Set endowed with two topologies
In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is
and the topologies are
and
then the bitopological space is referred to as
. The notion was introduced by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric.
A map
from a bitopological space
to another bitopological space
is called continuous or sometimes pairwise continuous if
is continuous both as a map from
to
and as map from
to
.
Bitopological variants of topological properties
[edit] 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. In this case,
must contain at least one member from
and at least one member from ![{\displaystyle \tau _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/236a07f6109f6e8c7dd3d9229ab240a983045cd8)
- A bitopological space
is pairwise Hausdorff if for any two distinct points
there exist disjoint
and
with
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 ![{\displaystyle \scriptstyle G_{\sigma }\cap G_{\tau }=\emptyset .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81960543b25b87ed4215a58ceab3bdf5a33f2b40)
- Kelly, J. C. (1963). Bitopological spaces. Proc. London Math. Soc., 13(3) 71–89.
- Reilly, I. L. (1972). On bitopological separation properties. Nanta Math., (2) 14–25.
- Reilly, I. L. (1973). Zero dimensional bitopological spaces. Indag. Math., (35) 127–131.
- Salbany, S. (1974). Bitopological spaces, compactifications and completions. Department of Mathematics, University of Cape Town, Cape Town.
- Kopperman, R. (1995). Asymmetry and duality in topology. Topology Appl., 66(1) 1--39.
- Fletcher. P, Hoyle H.B. III, and Patty C.W. (1969). The comparison of topologies. Duke Math. J.,36(2) 325–331.
- Dochviri, I., Noiri T. (2015). On some properties of stable bitopological spaces. Topol. Proc., 45 111–119.