Bitopological space

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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 .

Continuity[edit]

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
  • 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[edit]

References[edit]

  • 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.