Bitopological space
Encyclopedia
In mathematics
, a bitopological space is a set endowed with two topologies
. Typically, if the set is
and the topologies are
and
then we refer to the bitopological space as
.
from a bitopological space
to another bitopological space
is called bi-continuous if
is continuous both as a map from
to
and as map from
to
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a bitopological space is a set endowed with two topologies
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. Typically, if the set is




Bi-continuity
A mapMap (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...








Bitopological variants of topological properties
Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.- A bitopological space
is pairwise compact if each cover
of
with
, contains a finite subcover.
- A bitopological space
is pairwise Hausdorff if for any two distinct points
there exist disjoint
and
with either
and
or
and
.
- A bitopological space
is pairwise zero-dimensional if opens in
which are closed in
form a basis for
, and opens in
which are closed in
form a basis for
.
- A bitopological space
is called binormal if for every
-closed and
-closed sets there are
-open and
-open sets such that
, and