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.

Continuity

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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

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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  
  • 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  

Notes

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References

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  • 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.