Bitopological Essence of a Cotopology and Convenient Insertion
by
B. P. Dvalishvili
Tbilisi State University

The following abbreviations will be used: BS for a bitopological space, TS for a topological space, A-(2, 1)-BrS for an almost (2, 1)-Baire space and (2, 1)-BrS for a (2, 1)-weak Baire space [1]. If (X, \tau₁, \tau₂) is a BS and P is some topological property, then (i, j)-P-denotes the analogue of this property for \tau_i with respect to \tau_j, where i, j in {1, 2}, i =/= j, and p, -P denotes the conjunction (1, 2)-P /\ (2, 1)-P; also note that (X, \tau_i) has a property P --> (X, \tau₁, \tau₂) has a property i-P, and (X, \tau₁ < \tau₂) denotes the BS (X, \tau₁, \tau₂) with \tau₁ subset \tau₂.

Below the capacious notions of cotopology [2], of closed neighbourhoods condition [3] and of subordination of convex topologies [4] are formulated in bitopological terms.

Definition 1. We shall say that a bitopology , i.e., an ordered pair (\tau₁, \tau₂) of topologies on a set X has the (i, j)-A-insertion property, where A is any subfamily of the power set 2^X of X, if for every subset A subset X there exists a set G in A such that \tau_i int A subset G subset \tau_j cl A.

Definition 2. Let (X, \tau₂) be a TS. A topology \tau₁ on X is called a cotopology of \tau₂ and (X, \tau₁) is cospace of (X, \tau₂) if the following conditions are satisfied:

1. \tau₁ is weaker than \tau₂.
2. For each point x in X and any 2-closed neighbourhood M(x) there is a 1-closed neighbourhood N(x) such that N(x) subset M(X) [2].

Theorem. In a BS (X, \tau₁ < \tau₂) the topology \tau₁ is the cotopology of the regular topology \tau₂ if and only if (X, \tau₁, \tau₂) is (2, 1)-regular.

Corollary 1. If for a BS (X, \tau₁ < \tau₂) the dimension (2, 1)-rm ind X [5] is finite, then \tau₁ is the cotopology of the regular topology \tau₂.

Corollary 2. If (X, \tau₁, \tau₂) is a p-regular BS and (\tau₁, \tau₂) has the (1, 2)-\tau₂-insertion property, then \tau₁ is the cotopology of the regular topology \tau₂.

Corollary 3. Let \tau₁ and \tau₂ be two locally convex topologies on a vector space X. Then \tau₂ is subordinate to \tau₁ if and only if \tau₁ is the cotopology of the regular topology \tau₂.

Corollary 4. Let \tau₁ and \tau₂ be two locally convex topologies on a vector space X. Then the following conditions are equivalent:

1. \tau₂ is subordinate to \tau₁.
2. (\tau₁, \tau₂) satisfies the closed neighbourhoods condition.
3. \tau₁ is the cotopology of \tau₂.

Corollary 5. Let (X, \tau₁, \tau₂) be a (2, 1)-regular BS. Then the following statements hold:

1. If (X, \tau₁, \tau₂) is 2-PT₂ and 2-locally compact, then (X, \tau₁, \tau₂) is 1-compact.
2. If (X, \tau₁, \tau₂) is 2-quasi-regular and 1-compact, then (X, \tau₁, \tau₂) is 2-BrS and thus A-(2, 1)-BrS and (2, 1)-WBrS.

References

[1] B. Dvalishvili, Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications. Monograph (to appear).

[2] J. M. Aarts, J. de Groot and R. H. McDowell, Cotopology for metrizable spaces. Duke Math. J. 37(1970), 291-295.

[3] M. C. H. Cook, Sur deux problémes des sous-espaces ayant une topologie mixte. C. R. Acad Sci. Paris, Sér A-B 277(1973), A1095-A1097.

[4] G. Godefroy, Topologies subordonnées, Sém. Choquet (Initiation à l'analyse), 15e année, 1975/76. Comm. C10.

[5] B. Dvalishvili, On dimension of bitopological spaces. (Russian) Bull. Acad. Sci Georgian SSR 76, 1(1974), 49-52.

Date received: July 9, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeh-64.